Integrated optical circulator by stimulated Brillouin scattering induced non-reciprocal phase shift

Wei Fu; Fang-Jie Shu; Yan-Lei Zhang; Chun-Hua Dong; Chang-Ling Zou; Guang-Can Guo

doi:10.1364/OE.23.025118

1. Introduction

In recent years, great interests have been attracted to photonic integrated circuits (PIC) [1, 2], where lights are confined and guided in compact dielectric waveguides and resonators. The design of PIC devices are very different from their counterparts in free space, since the material properties and nano-fabrications should be both taken into account. A variety of efficient and novel integrated devices have been proposed and demonstrated in experiments, such as CMOS-compatible multiple-wavelength oscillator [3], phase shifter [4], single-photon sources [5, 6], polarization beam splitter [7] and absorber [8]. The non-reciprocal devices such as isolator [9–11 ] and circulator [12, 13] which break the time-reversal symmetry of lights, play an essential role in all the PIC devices for the ultrafast all-optical signal processing and optical computing systems [14–17 ]. However, due to the difficulties of preparation and fabrication of magneto-optic materials [18, 19], the non-reciprocal devices are still challenges for PIC.

To integrate the non-reciprocal devices into the PIC, many tentative methods without magneto-optical material have been proposed. For instance, indirect interband photonic transitions [20], optical excitation of guided acoustic mode in an optical crystal fiber core [21], strong optomechanical interaction in microring resonators [22] have been proposed and discussed. Besides, stimulated Brillouin scattering (SBS) has recently aroused great interests for its non-reciprocal light absorption [23]. Especially, it’s demonstrated that the SBS can be greatly enhanced in the whispering-gallery (WG) microresonators [24–26 ], and enable non-reciprocal light storage and absorption with lower control power [27–30 ].

In this paper, we proposed an all-optical circulator based on SBS induced non-reciprocity. As shown in Fig. 1, the circulator consists of an integrated Mach-Zehnder (MZ) interferometer with a non-reciprocal phase shifter coupled to its lower arm. The non-reciprocity results from the requirement of phase-matching condition for the interconversion between the acoustic traveling wave and optical photon when the control laser is launched in the forward direction (from left to right). Thus the phase shift is 0 for the forward propagation photons and is π for the backward (from right to left) light. Combining with the MZ interferometer, the non-reciprocal phase shift leads to non-reciprocal light transmission as circulator. Due to the narrow-band nature of our system, the bandwidth of our circulator is relatively small and limited by parameters such as cooperativity and mechanical decay rate. We discussed it under different parameters and provide solution to optimize it. In the end, the impacts of the backscattering are also analyzed. The proposed magnetic-free all-optical integrated circulator can be realized in silicon [31, 32] and chalcogenide photonic chips [33], may play a significant role for future classical and quantum information processing based on PIC.

Fig. 1 Schematic illustration of a integrated all-optical circulator: a whispering-gallery microresonator is coupling to one arm of a four-port Mach-Zehnder interferometer. The microresonator enables non-reciprocal SBS, which is serving as non-reciprocal phase shifter. The phase modulator on the upper arm can modulate paths lenth difference of two arm.

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2. Principle of optical circulator

First of all, we illustrate the basic principle of the proposed circulator. The light launched at Port 1 of the circulator is divided into two parts with equal amplitude by the beam splitter (directional coupler) A, and combined by the beam splitter B at the output ports. For such a MZ interferometer, the phase difference between the upper and lower paths determines which port the light outputs. Suppose the length difference and additional phase modulator [34] induce a phase difference between two paths of π, while the strong over-coupled microresonator induces a π phase shift at lower path, the overall phase difference is 0, light input from Port 2 will output at Port 3 (red arrow in Fig. 1). Similarly, light input from Port 4 will output at Port 1. If there is no non-reciprocal phase shift, the reversal of above process indicates that input from Port 1(3) will output at Port 4(2). Here, we pump the microresonator to stimulate the Brillouin scattering, and only the signal light satisfying the phase match conditions can couple the acoustic wave [27], then undergoes an additional phase due to the Brillouin scattering induced transparency (BSIT). For example, for forward Brillouin scattering, letting the pump laser couple to the clockwise whispering gallery mode in the resonator, the light from left to right (green arrow in Fig. 1) will gain 0 phase. Therefore, the light from Port 1(3) will output at Port 2(4). To sum up, the light circulates as 1 → 2 → 3 → 4 → 1; therefore, this structure can serve as an optical circulator.

