## Abstract

Indirect interband photonic transition provides a nonmagnetic and linear scheme to achieve optical isolation in integrated photonics. In this paper, we demonstrate that the nonreciprocal transition can be induced through two pathways respectively by different modulation designs. At the end of those pathways, the two final modes have *π* phaseshift. We call this phenomenon jumping phase control since this approach provides a method to control the mode phase after the conversion. This approach also yields a novel way to generate nonreciprocal phaseshift and may contribute to chip-scale optoelectronic applications.

© 2014 Optical Society of America

## 1. Introduction

Due to the Lorentz reciprocity theorem, achieving chip-scale optical nonreciprocal components such as isolators and circulators remains to be a key challenge in integrated photonics. Breaking the time-reversal symmetry of light in bulk optics is typically accomplished with magneto-optical materials [1–4]. Driven by magnetic fields, complex antisymmetric off-diagonal dielectric tensor elements are produced to enable Faraday rotation effect. In sophisticated geometries, the magneto-optical effects can also result in nonreciprocal phase shift, or exotic topologically protected one-way waveguides [5, 6]. Unfortunately, the fact that magneto-optical effect is typically weak and incompatible with standard optoelectronic materials has so far prevented full on-chip integration. One of the nonmagnetic approaches to evading optical reciprocity is to exploit nonlinear effects [7, 8]. The intensity-dependent light-material interaction induces nonreciprocal modifications for light travelling in opposite directions with different intensities. However the nonreciprocity of nonlinear schemes is power dependent. High isolation occurs only for a limited range of intensity, which limits the practical use of nonlinear materials.

Recently, a new nonmagnetic linear approach based on indirect interband photonic transition has been proposed [9–13]. Temporal and spatial modulation of the refractive index are implemented to form a traveling-wave perturbation which travels in a specific direction in silicon waveguide to break the time-reversal symmetry. Light propagating in the forward direction at mode I at frequency *ω*_{1} and propagation constant *k*_{1} will be resonantly converted to mode II at frequency *ω*_{2} and propagation constant *k*_{2} by the perturbation. While such dynamic photonic structure will have no effect on the signals propagating in the backward direction. This approach which has been demonstrated experimentally in silicon slotted waveguide [14, 15], is not only compatible with CMOS (complementary metal-oxide-semiconductor) processing, but also provides complete optical isolation. So it is important to improve this so far the most practical scheme.

In this paper, we show that the indirect interband photonic transition can be excited through two different pathways separately. At the end of those pathways, the two final modes II to which the light injected as mode I jumps have *π* phaseshift. We call this phenomenon jumping phase control in interband photonic transitions since this approach provides a way to control the mode phase after the conversion, which does not happen in electronic transitions in semiconductors. The paper is organized as follows:In Section II, after a brief theoretical review of indirect interband photonic transition, we propose an optimal design of perturbation profile to enhance coupling efficiency. And then we show that the profile of modulation creates new degrees of freedom to control the phase of final mode II after the transition. In Section III, we verify our viewpoint numerically with FDTD simulation and use this jumping phase control method in photonic transition to propose an isolator based on Mach-Zehnder interferometer [16].

## 2. Theoretical background and analysis

The indirect interband photonic transition requires a photon structure supporting two distinct optical modes. For simplicity, we discuss the transition process in a silicon slab waveguide which possesses a band structure of TE mode with even modes in first band and odd modes in second, as shown in Fig. 1(a). In order to induce photonic transition, a traveling-wave index modulation is applied to impart frequency and momentum shifts of photon states simultaneously. The modulation implemented along light propagation in the z direction can be written as

where*δ*(

*x*) is the modulation amplitude transverse to the waveguide, the Ω is the modulation frequency and the Λ is the modulation wave vector. In the modulated waveguide, the electric filed of light is given by

*E*

_{1,2}(

*x*) are normalized modal profiles of mode I and mode II, and the corresponding power flows along the waveguide of these two modes are |

*A*

_{1,2}(

*z*)|

^{2}

*W/m*, respectively. The traveling-wave modulation propagating along −z direction given by Eq. (1) makes the transition from mode I to mode II only occur for the light propagating along +z direction if the phase matching conditions are satisfied.

Assuming a slowly varying approximation and perfect phase-matching conditions (Ω = *ω*_{2} − *ω*_{1} and Λ = *k*_{1} − *k*_{2}), the coupled mode Eqs. for +z direction propagation can be derived as

*A*

_{1,2}(

*z*) can be expressed as

*K*= (

*C*

_{1}

*C*

_{2})

^{1/2}.

*ϕ*

_{1}and

*ϕ*

_{2}are the initial phase of

*A*

_{1,2}. The maximum amplitudes of

*A*

_{1,2}are chosen to be

*a*

_{1,2}respectively. After propagating over a distance of coherence length

*l*=

_{c}*π*/(2

*K*), the mode I at (

*ω*

_{1},

*k*

_{1}) will undergo a complete transition to mode II at (

*ω*

_{2},

*k*

_{2}). While the modes propagating along −z direction is not affected since the coupled mode Eq. (3) only works for +z direction propagation.

