Integrated polarization rotator/converter by stimulated Raman adiabatic passage

Xiao Xiong; Chang-Ling Zou; Xi-Feng Ren; Guang-Can Guo

doi:10.1364/OE.21.017097

1. Introduction

Great progresses have been achieved recently in the photonic integrated circuits (PICs), since the classical and quantum information processing in PICs is more compact, low-cost and stable [1, 2]. In particular, qubits encoded with the polarization states of photons, can be used for quantum computation [3, 4] and quantum simulations [5, 6]. Therefore, unitary and reversible gates to manipulate the polarization states of photons in PICs are desired [7]. For example, the polarization rotator is a fundamental Pauli-X gate, which is as important as other quantum gates, such as Hadamard gate and Controlled-NOT (CNOT) gate. Besides, the polarization rotator has also played an important role in the generation of integrated polarization-entangled photon pair source on chip [8]. Hence, for polarization-encoded information processing in PICs [9], polarization rotator is an indispensable component.

However, the manipulation of photon polarizations is still a challenge for PICs [10]. During the last years, there has been a growing attention paid to the polarization rotators, and various proposals have been presented, which can be classified into three types. (1) The most common ones are waveguides of asymmetric cross-sections, which are based on direct coupling between horizontal and vertical modes. For example, rotator with triangle [11, 12] or L-shaped [13–16] waveguide can be as short as 2 μm, but is very sensitive to the etching error. Waveguide with slant sidewall [17, 18] or trenches [19, 20] enables high conversion efficiency and good fabrication tolerance, while the etching process of such structures is rather complex and the design freedom is limited. Even if low-loss and efficient polarization converter can be fabricated with improved process [21,22], the device length is quite long. The polarization rotators above usually require fabrication of relatively high accuracy and cannot be broadband. (2) Another principle ever adopted is mode evolution [23–31], which rotates the photon polarization via slowly varying the waveguide geometry. Though it enables broad bandwidth, which is demanded for pulse operation, the conversion is generally effective for a specific polarization. Such unidirectional conversion is not suitable for quantum information processing. (3) In addition, polarization rotator with the aid of surface plasmon polaritons (SPPs) is also proposed [32–34], since SPPs show a selectivity for photon polarization states [35]. Such converters can be ultra-compact, but suffer large loss due to metallic absorption.

In this Letter, we propose an integrated polarization rotator, which is adiabatic, highly efficient and robust against fabrication error. The key idea comes from the atomic stimulated Raman adiabatic passage (STIRAP) [36] in quantum optics. By introducing an ancillary waveguide, an optical three-mode analogue to atomic three-level system is constructed. With a revised adiabatic evolution model, two-way polarization conversion in a single waveguide is realized. And the conversion efficiencies are more than 99% for both polarizations, which makes the rotator potential for unitary and reversible gate in quantum information technology. Besides the application value as device, this polarization rotator provides a platform for simulation of coherent quantum STIRAP phenomenon [37, 38]. In addition, our study is also an excellent example of photonic device design inspired by quantum physics. We expect that more novel and advanced devices would benefit from the ideas introduced from interdiscipline.

2. Model

In the simplest form, STIRAP [36] involves a three-level, two-photon Raman process and allows complete population transfer between two long-lived quantum states relying upon two coherent light pulses. Shown in Fig. 1(a) is the Λ-type three-level system. Direct electric dipole transition between initial state |1〉 and final state |3〉 is forbidden, but they are both optically connected with intermediate state |2〉 via pump pulse and Stokes pulse. Assuming that the population initially occupies state |1〉, it can be completely transferred to state |3〉 without any decay from state |2〉, which is accomplished with a temporal sequence of Stokes and pump pulses [36]. The evolution of system follows Schrödinger equation $i \bar{h} \frac{\partial}{\partial t} | ψ 〉 = ℋ_{0} | ψ 〉$ , and the Hamiltonian of this atom system reads

ℋ_{0} / \bar{h} = ω_{1} | 1 〉 〈 1 | + ω_{2} | 2 〉 〈 2 | + ω_{3} | 3 〉 〈 3 | + (Ω_{p} e^{- i ω_{p} t} | 1 〉 〈 2 | + Ω_{s} e^{i ω_{s} t} | 2 〉 〈 3 | + h . c .) .