To make the discussion more convenient, we introduce a transmission matrix M. Then input-output relation can be written in matrix form

{\vec{E}}_{out} = M {\vec{E}}_{in},

where the vector E⃗ _in(out) is the electric field input (output) from ports 1, 2, 3 and 4. The elements of the transfer matrix of the MZ interferometer can be solved as

M_{21 (43)} = \frac{1}{2} [1 + \sqrt{T_{f}} e^{i ϕ_{f}}]

M_{12 (34)} = \frac{1}{2} [1 + \sqrt{T_{b}} e^{i ϕ_{b}}]

M_{41 (23)} = \frac{1}{2} [1 - \sqrt{T_{f}} e^{i ϕ_{f}}]

M_{12 (34)} = \frac{1}{2} [1 - \sqrt{T_{b}} e^{i ϕ_{b}}] .

Where T _f(b) is the transmittance for the nonreciprocal phase shifter when propagating forwardly (backwardly), and ϕ _f(b) is the phase shift for forward (backward) direction. For ideal situation T _f(b) = 1, ϕ_f = 0 and ϕ_b = π, and together with the assumption that there is no elastic back-scattering M_ii = 0 and M ₁₃ = M ₃₁ = M ₂₄ = M ₄₂ = 0, the transmission matrix should be

M_{ideal} = (\begin{matrix} 0 & 0 & 0 & a \\ b & 0 & 0 & 0 \\ b & c & 0 & 0 \\ 0 & 0 & d & 0 \end{matrix}),

where |a| = |b| = |c| = |d| = 1 for perfect circulator. Thus, our optical circulator works when the non-reciprocal phase shift is π and its transmittance is near to unit. In practical experiments, there are absorption and elastic backscattering that make the isolator imperfect. In the following, we study the model and take these effects into account.

3. Brillouin scattering induced non-reciprocal phase shift

The nonreciprocal phase shifter is constructed by a microresonator, in which traveling light and acoustic wave are supported. In the dielectric, the light and acoustic waves interact with each other when propagating in the same (opposite) direction. The interaction primarily arises from electrostriction and photoelastic effects [35]. The WG microresonator supports two degenerate modes (clockwise and counter-clockwise modes) for each optical (control and probe light) modes and the WG mode. So, there are three modes involved, which are called control mode (frequency ω ₁ and momentum k ₁), probe mode (ω ₂, k ₂) and acoustic mode (Ω, k_a). When the acoustic and optical modes satisfy the conservation of energy (Ω = ω ₂ − ω ₁) and momentum (k_a=k ₂−k ₁), photons can be scattered between the optical modes through Brillouin scattering [27], as shown in Fig. 2(a). This SBS process is very feasible for experiments that have demonstrated in various WG microresonators [24–28 ].

Fig. 2 (a). Non-reciprocal phase shifter: A strong control laser in the forward direction causes the difference of phase shift between forward and backward laser. Transmission and phase shift of the nonreciprocal phase shifter for forward and backward direction in critically coupled regime (b) and in over-coupled regime (c) with C = 100 and Γ = 0.02κ ₀.

