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Unidirectional Transmission Based on a Passive PT Symmetric Grating With a Nonlinear Silicon Distributed Bragg Reflector Cavity Volume 6, Number 1, February 2014 Ye-Long Xu Liang Feng Ming-Hui Lu Yan-Feng Chen

DOI: 10.1109/JPHOT.2013.2295462 1943-0655 Ó 2013 IEEE

IEEE Photonics Journal

Transmission Based on a Passive PT Grating

Unidirectional Transmission Based on a Passive PT Symmetric Grating With a Nonlinear Silicon Distributed Bragg Reflector Cavity Ye-Long Xu, 1 Liang Feng, 2 Ming-Hui Lu, 1 and Yan-Feng Chen 1 1

2

National Laboratory of Solid State Microstructures and Department of Materials Science and Engineering, Nanjing University, Nanjing 210093, China Department of Electrical Engineering, The State University of New York at Buffalo, Buffalo, NY 14260-2500 USA

DOI: 10.1109/JPHOT.2013.2295462 1943-0655 Ó 2013 IEEE. Translations and content mining are permitted for academic research only. Personal use is also permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

Manuscript received October 16, 2013; revised December 11, 2013; accepted December 11, 2013. Date of publication December 20, 2013; date of current version January 15, 2014. This work was supported by the National Basic Research of China under Grants 2012CB921503 and 2013CB632702, by the National Natural Science Foundation of China under Grant 11134006, by the Priority Academic Program Development of Jiangsu Higher Education, and by FANEDD of China. Corresponding author: M.-H. Lu (e-mail: [email protected]).

Abstract: A silicon waveguide consisting of passive parity-time (PT) symmetry optical potentials connected with a distributed Bragg reflector (DBR)-based resonator is proposed to achieve nonreciprocal waveguide transmission. The unidirectional reflectionless waveguide implementing PT symmetry is discussed by consistent coupling mode theory and finitedifference time-domain (FDTD) simulation results. Due to the high field confinement in the DBR cavity, transmission of a light pulse through the device is analyzed in the nonlinear optical regime using FDTD simulations. The combination of the nonlinear DBR cavity and the passive PT symmetric grating results in unidirectional transmission in the silicon waveguide, while still keeping the reflection minimum in the desired direction. The numerical simulation shows an extinction ratio of about 20 dB at the telecom wavelength of around 1550 nm. Index Terms: Silicon waveguide, unidirectional reflectionless parity-time (PT) grating, nonlinear distributed Bragg reflector (DBR) cavity.

1. Introduction Nowadays, Si photonics has become a crucial technology in optical communications and computing due to the recently demonstrated on-chip integration of various compact optical devices [1]–[3]. It is of great importance to overcome challenges in designing on-chip integrated devices that can control light flows in different spatial directions as desired. For example, it requires the breaking of time reversal symmetry and Lorentz-reciprocity of an optical system to achieve one-way nonreciprocal light transmission, which is typically associated with the Faraday effect [4], [5] where external magnetic fields are applied to break the space–time symmetry. However, this approach requires materials with appreciable Verdet constants, and it may not be compatible with complicated on-chip integration schemes and fabrication. To address this issue, alternative non-magnetic proposals have been suggested recently, including optical diodes based on second harmonic generation in asymmetric waveguides [6], nonlinear photonic band-gap materials [7], and asymmetric nonlinear

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Fig. 1. Schematic diagram of periodically arranged passive PT optical potentials with the modulated dielectric permittivity of " ¼ expðiqzÞ ð2n=q z 2n=q þ =qÞ are introduced into a 0. 23 m-wide Si waveguide. The gray regimes are the passive modulation regimes. The PT optical Bragg grating consist of 20 modulation periods of 2=q ¼ 0:27 m. The spectra of numerical reflectance and transmittance of the PT grating over the studied wavelength range from 1540 nm to 1560 nm in (a) forward and (b) backward directions, respectively.