Here ω₁, ω₂, ω₃ stand for the three energy levels, and ω_p(s) and Ω_p(s) are the frequency and Rabi frequency of pump (Stokes) pulse, respectively. When pump and Stokes pulses are on resonance with |2〉 → |1〉 and |2〉 → |3〉 transitions (ω_p = ω₂ − ω₁ and ω_s = ω₂ − ω₃), the Hamiltonian can be simplified in the interaction picture as

ℋ_{0} / \bar{h} = Ω_{p} | 1 〉 〈 2 | + Ω_{s} | 2 〉 〈 3 | + h . c ..

Written in the matrix form, it becomes

ℋ_{0} / \bar{h} = (\begin{matrix} 0 & Ω_{p} & 0 \\ Ω_{p} & 0 & Ω_{s} \\ 0 & Ω_{s} & 0 \end{matrix}) .

We can obtain three eigenstates of ℋ₀, which are linear combinations of states |1〉, |2〉, and |3〉. Specially, we focus on the so-called “dark” state

| ψ_{D} 〉 = cos θ | 1 〉 - sin θ | 3 〉,

where the excited state |2〉 is not involved. Here, tanθ = Ω_p/Ω_s. Via adjusting θ, which is dependent on the energy of pump and Stokes pulses, |ψ_D〉 = |1〉 (θ = 0) can be evolved to |ψ_D〉 = |3〉 (θ = π/2). That is to say, adiabatically controlling the coupling strengths will realize complete population transfer from state |1〉 to state |3〉.

Fig. 1 (a) STIRAP model in atomic system. (b) Analogue of STIRAP in waveguide system, with the white arrows indicating the electric field directions of the three modes. (c) The effective refractive index difference between V(H) and O modes along z direction. (d) The coupling strengths of V and H modes with O mode along z direction. (e) and (f) are the dynamics of the eigenstates of ℋ along z direction, which realize polarization conversions.

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Inspired by the atomic STIRAP, we can construct a similar waveguide system to realize adiabatic population (energy) transfer between different waveguide modes. For a rectangular signal waveguide (SW), it supports Horizontal (H) and Vertical (V) polarization modes, which are orthogonal to each other, indicating that direct polarization conversion is forbidden in SW. Introduce another ancillary waveguide (AW) in vicinity to the SW, the Oblique (O) mode in AW can couple with both H and V modes in SW. As shown in Fig. 1(b), H, V and O modes compose a Λ-type three-level waveguide system.

Generally, the evolution of waveguide system follows Schrödinger equation as well: $i \frac{\partial}{\partial z} | Ψ 〉 = ℋ | Ψ 〉$ , with z being the propagation direction of light, and the Hamiltonian reads

ℋ / k_{0} = n_{V} | V 〉 〈 V | + n_{H} | H 〉 〈 H | + n_{O} | O 〉 〈 O | + g_{V} (| V 〉 〈 O | + | O 〉 〈 V |) + g_{H} (| H 〉 〈 O | + | O 〉 〈 H |),

where n_V, n_H, n_O are the effective refractive indices of V, H and O modes, g_V(H) is the coupling coefficient between V(H) and O modes, and k₀ is the wave-vector in vacuum. In the interaction picture, this Hamiltonian is simplified as

ℋ / k_{0} = Δ n_{V} | V 〉 〈 V | + Δ n_{H} | H 〉 〈 H | + g_{V} (| V 〉 〈 O | + | O 〉 〈 V |) + g_{H} (| H 〉 〈 O | + | O 〉 〈 H |) .