Although most parameters, such as Ω, Λ, in the theory are determined by the frequencies and wavevectors of two modes, there are still three free parameters can be exploited to control the transition process. Firstly, the transverse modulation amplitude *δ*(*x*) which couples the two orthogonal electric fields should be designed to maximize the couple strength. In previous works, the modulation is applied only to half the waveguide width, since the different symmetry of these two modes will make the overlap integral <*E*_{1}|*δ*|*E*_{2}> be zero if the transverse modulation profile is uniform across the whole waveguide cross section. As a matter of fact, the symmetry difference of two modes should be an advantage to be taken to enhance the couple efficiency and the modulation should occupy the whole cross section of waveguide despite that it can lead to practical challenges to be overcome when trying to implement it.The optimal modulation profile in piecewise form, as shown in Fig. 1(b), is given as

The *π* phase shift between modulations applied to the upper half and bottom half of the waveguide makes sure that the contributions to the overlap integral from these parts will be in phase. Therefore, using the modulation scheme as given by Eq. (5), the couple strength is twice as big as employing the modulation design which only applied to half waveguide.

Furthermore, we would like to emphasize that the other two free parameters, *ϕ*_{1} and *ϕ*_{2}, bring more interesting physical phenomena in transition process. With an initial condition of *A*_{1} = *a*_{1} and *A*_{2} = 0, two different sets of possible values can be assigned to *ϕ*_{1} and *ϕ*_{2}. The first set of possible values is *ϕ*_{1} = 0 and *ϕ*_{2} = −*π*/2, and the second is *ϕ*_{1} = 0 and *ϕ*_{2} = *π*/2, indicating that light at mode I can jump to two modes II with different final phases through two distinct pathways. In Fig. 2, it is obvious to see that the two final modes II of these two transitions have *π* phase shift which stems from the *π* phase shift of two possible *ϕ*_{2}. We call this phenomenon jumping phase control in interband photonic transitions. To demonstrate the approach which could be used to induce these two kinds of transition, it is important to note that the two first derivatives of *A*_{2}(*z*) with respect to *z* in different transitions have opposite signs as indicated in Fig. 2. Therefore, the sign of parameter *C*_{2} which determines the sign of the first derivative of *A*_{2}(*z*) with respect to *z*, as shown in Eq. (3), plays the pivotal part in jumping phase control. We here point out that the modulation profile could be used to determine the sign of *C*_{2}. In the condition of

*C*

_{2}comparing to the situation which the modulation is applied using the scheme as given by Eq. (5). Thus, the two modulation designs proposed as Eq. (5) and Eq. (6) can be used to induce the transitions to two final modes II with

*π*phase shift as shown in Fig. 3.

## 3. Numerical demonstration

To validate the theoretical analysis above, we utilize finite difference time-domain (FDTD) method to simulate the photonic transition process on a 2-D silicon slab waveguide. The width of the waveguide is chosen to be 0.22*a* (*a* is normalized length) and the dielectric constant to be 12.25. With the choice, we have an even mode I on the first band at (*ω*_{1} = 0.6468(2*πc/a*), *k*_{1} = 1.836(2*π/a*)) and an odd mode II on the upper band at (*ω*_{2} = 0.8879(2*πc/a*), *k*_{2} = 1.367(2*π/a*)). The normalized electric field profiles of these two mode are shown in Fig. 3. The dynamic modulation whose length is chosen to be the coherence length *l _{c}* = 5.02

*a*, is applied to the whole waveguide as given by Eq. (5), satisfying the perfect phase-matching conditions to couple the two modes. The modulation strength

*δ*is chosen to be 1.0. As shown in Fig. 4, Light traveling from left to right at symmetric mode I is converted to antisymmetric mode II whereas the light propagating at the opposite direction is unaffected.

After propagating through the modulated region with length of twice the coherence length *L* = 2*l _{c}* from left to right, light will return to the initial mode I with an nonreciprocal

*π*phase-shift as shown in Fig. 2. The use of this nonreciprocal phaseshift allows one to construct devices for optical isolation, but the length of these devices have to reach 2

*l*. We would like to emphasize that the jumping phase control method is the more efficient way to generate nonreciprocal phaseshift since the method reduces the action length by half. We propose an isolator using nonreciprocal phaseshift generated by the jumping phase control method in interband photonic transition. Figure 5(a) shows the design of a Mach-Zehdner interferometer in which two waveguide arms are subject to the dynamic modulation simultaneously. In order to induce the transition through different paths, the modulation profile applied on the upper/lower arm is decided by the Eq. (5)/(6), respectively. Therefore, the conversion through the upper arm is completely outphase with the transition through the lower arm, leading a great loss at the end of the interferometer when light is propagating form left to right, as shown in Fig. 5(b). In contrast, the structure has a high transmission coefficient in the opposite direction, due to the fact that light injected in from the opposite side was unaffected by the modulations[Fig. 5(c)]. With optimization of geometry, such interferometer can work as an optical isolator using nonreciprocal phaseshift generated by jumping phase control method.

_{c}## 4. Conclusions

In this paper, we demonstrate that the indirect interband photonic transition can be excited through two pathways respectively by different modulation schemes. Different path leads to distinct final phase. After transition, the two final modes II are completely outphase. We call this phenomenon jumping phase control and use this method to proposal an optical isolator based on on Mach-Zehdner interferometer. In conclusion, the jumping phase control method provides a novel way to generate nonreciprocal phaseshift in indirect interband photonic transition. We envision this approach may contribute to chip-scale optoelectronic applications when traveling-wave silicon modulation becomes practical.

## Acknowledgments

This work was supported by the National Natural Science Foundation of China (Grant No. 61177056 and 61205052) and sponsored by Shanghai Pujiang Program.

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