structures [8]. However, although with no need of magnetic fields, these schemes are also not suitable for chip-scale applications because of silicon’s low second-order susceptibility. Recently, nonreciprocal transmission has been realized in Si photonics using time-dependent index modulations [9], [10] and asymmetric Si micro-ring resonators [11]. Another approach to onchip unidirectional optical transport is also of great interest, involving parity-time (PT) symmetric optical materials with optical Kerr nonlinearity [12], [13]. Because of Si’s considerable Kerr coefficient [14], it is feasible to realize a new class of optical diodes and other novel unidirectional photonic elements on the silicon photonic circuits by implementing PT optical materials. The PT invariant Hamiltonians was originally proposed by Bender and Boettcher [15] and has recently been introduced into classical optics by the combination of delicately balanced amplification and absorption regions with the desired modulation of the refractive index [16]–[18]. Such complex PT optical potentials can be realized in a synthetic manner and demonstrate a series of intriguing features, such as PT phase transition [17], [18]. More interestingly, at the PT phase transition point, also called spontaneous PT phase breaking point, a unidirectional invisible medium was realized [19]–[21]. At the PT-symmetry breaking point, the PT symmetric complex crystal is almost reflectionless when probed from one side, whereas from the other side, reflection is enhanced. A similar unidirectional reflectionless phenomenon was realized on a Si chip by implementing sinusoidally shaped index and loss modulations in the lengthy direction of a Si waveguide [22].


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In this paper, we will combine the demonstrated unidirectional reflectionless PT grating with a nonlinear silicon distributed Bragg reflector (DBR) cavity to realize unidirectional transmission at the telecom wavelength of around 1550 nm while still keeping the reflection minimum in the desired direction. In Section 2, we will analyze the unidirectional reflectionless PT symmetric grating using the coupled mode theory and finite-difference time-domain (FDTD) simulations. Their respective results well agree with each other. In Section 3, the transmission property of the DBR cavity will be analyzed using FDTD simulations at different power amplitudes of the incident light pulse. In Section 4, by combining the unidirectional reflectionless PT grating with the nonlinear DBR cavity, we have successfully realized unidirectional transmission with an extinction ratio of 20 dB at around 1550 nm with negligible reflection in the desired direction. Finally, the conclusion is given in Section 5.

2. Unidirectional Reflectionless PT Symmetric Grating The considered silicon waveguide ð"Si ¼ 12:25Þ is a two-dimensional (2D) waveguide. Although a three-dimensional design is more practical, our 2D design is more straightforward to demonstrate the principle of the device. The designed waveguide has a width of w ¼ 0:23 m as shown in Fig. 1(a), such that the waveguide supports a single mode with a wavevector of k1 ¼ 2:87k0 at the wavelength of 1550 nm, where k0 ¼ ð2Þ=. Therefore, to form the Bragg grating for the single mode of guided light, the introduced modulation of the dielectric permittivity along the waveguide is " ¼ cosðqzÞ þ isinðqzÞ, where q ¼ 2k1 . Since there is no optical gain in Si at telecom wavelengths, in this work, we only implement the corresponding passive modulation. Consequently, the modulation is limited to the absorption regime of 2n=q z 2n=q þ =q on silicon photonic circuits. Light propagation in such a passive PT Bragg grating can be analyzed using the coupled mode theory. In the modulated regime, electric field can be written as E ðx ; zÞ ¼ AðzÞE ðx Þe ik1 z þ BðzÞE ðx Þe ik1 z , where AðzÞ and BðzÞ are the amplitudes of forward and backward modes, respectively. With slowly varying approximation, the coupled-mode equations can be derived as ( dAðzÞ 1þ dz ¼ AðzÞ þ i 4 BðzÞ (1) dBðzÞ 1 dz ¼ i 4 AðzÞ þ BðzÞ R R where ¼ nimag k0 and ¼ ðk02 =ð2k1 ÞÞðð E "Edx Þ=ð jE j2 dx ÞÞ. At the spontaneous PT symmetry breaking point where ¼ 1, the coupled mode equations can be simplified as ( dAðzÞ dz ¼ AðzÞ þ i 2 BðzÞ (2) dBðzÞ dz ¼ BðzÞ: Therefore, the corresponding transmission and reflection coefficients are T ¼ expððð2LÞ=ÞÞ, Rf ¼ 0, and Rb ¼ ðð2 2 Þ=ð42 ÞÞsinh2 ððLÞ=Þexpððð2LÞ=ÞÞ, where L is the modulation length of the grating. It is worth mentioning that Rf is always zero for the PT grating regardless of its length if its length is an integer multiple of the modulation period, which has been discussed in detail in Ref. [22]. With derived analytical formulas for transmission and reflection, the attenuation coefficient can be determined as ¼ 0:5944 m1 , and the coupling coefficient is ¼ 0:5944 m1 . Specifically in this work, the passive PT Bragg grating has 20 modulation periods. The corresponding transmission and reflection coefficients in the forward and backward directions are obtained: T ¼ 0:13, Rf ¼ 0, Rb ¼ 0:47. Additionally, we also numerically calculated transmission and reflection spectra of the passive PT grating over the studied wavelength range from 1540 nm to 1560 nm in both forward and backward directions using FDTD simulations (Lumerical FDTD 7.5) shown in Fig. 1. The transmission and reflection coefficients at 1550 nm are consistent with the analytical results from the coupled mode theory. It is worth noting that the studied passive PT grating can be realized by an equivalent guided-mode modulation, which has been realized using additional sinusoidal-shaped structures on top of the waveguide to mimic the microscopic PT modulation at its spontaneous PT symmetry breaking point " ¼ cosðqzÞ þ isinðqzÞ [22].