Or, it can be represented in the matrix form

ℋ / k_{0} = (\begin{matrix} Δ n_{V} & g_{V} & 0 \\ g_{V} & 0 & g_{H} \\ 0 & g_{H} & Δ n_{H} \end{matrix}),

where Δn_V(H) = n_V(H) − n_O is determined by the geometric parameters of SW and AW, while g_V(H) depends on the gap between SW and AW.

Similar to the atomic STIRAP, we can control the eigenstates of the waveguide system from |Φ〉 = |V〉 to |Φ〉 = |H〉 by adjusting the parameters of ℋ. In other words, we expect to convert the light polarization adiabatically by adjusting the waveguide geometries during light propagation. However, g_V and g_H cannot be controlled separately, since they simultaneously change with the gap between AW and SW. Thus, we need to control Δn_V and Δn_H additionally.

According to these limitation, we proposed a new adiabatic polarization conversion model. As engineered in Figs. 1(c) and 1(d), Δn_V and Δn_H cross with each other along z direction, while g_V and g_H change in accordance with Gaussian lineshape, with the maximums located at the intersection (In our primary model, g_V = g_H). Solving the eigenstates of corresponding ℋ at each point z, we obtained Figs. 1(e) and 1(f), which demonstrates the conversions of eigenstates |V〉 → |H〉 and |H〉 → |V〉. In the following, we’ll carry out this adiabatic model with practical silicon-on-insulator (SOI) waveguides.

3. STIRAP polarization rotators

From the perspective of experiment, we focus our studies on λ = 1550 nm and perform all the simulations with Finite Element Method (FEM), with the refractive indices of silicon and silica being n_Si = 3.5 and n_SiO₂ = 1.444. The inset in Fig. 2(c) illustrates the cross-section of the polarization rotator we proposed. The left SW supports H and V modes, with width w and height 400 nm. And the right one is AW, with width 410 nm, height 400 nm, and an unfilled corner of 250 nm × 150 nm, supporting O mode (n_O = 2.4). The gap between two waveguides is denoted as d.

Fig. 2 (a) The effective refractive indices of V and H modes against the waveguide width, respectively. (b) The coupling strengths of V and H modes with O mode against the gap between SW and AW, respectively. (c) Schematic illustration of the polarization rotator, with a SW of varying width and an AW of curved trend. Inset: the cross-section view of this rotator.

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Fixing the height of SW and AW at 400 nm, we obtained the effective refractive indices of V and H modes against w, as shown in Fig. 2(a). The hollow circles are data from numerical simulations, while solid curves are fitted from them. The two curves cross with each other at 412 nm, which is exactly where we expect the polarization conversion to happen. As for the coupling strengths g_V and g_H, they are extracted from numerical simulations as well. When V and O modes are resonant, they are split into two eigenmodes as their hybridization, and the index difference between these two eigenmodes reaches a minimum as δ = 2g_V/k₀ (k₀ = 2π/λ) [39]. Hence, by calculating the effective refractive indices of eigenmodes, g_V is derived. In addition, considering the transverse electric fields in waveguides decay exponentially along lateral direction and the coupling strength is the integral of the overlap between V and O modes, $g_{V} \propto e^{- \frac{d}{D}} / \sqrt{w}$ with D being a constant. And the same applies to H mode. In Fig. 2(b), fitted g_V and g_H are displayed together with the numerical data, showing good agreement. g_V doesn’t equal to g_H since the transverse field distributions of V and H modes are not the same.

Figure 2(c) shows the schematic of the three-level waveguide system. The left SW of varying width has length of L, while the right one is AW with radius of curvature R. The coordinate axis z is established along the light propagation direction, with the origin o fixed in the center. The gap and the SW width at the origin are d₀ and w₀, respectively, with w₀ fixed as w₀ = 412 nm. To realize polarization rotation, w varies linearly along z direction, with the width difference between input and output ends being δw. If there exists fabrication error, leading w₀ located at z = δz, w as the function of z becomes

w (z) = w_{0} + \frac{δ w}{L} \cdot (z - δ z) .

As for d, under the approximation z ≪ R, we have

d (z) \approx d_{0} + z^{2} / R .

