## 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 (*2*–*4*) 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 (*7*–*11*), 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 *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 (*17*–*19*). 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 *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 (*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 *22*), suggesting noncommutative binary operations to the Hamiltonian

More intuitively, the introduced one-way wave vector *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

However, the approach so far demonstrated to create the exponentially modulated *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

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

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

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.

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

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**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.