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Fig. 2. (a) Schematic of the nonlinear silicon DBR cavity. The dark blue and light blue regimes indicate the high and low refractive index layers, respectively. The purple regime at the center denotes the nonlinear defect layer. (b) The transmission spectra of the nonlinear cavity at different incident electric amplitudes of 1:3 108 V/m (red) and 1 109 V/m (blue). (c) The corresponding contrast ratio of transmission.

3. Optical Response of the Nonlinear Silicon Waveguide DBR Cavity The structure of the nonlinear silicon waveguide DBR cavity in this work is schematically shown in Fig. 2(a). In this structure, the lattice constant is =2neff ¼ 0:27 m. The binary alternative materials to form the DBR are assumed: "1 ¼ 13:25 and "2 ¼ 11:25, respectively. A defect is then introduced at the center of the structure, and the length of the defect is assumed 11=2neff ¼ 2:97 m. Because of the expected high field confinement in the DBR cavity, light may propagate through the device in the nonlinear regime. The third-order nonlinearity susceptibility considered in this work is 2:5 1019 m 2 =V 2 , which corresponds to a typical value of a nonlinear optical material, such as silicon [14]. In practice, such silicon waveguide resonators can be realized by implementing nanostructured sidewall modulations to form two DBRs on both sides of the cavity [23], [24]. A theoretical model [25], [26] is then adopted to analyze the optical response of the silicon waveguide DBR cavity. To simplify the problem, effects resulting from two photon absorption, freecarrier dispersion, and thermal expansion are neglected in this work. Of particular interest, the steady-state relationship between the input and output light intensities of the cavity can be expressed as It 1=T ¼ 2 4R 1 Ii 1 þ T 2 sin 2 ð0 þ ð4n2 !l=cÞIt Þ

(3)

where Ii is the input light intensity into the cavity; It is the intensity transmitted through the cavity; R and T are the reflectance and transmittance of the DBR, respectively; l is the length of the resonant cavity; 0 is the phase shift of a linear contribution acquired in a round trip through the cavity; and n2 is the effective nonlinear index coefficient of the guided mode. From the phase shift of the nonlinear contribution proportional to intensity in the denominator of Eq. (3), we can see that different incident intensities will induce different resonant wavelengths as well as result in different output intensities, which has been explicitly investigated [26]. We applied FDTD simulations to further discuss this effect in the designed DBR cavity. In the simulation, the number of periods on each side of the cavity is N ¼ 39. The incident pulse is centered at 1550 nm with a pulse length of 200 fs in time. The peak wavelength of the output pulse is located at 1549.153 nm when the incident electric amplitude is 1:3 108 V/m and at 1549.166 nm if the electric amplitude is 1 109 V/m. The simulated transmission spectra for both peak powers are plotted in Fig. 2(b). The contrast ratio of transmission between these two power levels ðC ¼ jðT1 T2 Þ=ðT1 þ T2 Þj [13]) is also plotted, as shown in


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Fig. 3. Pulsed spectra for the unidirectional reflectionless device with the nonlinear silicon cavity. (a) The pulse spectra of incident, transmission and reflection for forward incidence. (b) The pulse spectra of incident, transmission and reflection for backward incidence.