Hence, the coupling strengths g_V(H) can be approximated in the form of Gaussian function as

g_{V (H)} (z) \approx \frac{g_{V (H)} (0)}{\sqrt{w (z)}} e^{- z^{2} / D R},

where g_V(H)(0) is fitting constant related to d₀.

According to the relationship of ℋ with Δn_V, Δn_H, g_V and g_H, which depend on w and d, the performance of the polarization rotator is totally determined by δw, d₀, δz, R and L. By adjusting these parameters, we obtained the results shown in Figs. 1(c) and 1(f), which verify that this polarization rotator model is indeed workable.

4. Theoretical calculation

Since the size of waveguides and the gap between them is engineered along z direction, ℋ is supposed to be the function of z. Denoting the field amplitudes of V, O and H modes at z as a_V(z), a_O(z), a_H(z), their evolutions follow the Schrödinger equation

i \frac{d}{d z} | Ψ (z) 〉 = ℋ (z) | Ψ (z) 〉,

where |Ψ(z)〉 = {a_V(z), a_O(z), a_H(z)}^T. Then, the action of ℋ on this system from 0 to z can be described with a transfer matrix 𝔗 as

| Ψ (z) 〉 = 𝔗 | Ψ (0) 〉 = e^{i \int_{0}^{z} ℋ (s) d s} | Ψ (0) 〉 .

Since the function of polarization rotator can be expressed as

𝔗 | V 〉 = e^{i ϕ_{1}} | H 〉, 𝔗 | H 〉 = e^{i ϕ_{2}} | V 〉, 𝔗 | O 〉 = e^{i ϕ_{3}} | O 〉,

with |V〉 = {1, 0, 0}^T, |H〉 = {0, 0, 1}^T, |O〉 = {0, 1, 0}^T, the transfer matrix of a perfect polarization rotator is expected to be

𝔗 = (\begin{matrix} 0 & 0 & e^{i ϕ_{1}} \\ 0 & e^{i ϕ_{3}} & 0 \\ e^{i ϕ_{2}} & 0 & 0 \end{matrix}) .

Here, ϕ₁, ϕ₂ and ϕ₃ are the phase changes induced by mode propagation from input to output ends. Regardless of the additional phases, the unitary matrix 𝔗 is a reversible quantum operation, which can transfer initial state |V〉 adiabatically to final state |H〉, and vice versa.

Here, the transfer matrix $𝔗 = e^{i \int_{0}^{z} ℋ (s) d s}$ is theoretically calculated with fourth order Runge-Kutta method. As an example, we examined the adiabatic polarization rotator with optimized parameters δw = 0.016, d₀ = 0.03, δz = 4.5, R = 150² and L = 150 (in the unit μm), and obtained the absolute value of the transfer matrix

| 𝔗 | = (\begin{matrix} 0.0772839 & 0.00844906 & 0.996973 \\ 0.0268221 & 0.999597 & 0.00928259 \\ 0.996648 & 0.0270963 & 0.0771882 \end{matrix}) .

Denoting the efficiencies of |V〉 → |H〉 and |H〉 → |V〉 conversions as η_V_→_H and η_H_→_V, this matrix demonstrates two-way conversion efficiency over 99%, where

η_{V \to H} = {| 𝔗 |}_{13}^{2}

,

η_{H \to V} = {| 𝔗 |}_{31}^{2}

, |𝔗|₁₃ and |𝔗|₃₁ are matrix elements of |𝔗|. Note, that the best conversion does not happen at δz = 0 is due to the inequality of coupling strengths.

5. Results and discussions

For a better understanding of the behavior of this polarization rotator, we defined an error function as

Error = 1 - \sqrt{η_{V \to H} \times η_{H \to V}},

which takes the two-way performance of this polarization rotator into consideration. Obviously, Error is the function of L, δw, R, d₀ and δz. So we calculated the dependence of Error on L with different δw, R, d₀ and δz, as displayed in Fig. 3.