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This stimulated Brillouin scattering in this triple-resonant system can be described by Hamiltonian

H = H_{modes} + H_{int} + H_{in},

where the Hamiltonians for optical and acoustic modes, interaction and input are (h̄ = 1)

H_{modes} = ω_{1} a_{c, f}^{†} a_{c, f} + ω_{2} a_{p, f}^{†} a_{p, f} + ω_{2} a_{p, b}^{†} a_{p, b} + Ω b^{†} b,

H_{int} = g (a_{p, f}^{†} b a_{c, f} + a_{p, f} b^{†} a_{c, f}^{†}),

\begin{array}{l} H_{in} & = & i \sqrt{κ_{ex}} E_{c, f, in} (a_{c, f}^{†} e^{- i ω_{c} t} - a_{c, f} e^{i ω_{c} t}) \\ + i \sqrt{κ_{ex}} E_{p, f, in} (a_{p, f}^{†} e^{- i ω_{p} t} - a_{p, f} e^{i ω_{p} t}) \\ + i \sqrt{κ_{ex}} E_{p, b, in} (a_{p, b}^{†} e^{- i ω_{p} t} - a_{p, b} e^{i ω_{p} t}) . \end{array}

Here, b is acoustic Boson operator, a_p,f, a_c,f and a_p,b are optical Boson operators for forward control light, forward and backward probe light, respectively. g is the Brillouin scattering coupling strength, and E_c,f,in, E_p,f,in and E_p,b,in is the amplitude of input optical fields normalized to photon flux. Furthermore, κ_ex is the external coupling strength between the optical WGM to the waveguide.

The Langevin equation for the control light is

\frac{d}{d t} a_{c, f} = (- i ω_{1} - κ / 2) a_{c, f} - i g a_{p, f} b^{†} + \sqrt{κ_{e x}} E_{c, f, in} e^{- i ω_{c} t} .

Here, κ is the total optical decay rate and κ = κ ₀ + κ_ex and κ ₀ is intrinsic decay rate. Assume that E_c,in (control) is much stronger than E_p,in (probe), we apply the mean field approximation and describe the control light classically as complex number which is solved as

〈 a_{c, f} 〉 = \sqrt{N_{c}} e^{- i ω_{c} t} = | \frac{\sqrt{κ_{ex}} E_{c, f, in}}{i Δ_{1} - κ / 2} | e^{- i ω_{c} t} .

Where Δ₁ = ω_c − ω ₁ is the detuning and N_c is the photons number of intracavity control light. By substituting this result into the Hamiltonian and simplifying it in a frame rotating with

H_{0} = ω_{c} a_{c, f}^{†} a_{c, f} + ω_{p} a_{p, f}^{†} a_{p, f} + ω_{p} a_{p, b}^{†} a_{p, b} + (ω_{p} - ω_{c}) b^{†} b

, the linearized system Hamiltonian can be simplified as

\begin{array}{l} \tilde{H} = & - Δ_{1} a_{c, f}^{†} a_{c, f} - Δ_{2} a_{p, f}^{†} a_{p, f} - Δ_{2} a_{p, b}^{†} a_{p, b} - δ ω b^{†} b + g \sqrt{N_{c}} (a_{p, f}^{†} b + a_{p, f} b^{†}) \\ + i \sqrt{κ_{ex}} E_{p, b, in} (a_{p, b}^{†} - a_{p, b}) + i \sqrt{κ_{ex}} E_{p, f, in} (a_{p, f}^{†} - a_{p, f}) . \end{array}

Then, the dynamics of photon and acoustic phonon can be described by Langevin equations

\frac{d}{d t} a_{p, f} = [i Δ_{2} - κ / 2] a_{p, f} - i g \sqrt{N_{c}} b + \sqrt{κ_{ex}} E_{p, f, in},

\frac{d}{d t} a_{p, b} = [i Δ_{2} - κ / 2] a_{p, b} + \sqrt{κ_{ex}} E_{p, b, in},

\frac{d}{d t} b = [i δ ω - Γ / 2] b - i g \sqrt{N_{c}} a_{p, f},

where Δ₂ = ω_p − ω ₂ is frequency mismatch between probe light and its mode frequency and δω = ω_p − ω_c − Ω = Δ₂ − Δ₁ is frequency mismatch between acoustic phonon and acoustic mode frequency. Decay rates of mechanical mode is Γ. It is convenient to introduce the cooperativity factor as

