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Nonreciprocal Light Propagation in a Silicon Photonic Circuit

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Science  05 Aug 2011:
Vol. 333, Issue 6043, pp. 729-733
DOI: 10.1126/science.1206038

Abstract

Optical communications and computing require on-chip nonreciprocal light propagation to isolate and stabilize different chip-scale optical components. We have designed and fabricated a metallic-silicon waveguide system in which the optical potential is modulated along the length of the waveguide such that nonreciprocal light propagation is obtained on a silicon photonic chip. Nonreciprocal light transport and one-way photonic mode conversion are demonstrated at the wavelength of 1.55 micrometers in both simulations and experiments. Our system is compatible with conventional complementary metal-oxide-semiconductor processing, providing a way to chip-scale optical isolators for optical communications and computing.

An example of nonreciprocal physical response, associated with the breaking of time-reversal symmetry, is the electrical diode (1). Stimulated by the vast application of this one-way propagation of electric current, considerable effort has been dedicated to the study of nonreciprocal propagation of light. The breaking of time-reversal symmetry of light is typically achieved with magneto-optical materials that introduce a set of antisymmetric off-diagonal dielectric tensor elements (24) or by involving nonlinear optical activities (5, 6). However, practical applications of these approaches are limited for the rapidly growing field of silicon (Si) photonics because of their incompatibility with conventional complementary metal-oxide-semiconductor (CMOS) processing. Si optical chips have demonstrated integrated capabilities of generating (711), modulating (12), processing (13) and detecting (14) light signals for next-generation optical communications but require on-chip nonreciprocal light propagation to enable optical isolation in Si photonics.

Parity-time (PT) symmetry is crucial in quantum mechanics. In contrast to conventional quantum mechanics, it has been proposed that non-Hermitian Hamiltonians where H^+H^ can still have an entirely real spectrum with respect to the PT symmetry (15, 16). Due to the equivalence between the Schrödinger equation in quantum mechanics and the wave equation in optics, PT symmetry has been studied in the realm of optics with non-Hermitian optical potentials (1719). The breaking of PT symmetry has recently been experimentally observed, showing asymmetric characteristics transverse to light propagation above the PT threshold (20, 21). Here, we have designed a Si waveguide integrated with complex optical potentials that have a thresholdless broken PT symmetry along the direction of light propagation, thus creating on-chip nonreciprocal light propagation.

On a Si-on-insulator (SOI) platform, the designed two-mode Si waveguide is 200 nm thick and 800 nm wide, allowing a fundamental symmetric quasi-TE mode with a wave vector of k1=2.59k0 and a higher-order antisymmetric mode with a wave vector of k2=2k0 at the wavelength of 1.55 μm. Periodically arranged optical potentials are implemented in the Si waveguide and occupy half of the waveguide width in the x direction (Fig. 1A). The optical potentials have a complex modulation in their dielectric constants along light propagation in the z direction compared with the Si waveguide background (εSi=12.11), as shown in Fig. 1B Δε=exp[iq(zz0)](1)where q=k1k2, and z0 is the starting point of the first modulation region. This complex exponential variation of Δε along the z direction introduces a one-way wave vector that is intrinsically not reciprocal because its corresponding Fourier transform is one-sided to the guided light inside the Si waveguide. These complex optical potentials are located in phase with each other with a spacing of 2π/q (or multiples of 2π/q) in between, such that light modulation always remains in phase with and is consistently applied to guided light. We chose the dielectric constant modulation to be completely passive in order to make the experiment easier to perform, meaning that the modulation length of each optical potential is π/q. Therefore, no gain is required to construct these optical potentials. From a quantum mechanics analysis, these optical potentials have a spontaneously broken PT symmetry with a non-Hermitian Hamiltonian H^+(x,z)H^(x,z) or H^+(x,z)H^(x,z) (22), suggesting noncommutative binary operations to the Hamiltonian PTH^+H^PT. In our system, this is observed as nonreciprocal light propagation through the optical potentials in the Si waveguide.

Fig. 1

(A) Nonreciprocal light propagation in a Si photonic circuit. Based on a SOI platform, PT optical potentials with exponentially modulated dielectric constants, as depicted in (B) where blue and red curves represent the real and imaginary parts of Embedded Image, respectively, are embedded in the Si waveguide to introduce an additional wave vector Embedded Image to guided light. (C) Band diagram for TE-like polarization of the Si waveguide, where the frequency and wave vector are normalized with a = 1 μm. At the wavelength of 1.55 μm, if incoming light is a fundamental symmetric mode, one-way mode conversion is only expected for backward propagation where the phase-matching condition is satisfied as indicated by arrows.