Fig. 3 (a) Dependence of Error on L for different δw, with R = 150², d₀ = 0.03, δz = 4.5. (b) Dependence of Error on L for different R, with δw = 0.016, d₀ = 0.03, δz = 4.5. (c) Dependence of Error on L for different d₀, with δw = 0.016, R = 150², δz = 4.5. (d) Dependence of Error on L for different δz, with δw = 0.016, R = 150², d₀ = 0.03. All the numbers are in the order of μm.

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Figure 3(a) shows the dependence of Error on L for different δw. Each curve gradually decreases at first, then reaches a minimum, finally increases to a definite value and gets saturated. This can be explained as following: When all the geometric parameters except L are fixed, longer coupling distance brings higher conversion efficiency. As L keeps on increasing, the width difference of SW within effective coupling distance δw_eff is smaller than its set value δw, leading to incomplete conversion. That’s why the best L emerges. When L gets longer furthermore, δw_eff will get saturated towards zero and cause the saturation of Error. For smaller δw, L needed to reach maximum polarization conversion efficiency is shorter, as verified by the curves of different colors. This is reasonable, since larger δw corresponds to larger Δn_V and Δn_H, which means a longer evolution distance is needed. In addition, according to Eq. 8, the best L should be proportional to δw.

In Fig. 3(b), the dependence of Error on L for different R is shown. They all drop and then hold steady at some numbers. As R increases, the coupling distance is longer, and the conversion procedure turns to be more adiabatic according to Eq. 10. But in the meantime, the rise of effective coupling distance will add the variance of Δn_V(H), which makes the relationship of Error with L and R quite complicated. Therefore, for some large R, the curves go back slightly. For a conversion efficiency over 99%, there exists an optimal R between R = 150² and R = 170². If R < 150², the effective region for SW to couple with AW is insufficient to realize polarization conversion. And when R > 170², the variance of d is quite small, which means that the coupling strengths tend to be constant and the coefficient of |O〉 is unable to reach zero. In such a situation, a larger δw is favorable so that longer L matching with R can be obtained to enlarge the variance of d.

Figure 3(c) displays the dependence of Error on L for different d₀. Like the curves for different R, these curves reach their minimums when L is small and are fixed at some values for larger L. But different from Fig. 3(b), the slight going back after the minimum happens for small d₀. This is because when the two waveguides get closer to each other, the effective coupling distance is longer, the effect of which is exactly equivalent to the increase of R. It is suggested that d₀ has optimal value as well, which can be intuitively understood. If the two waveguides get too far from each other, the coupling strengths between them are not enough for the energy to transfer from one polarization to another. But if they are too close, the originally adiabatic process will be broken since their couplings are too strong. Thus, the optimal value ranges from d₀ = 0.022 to d₀ = 0.034 for a conversion efficiency over 99%.

Finally, from a practical point of view, we also took fabrication error into consideration and calculated the influence of δz on Error, as illustrated in Fig. 3(d). It’s very easy to understand that the absolute value of δz should be as small as possible, since the point w₀ = 412 nm is where we expect the polarization conversion to happen. No matter this point is located too far to the left or to the right, the effective distance for polarization to convert is too short. However, since g_H and g_V are inequal, this point needs to be a little away from the origin. As for the slightly going back for δz = 5 and δz = 10, it arises from similar reason for δw explained above.