C = \frac{4 g^{2} N_{c}}{κ Γ}

, representing the coherent interaction strength between photon and phonon. We are specially interested in the case that the control laser is on its resonant frequency, that is to say Δ₁ = 0. We can easily solve the equation for steady state, and the probe mode intensity by forward probe light is

a_{p, f} = - \frac{\sqrt{κ_{ex}} E_{p, f, in}}{i Δ_{2} - κ / 2 + \frac{g^{2} N_{c}}{i Δ_{2} - Γ / 2}} .

For the backward direction, the probe light intensity is

a_{p, b} = - \frac{\sqrt{κ_{ex}} E_{p, b, in}}{i Δ_{2} - κ / 2} .

We can see from the Eq. (17) and Eq. (18) that the external control laser triggers an asymmetry between forward light and backward light. For the transmission of probe lights, we use the input-output formalism:

E_{p, out} = E_{p, in} - \sqrt{κ_{ex}} a_{p} .

For both directions, we define phase shift as

ϕ = arg (\frac{E_{p, out}}{E_{p, in}}),

and the normalized transmission as

T = {| \frac{E_{p, out}}{E_{p, in}} |}^{2} .

We calculated transmission and phase shift in critically coupled regime (κ_ex ≃ κ ₀) and over-coupled regime (κ_ex ≫ κ ₀), which is determined by the length of the coupling region and air gap between waveguide and microresonator. In the critically coupled regime, non-reciprocal transmission can be achieved as shown in Fig. 2(b), which have been discussed in the previous work for non-reciprocal light storage [22, 27]. In this work, we focused on the over-coupled regime where the phase shift for different direction are different as Fig. 2(c) while the transmittance for both direction are near to unit, which result in high visibility for M–Z interferometer, thus good performance of circulator can be achieved. We can easily learn from this figure that at Δ₂ = 0 the nonreciprocal phase shift is π, which is what we want for perfect circulator as discussed in the previous section.

4. Circulating performance

In this section, we study the performance of our optical circulator. Putting the dispersion relationship and transmission [Fig. 2(c)] into the formalism Eqs. (2)–(5), we can get the transmission property for the designed circulator. As shown in the Fig. 3(a), the transmission for forward light shows an EIT-like lineshape while it for backward light is a Lorentz curve. Therefore, the transmittance at resonance for allowed light path (1 → 2, 2 → 3, 3 → 4, 4 → 1) is 1 while it for forbidden light path (2 → 1, 3 → 2, 4 → 3, 1 → 4) is 0. We can learn easily that our proposed system can function as a circulator.

Fig. 3 (a):Transmittance of this interfering system when operates as an optical circulator, and for this plot κ_ex = 100κ ₀, C = 1000. (b) The circulator bandwidth of 30 dB is approximately linear to the C factor. (c) The circulator bandwidth of 30 dB and loss after light intensity of two arm are balanced.

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Bandwidth is an essential parameter for optical device. For optical circulator, bandwidth is limited by its isolation (−10log|M_ij/M_ji|²), because isolation decreases with the increasing of detuning. As shown in Fig. 3(a), isolation for light from Port 1(3) to Port 4(2) decreases most rapidly. So we can learn the bandwidth of circulator by discussing light at this path. The bandwidth of 30 dB isolation is calculated and shown as Fig. 3(b). It’s shown that isolation of 30 dB can be achieved with C larger than 30, which is infeasible for experiments (C of 5.9 is realized in our previous work [27]).

Smaller C would result in reduced transmission of the microcavity and the interference visibility deteriorate, due to the energy absorption by the cavity. By simple calculation, it can be demonstrated that perfect isolation can not be realized for unbalanced losses in two paths. To solve this problem, we can add a loss in the upper arm to balance the light intensity in two paths. After balanced light intensity in two arms, the output electric field is

E_{out} (Δ) = \frac{1}{2} E_{p, in} [L^{1 / 2} - T^{1 / 2} (Δ) e^{ϕ_{f} (Δ)}] .