More intuitively, the introduced one-way wave vector q shifts the incoming photons of the symmetric mode with an additional spatial frequency: k1+q for forward propagation and k1+q for backward propagation. The mode transition between the symmetric mode and the antisymmetric mode can happen only when the phase-matching condition is approximately satisfied Δk=±(k1k2)+q0, where + and – represent forward and backward propagation, respectively. In our case, for an incoming symmetric mode the phase-matching condition is only valid in the backward direction, supporting a one-way mode conversion from k1 to k2 (Fig. 1C). In the modulated regime, the electric field of light is given byE(x,z,t)=A1(z)E1(x)ei(k1zωt)+A2(z)E2(x)ei(k2zωt)(2)where E1,2(x) are normalized mode profiles of two different modes, and A1,2(z) are the corresponding normalized amplitudes of two modes, respectively. Assuming a slowly varying approximation, the coupled-mode equations can be expressed as follows:ddzA1(z)=iB1exp(iqz)A1(z)+iC1A2(z)ddzA2(z)=iC2exp(i2qz)A1(z)+iB2exp(iqz)A2(z)(3)for forward propagation andddzA1(z)=iB1exp(iqz)A1(z)iC1exp(i2qz)A2(z)ddzA2(z)=iC2A1(z)iB2exp(iqz)A2(z)(4)for backward propagation, where B1=12k1ω2c2E1(x)E1(x)dx|E1(x)|2dx, C1=12k1ω2c2E1(x)E2(x)dx|E1(x)|2dx, C2=12k2ω2c2E2(x)E1(x)dx|E2(x)|2dx, and B2=12k2ω2c2E2(x)E2(x)dx|E2(x)|2dx. The mode transition can happen only when the phase-matching condition is satisfied as the exponential term disappears because exp(iΔkz)=1. Therefore, it is evident that with an initial condition of A1=1 and A2=0, photons from the symmetric mode can be converted to the antisymmetric mode only for backward propagation, whereas A2 remains 0 for forward propagation, indicating negligible mode conversion. The nonreciprocity of the mode transition here results from the spontaneous breaking of the PT symmetry of guided light by the engineered complex optical potentials. It is worth emphasizing that this nonreciprocal unidirectional mode transition is always valid with any modulation intensity, indicating a completely thresholdless breaking of PT symmetry (22), in stark contrast to previous work on threshold PT symmetry breaking (20, 21).

Fully vectorial three-dimensional (3D) finite element method simulations have been performed to validate the proposed nonreciprocal propagation of guided light at the telecom wavelength of 1.55 μm. With a TE-like symmetric incident mode, after forward propagating through the PT optical potentials where Δε follows the exponential modulation, guided light does not meet any phase-matching condition with the additional wave vector q and therefore retains the same symmetric mode profile. However, for backward propagation, it is evident that the antisymmetric mode is converted from the incoming symmetric mode due to the phase matching with the additional wave vector q, showing a one-way mode transition (Fig. 2A). The nonreciprocal light propagation can also be analytically calculated using the coupled-mode theory from Eqs. 2 to 4 (fig. S1), consistent with the simulated results.

Fig. 2

Evolution of PT optical potentials in the Si waveguide (top) and their corresponding field distributions of Embedded Image for forward (middle) and backward (bottom) propagation with an incoming symmetric mode. (A) Original PT optical potentials with exponentially modulated dielectric constant. (B) Two different kinds of optical potentials with real cosine and imaginary sinusoidal modulated dielectric constants. (C) Optical potentials with the real part modulation in (B) are shifted Embedded Image in the z direction.

However, the approach so far demonstrated to create the exponentially modulated Δε (21) is difficult to integrate with Si photonics. It is therefore necessary to design an equivalent guided-mode modulation that at a macroscopic scale mimics the intrinsically microscopic exponential modulation of the PT optical potentials. To simplify fabrication, each complex exponential modulation is separated into two different modulation regions: one providing only the imaginary sinusoidal modulation of the dielectric constant covering one transverse half space (bottom) of the waveguide, and the other creating the real cosine modulation occupying the other transverse half space (top) of the waveguide (Fig. 2B), as follows

Δεreal=cos[q(zz0)]Δεimag=isin[q(zz0)](5)

Although individual sinusoidal or cosine modulation does not contribute to the breaking of PT symmetry, simultaneous modulations of both cause an equivalent nonreciprocal one-way mode transition. Guided light in different half spaces experiences complementary mode modulation from each other and therefore behaves as if the PT optical potentials do exist. Moreover, to have only the positive Δεreal of the modulations for ease of fabrication, regions of Δεreal are shifted π/2q in the z direction: Δεreal=sin[q(z(z0+π/2q))] (Fig. 2C). The resulting one-way mode transition of guided light consequently remains the same.