In order to investigate the robustness of this polarization rotator, we calculated the tolerances of δw, R, d₀ and δz for a polarization rotator with δw = 0.016, R = 150², d₀ = 0.03, δz = 4.5, L = 150. (The following discussions are all for incident wavelength λ = 1550 nm). As shown in Fig. 4(a), δw can vary in the range from δw = 0 to δw = 0.018 while maintaining 99% conversion efficiency. As soon as the point w₀ = 412 nm is included in SW, polarization conversion can be completed even for δw = 0. However, when δw > 0.018, the length needed for polarization evolution is longer than L, causing the rising of error function. As regards the tolerance of R, which is displayed in Fig. 4(b), its region for polarization conversion efficiency over 99% starts from R = 147² to R = 174², symmetric about R = 160². This indicates that the tolerance range of R is quite considerate to fabrication process. Similar to the tolerance of R, the curve for d₀ shown in Fig. 4(c) is also symmetric, about d₀ = 0.027. From d₀ = 0.0235 to d₀ = 0.0305, the polarization conversion efficiency always keeps over 99%. It implies a fabrication tolerance of 23% with respect to d₀ = 0.027, although the absolute varying range of d₀ is only 7nm. This imperfect behavior is due to the nature that coupling is strongly dependent on the waveguide gap. Figure 4(d) illustrates the tolerance of δz, where polarization conversion efficiency over 99% can be maintained between δz = 1 and δz = 8.15. Different from R and d₀, it is symmetric with respect to δz = 4.575, rather than δz = 0. This originates from the coupling strengths of V and H modes with O mode. Unequal g_V and g_H bring that the energy of V and H modes does not equal to each other when the effective refractive indices of them n_V(H) are the same. What’s more, we can also find that these four curves in Fig. 4 are quite gentle. It means that the tolerances of δw, R, d₀ and δz will be broaden a lot, if the restriction of polarization conversion efficiency is relaxed to 98%.

Fig. 4 (a) Tolerance of δw, with R = 150², d₀ = 0.03, δz = 4.5, L = 150. (b) Tolerance of R, with δw = 0.016, d₀ = 0.03, δz = 4.5, L = 150. (c) Tolerance of d₀, with δw = 0.016, R = 150², δz = 4.5, L = 150. (d) Tolerance of δz, with δw = 0.016, R = 150², d₀ = 0.03, L = 150. All the numbers are in the order of μm. And the grey area indicates a conversion efficiency over 99%.

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Besides, the operation bandwidth is analyzed. The tolerances of geometric parameters with incident wavelengths λ = 1545 nm and λ = 1555 nm are plotted together with λ = 1550 nm in Fig. 4, and the dependence of conversion efficiency on wavelength for a specific rotator is also illustrated in Fig. 5. From λ = 1535 nm to λ = 1565 nm, the polarization conversion efficiency is maintained over 99% except for λ = 1540 nm and λ = 1560 nm. Even for the worst point λ = 1560 nm, the efficiency is still 96%, which demonstrates that the performance of this polarization rotator is not that sensitive to incident wavelength. In addition, the couplings of higher-order waveguide modes have also been taken into consideration. We calculated the effective refractive index of higher order modes and their coupling strengths with O mode, and constructed the Hamiltonian including the couplings of higher-order modes. Then, the corresponding transfer matrix is obtained, which demonstrates that the higher order modes will not influence the conversion between the two ground modes.

Fig. 5 Dependence of Error on λ, with δw = 0.016, R = 150², d₀ = 0.03, δz = 4.5, L = 150, which are all in the order of μm.

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6. Conclusion

In summary, we propose a novel concept to realize conversion between two orthogonal polarization modes, and demonstrate its feasibility based on practical SOI waveguides. Numerical calculations prove that such polarization rotator is of ultra-high efficiency, very robust against fabrication error, and compatible to integrated photonic circuits. We introduce a new idea from quantum physics, based on which an essential integrated photonic device for both quantum and classical information processes in the future is innovated. In addition, we bring the purely theoretical model in quantum physics from the platform of fundamental researches to practical applications, such as device designs.

Acknowledgments

This work was funded by NBRP (grant nos. 2011CBA00200 and 2011CB921200), the Innovation Funds from the Chinese Academy of Sciences (grant no. 60921091), NNSF (grant nos. 10904137 and 10934006), the Fundamental Research Funds for the Central Universities (grant no. WK2470000005), and NCET.

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Integrated polarization rotator/converter by stimulated Raman adiabatic passage

Abstract

1. Introduction

2. Model

3. STIRAP polarization rotators

4. Theoretical calculation

5. Results and discussions

6. Conclusion

Acknowledgments

References and links

Cited By

Figures (5)

Equations (16)

Optics Express