Around the resonant frequency, this can be simplied as

E_{out} (Δ) = E_{out} (0) + E_{out}^{'} (0) Δ + o (Δ) \approx E_{out}^{'} (0) Δ .

When isolation equals 30 dB,

{| M_{41 (23)} |}^{2} = 10^{- 3} {| M_{14 (32)} |}^{2} .

Using Eq. (23), together with Eqs. (17)–(21), We calculated the bandwidth of 30 dB isolation for balancedlight intensity in two arms

Bandwidth = 10^{- 3 / 2} \frac{{(1 + C)}^{2} Γ}{C}

Bandwidth also been gained by directly find root of Eq. (4) without approximation, as shown in black line in Fig. 3(c) (with Γ = 0.02κ ₀). Though 30 dB isolation can be achieved with very small C, the loss (blue line in Fig. 3(c)) would be rather large. With C becoming larger, bandwidth increases and loss decreases. At C = 10, the bandwidth is 0.007κ ₀ and the loss is 1.57 dB. And at C = 20, the bandwidth is 0.013κ ₀ and the loss is 0.79 dB.

5. The impact of elastic backscattering

In this section we discuss a more realistic situation, where the intrinsic defects inside the resonator cause elastic backscattering for both direction and lead to coupling between forward and backward optical modes [36]. This coupling may degenerate the non-reciprocity of the phase shifter thus damage the performance of our optical circulator.

To study the impact of the backscattering, we firstly modify the Hamiltonian as

{\tilde{H}}_{op} = H_{op} + ω_{1} a_{c, b}^{†} a_{c, b}

{\tilde{H}}_{mech} = Ω b_{f}^{†} b_{f} + Ω b_{b}^{†} b_{b}

\begin{array}{l} {\tilde{H}}_{int} & = H_{int} + β a_{c, f}^{†} a_{c, b} + β^{*} a_{c, f} a_{c, b}^{†} + β a_{p, f}^{†} a_{p, b} + β^{*} a_{p, f} a_{p, b}^{†} \\ + g (a_{p, b}^{†} b_{b} a_{c, b} + a_{p, b} b_{c, b}^{†} a_{c, b}^{†}) \end{array}

Here, the backward control mode as well as backward acoustic mode caused by the backscattering of forward control laser are included. Furthermore, the coupling between forward and backward mode due to elastic backscattering are introduced, which are described by terms

β a_{f}^{†} a_{b}

and

β^{*} a_{f} a_{b}^{†}

with β representing the strength of elastic backscattering. The Langevin equations describing the system read

\frac{d}{d t} a_{c, f} = (i Δ_{1} - κ / 2) a_{c, f} - i g a_{p, f} b_{f}^{†} + \sqrt{κ_{ex}} E_{c, f, in} - i β^{*} a_{c, b}

\frac{d}{d t} a_{p, f} = (i Δ_{2} - κ / 2) a_{p, f} - i g a_{c, f} b_{f} + \sqrt{κ_{ex}} E_{p, f, in} - i β^{*} a_{p, b}

\frac{d}{d t} a_{c, b} = (i Δ_{1} - κ / 2) a_{c, b} - i g a_{p, b} b_{b}^{†} - i β a_{c, f}

\frac{d}{d t} a_{p, b} = [i Δ_{2} - κ / 2] a_{p, f} - i g a_{c, b} b_{b} + \sqrt{κ_{ex}} E_{p, b, in} - i β a_{p, f}

\frac{d}{d t} b_{f} = [- i δ ω - Γ / 2] b_{f} - i g a_{p, f} a_{c, f}^{†}

\frac{d}{d t} b_{b} = [- i δ ω - Γ / 2] b_{b} - i g a_{p, b} a_{c, b}^{†} .