Finally, to achieve sinusoidal optical potentials using microscopically homogeneous materials, sinusoidal-shaped structures are adopted on top of the Si waveguide for both real and imaginary modulations to mimic the modulations described in Eq. 5 (Fig. 3A). An 11-nm germanium (Ge)/18-nm chrome (Cr) bilayer structure is applied for the imaginary modulation Δεimag as guided modes have the same effective indices as Δε=i. For the real modulation Δεreal, additional 40-nm Si on top of the original Si waveguide achieves the same effective indices of guided modes as Δε=1. The length, period, and locations of these sinusoidal shaped structures follow those in Fig. 2C. The designed sinusoidal-shaped structures have almost the same effective indices of the waveguide modes, as if the real and imaginary function-like modulations exist in the waveguide (Fig. 3, B and C), such that the same unidirectional wave vector q can be introduced. Therefore, an equivalent one-way mode transition is realized, as shown in Fig. 3D: Forward propagating light remains in the symmetric profile, whereas mode conversion from the symmetric mode to the antisymmetric mode exists for backward propagation. It is thus evident that our classical waveguide system successfully mimics the quantum effect inherently associated with a broken PT symmetry. Overall, at different steps of the evolution of PT optical potentials from Fig. 2 to Fig. 3, guided light exhibits almost identical phase and intensity for both forward and backward propagation, further proving the equivalence of our classical design to the quantum PT potentials for guided light. To confirm the thresholdless condition of the breaking of PT symmetry in our system, we also designed and simulated different Si and Ge/Cr combinations corresponding to different modulation intensities from Δε=0.25exp[iq(zz0)] to Δε=0.75exp[iq(zz0)] (fig. S2). Although conversion efficiencies reduce as modulation intensities decrease, one-way mode conversion always exists, indicating that the breaking of PT symmetry in our system is spontaneous without any threshold.

Fig. 3

(A) Design of the metallic-Si waveguide to mimic the light modulation of PT optical potentials. (B) Effective indices of symmetric and antisymmetric modes with the imaginary part sinusoidal-modulated optical potential (red lines) and the sinusoidal-shaped Ge/Cr bilayer structure (blue dots). (C) Modes’ effective indices with the real part sinusoidal-modulated optical potential (red lines) and the sinusoidal-shaped Si structure (blue dots). Insets in (B) and (C) show the considered waveguide, and Δz starts from where modulation begins. (D) Numerical mappings of Embedded Image for forward (upper) and backward (lower) propagation with an incoming symmetric mode.

A picture of the fabricated device is shown in Fig. 4A. Nonreciprocal light propagation in the Si waveguide was observed using a near-field scanning optical microscope (23). In experiments, a tapered fiber was used to couple light into the waveguide. Although the fundamental symmetric mode is dominant in incidence, there also exists some power coupled to the antisymmetric mode as shown in Fig. 4B. Consistent with simulations, light remains predominantly the fundamental symmetric mode after propagating through the optical potentials for forward propagation. However, the symmetric-mode-dominant incoming light in backward propagation clearly shows mode conversion to the antisymmetric mode after the device. With phase information of guided light simultaneously obtained using our heterodyne system (22, 23), we applied the Fourier transform analysis to the measured field distribution. The results further confirm that the breaking of PT symmetry in our system as the conversion from the symmetric mode to the antisymmetric mode can be observed only in the backward direction (fig. S3). It is therefore evident that one-way mode conversion and nonreciprocal light propagation have been successfully realized in CMOS-compatible Si photonics, paving the way to on-chip optical isolation. Although the insertion loss of about 7 dB is observed through the optical potentials, it can be completely compensated by incorporating gain into the imaginary part modulation of the PT optical potentials in Eqs. 1 and 5. One can easily envision constructing chip-scale isolators by extending the demonstrated nonreciprocal light propagation. The excited antisymmetric mode can be removed in transmitted fields by implementing, next to the PT optical potentials, an optical mode filter that completely reflects the antisymmetric mode but allows the symmetric mode to transmit. Optical isolation with large extinction ratio can be achieved by controlling interference between incidence and reflection.

Fig. 4

(A) Scanning electron microscope image of the fabricated device. (B) Measured near-field amplitude distribution of light in the one-way mode converter for both forward (upper) and backward (lower) light propagation.

The nonreciprocal light propagation we have accomplished in a Si photonic circuit is expected to have strong impacts in both fundamental physics and device applications. The feasibility of mimicking complicated quantum phenomena in classical systems promises completely chip-scale optical isolation for rapidly growing Si photonics and optical communications. The proposed one-way system is completely linear and expected to have higher efficiencies and broader operation bandwidths than nonlinear strategies. Analogously the concept of this one-way propagation can be applied to other classical waves, such as sound, as a promising method to drastically increase the rectification efficiency over current nonlinearity-induced isolation (24).

Supporting Online Material

www.sciencemag.org/cgi/content/full/333/6043/729/DC1

Materials and Methods

Figs. S1 to S3

References

References and Notes

  1. Materials and methods are available as supporting material on Science Online.
  2. Acknowledgments: Support by NSF and NSF ERC Center for Integrated Access Networks grant EEC-0812072, Defense Advanced Research Projects Agency under the Nanoscale Architecture for Coherent Hyperoptical Sources program grant W911NF-07-1-0277, National Basic Research Program of China grant 2007CB613202, National Nature Science Foundation of China grants 50632030 and 10874080, and Nature Science Foundation of Jiangsu Province grant BK2007712. We thank Nanonics, Ltd., for extensive training and support in near-field scanning optical microscopy. M.A. acknowledges support of a Cymer Corp. graduate fellowship.
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