Fig. 2(c). It can be seen that different powers of the input light pulse lead to highly contrasted transmittances through the cavity. In the next section, we will use the passive PT Bragg grating to introduce different power levels for light coming into the cavity from forward and backward directions, respectively, and thus transform the power-related transmission spectra to be direction-dependent.

4. Unidirectional Transmission In this section, we will discuss the property of the final device that cascades the unidirectional reflectionless passive PT grating and the nonlinear DBR cavity. We perform FDTD numerical simulations to obtain the transmission and reflection spectra of the device. The incident pulse is centered at 1550 nm with a pulse length of 200 fs (about 20 nm bandwidth in wavelengths), and the peak electric amplitude is 1 109 V/m. As shown in Fig. 3(a), for forward incidence, due to the existence of the unidirectional reflectionless PT grating, the pulse reflection is almost negligible. After the forward incident pulse goes through the passive PT grating, the pulse amplitude reduces due to absorption in the PT grating and then goes through the nonlinear DBR cavity. The transmitted pulse has a bandwidth of about 0.02 nm centered at 1549.156 nm. For backward incidence [see Fig. 3(b)], the incident pulse is almost reflected and has even higher amplitude at 1549.171 nm because of the nonlinear wavelength conversion from the broadband incident pulse. As a comparison, the transmission spectra from 1549.0 nm to 1549.4 nm in both directions are shown in Fig. 4(a). The corresponding transmission contrast ratio is then plotted in Fig. 4(b). The maximum contrast ratio is 0.9810, corresponding to an extinction ratio of 20 dB at 1549.145 nm: Forward transmittance is about 0.0837, and backward transmittance is about 0.0008. It is therefore evident that by cascading the passive PT grating and the nonlinear DBR cavity, we can achieve unidirectional transmission in a Si waveguide, while still keeping reflection minimum in the desired direction. Moreover, to make the device power-wisely more practical, the insertion loss due to absorption from the passive PT grating may be compensated by a linear optical amplifier through on-chip integration of uniform optical gain materials [22].


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Fig. 4. (a) Transmission spectra of the device in forward (red) and backward (blue) directions from 1549.0 nm to 1549.4 nm. (b) The corresponding contrast transmission ratio between forward and backward directions.

5. Conclusion In this paper, we have proposed a device cascading a passive PT grating and a DBR cavity to realize the unidirectional transmission in a Si waveguide. The characteristics of the device were analyzed using both the coupled mode theory and FDTD simulations. Because of the high-field confinement inside the cavity, light propagation through the cavity has to be analyzed in a nonlinear regime. The introduced optical nonlinearity makes the PT grating show unidirectional behaviors in light transmission. The results show that, with an incident pulse with its peak amplitude of 1 109 V/m, unidirectional transmission with an extinction ratio of 20 dB at 1549.145 nm in the forward direction. Forward reflection from the device itself is also significantly suppressed due to the inherently associated unidirectional reflectionless light transport in the PT grating. Although the device here is designed for high energy pulses, the power of the input pulse can be reduced by improving the Q factor of the cavity or by hybrid integration with on-chip optical amplifiers, showing promise for its practical applications in silicon photonics. The demonstrated unidirectional transmission in the device may inspire more unique functionalities based on optical PT-symmetry materials. Additionally, other intriguing photonic devices implementing optical PT-symmetry materials may be realized for integrated photonic circuits. For example, by connecting a DBR resonator to a bus waveguide through a PT symmetrical grating, very low threshold lasing can be expected, which may also be implemented as an optical memory unit of replicating any input optical waveform [27].

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