In the equations above, E_p,f,in and E_p,b,in are input field operators for forward and backward probe light. As before we assume that the control laser for forward direction is much stronger than other light and we can solve the equation in steady state numerically. Then put them into the Eqs. (2–5) so we can get the performance of the optical circulator under the impact of backscattering as shown in the Fig. 4(a). Light propagating forward are almost not influenced by the backscattering while light propagating backward are greatly influenced especially for light propagating 2 → 3 and 4 → 1 due to the reflected control laser inside the cavity. As shown by the Red lines in Fig. 4(a), there are narrow peak and dip due to Brillouin scattering induced absorption and the phase shifter for backward light, and the narrow linewidth is due to weak control light intensity in the backward direction. Thus, the circulating performance deteriorate at resonance. As β and cooperativity C increasing, the backward control laser becomes stronger and the performance of the circulator is even damaged. Fig. 4(b) shows the intensity of the channel of probe light 2 → 1 or 4 → 3, which is most seriously influenced by backscattering, varying with backscattering factor β and cooperativity factor. Fortunately, in spite of the impact of backscattering, the red region (dip depth ≈ 1) can still be used for a good performing circulator. Moreover, considering the peak and dip are extremely sharp, we can still find ideal circulating performance detuned from the the peak and dip (Δ₂ ≠ 0).

Fig. 4 (a) The transmittance for our optical circulator when backscattering exists. For this plot β = 1 and C = 1000. (b) Contour for the dip in the transmission spectra of light 4-1 and 2–3, which is the most prominent impact of backscattering. (c) Probe light is backscattered and propagates back and for this plot C = 1000.

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In addition for the influence of the backscattered control light, the probe light is also directly backscattered. However, as the intensity is shown in Fig. 4(c), the EIT-like curve that the intensity is 0 at resonance. Little probe light is backscattered when it is on the cavity resonant frequency. So this process would not damage the performance of our optical circulator either cause unwanted feedback.

6. Conclusion

In this work, we have studied that stimulated Brillouin scattering can induce light non-reciprocity. Particularly, in the over-coupled regime (κ_ex ≫ κ ₀), the high transmittance and nonreciprocal phase shift are ideal property for optical circulator when the microresonator coupled with one arm of a Mach-Zehnder interferometer. From an application point of view, this system provides an solution for on-chip magnetic-free integrated optical circulator. By introducing of energy loss of the upper arm, complete isolation for forbidden light path can be realized. The bandwidth of this circulator is determined by the cooperativity factor and the coupling parameter. In the previous experiment [28], Kim et. al. estimated that strong coupling in a microsphere where $g N_{c}^{1 / 2} \approx κ$ can be achieved with less than 1mW control power. Under this condition, C ≈ 250 for Γ = 0.017 MHz and κ ₀ = 4.4 mHz, the corresponding 30 dB bandwidth for the circulator can reach 100 kHz. From an experimental perspective, our design can be realized in many different systems [24–28 ]. We have only discussed about forward SBS above, that is, acoustic wave is stimulated by co-propagating control and probe light. Backward SBS coupling between counter-propagating light can also be achieved the non-reciprocal phase shifter. Moreover, SBS can be excited in waveguide. However, without the resonant geometry, long device length is needed for enough coupling strength at a moderate pump power.

Acknowledgments

The work was supported by the Strategic Priority Research Program(B) of the Chinese Academy of Sciences (Grant No. XDB01030200), National Basic Research Program of China (Grant Nos. 2011CB921200 and 2011CBA00200) and the National Natural Science Foundation of China (Grant No. 61308079 and 11204169), Anhui Provincial Natural Science Foundation (Grant No. 1508085QA08), the Fundamental Research Funds for the Central Universities.

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Integrated optical circulator by stimulated Brillouin scattering induced non-reciprocal phase shift

Abstract

1. Introduction

2. Principle of optical circulator

3. Brillouin scattering induced non-reciprocal phase shift

4. Circulating performance

5. The impact of elastic backscattering

6. Conclusion

Acknowledgments

References and links

Cited By

Figures (4)

Equations (34)

Optics Express