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Abstract
One of the most prominent classes of non-reciprocal devices relies upon the effect of non-reciprocal polarization conversion, such as those observed in Faraday isolators. This effect is usually achieved with the use of magneto-optical materials. Here, we introduce a waveguide type optical isolator based on non-reciprocal polarization conversion, without the use of magneto-optical materials. Our isolator is based on spatial-temporal dynamic refractive index modulation, which is more readily amenable for on-chip integration. We numerically demonstrate our design with both first-principle multi-frequency electromagnetic simulations and the vectorial coupled mode theory formalism.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Optical isolators play an essential role in both on-chip and fiber optical communication systems [1–3]. By suppressing back reflections, optical isolators reduce noises and enhance the stability of optical systems. Constructing optical isolators requires the breaking of reciprocity [3,4]. The standard mechanism to construct optical isolator is to use magneto-optical materials, which possess an asymmetric permittivity tensor that breaks reciprocity.
Among all the different magneto-optic effects, the effects of non-reciprocal polarization rotation, including Faraday effects, form the basis for the most widely used optical isolators and therefore are arguably the most prominent [5–12]. For a structure consisting of a uniform magneto-optical material subject to a magnetic field along the $\hat {z}$ direction, a linearly polarized light with electric field polarization vector $\vec {E}$, incident along the $+\hat {z}$ direction, will experience polarization rotation as it propagates through the material, resulting in an output polarization $R \cdot \vec {E}$, where $R$ is a rotation matrix. This rotation is non-reciprocal: if one injects light along the $-\hat {z}$ direction, through the same structure, with an incident polarization $R\cdot \vec {E}$, the output polarization does not coincide with $\vec {E}$. Aside from its fundamental interest, such a non-reciprocal polarization conversion effect is attractive in practice since the magneto-optical structure can be combined with a reciprocal polarizer, which is widely available, to construct an isolator.
In recent years, there are significant efforts in seeking to reproduce the magneto-optic effects in non-magnetic systems, by utilizing dynamically modulated systems that also break reciprocity [13–17]. These efforts are motivated by the desire to achieve optical isolation using material systems that are more compatible with on-chip or in-fiber integration [17]. In most of these works, the non-reciprocity manifests either in terms of a non-reciprocal frequency conversion, or a non-reciprocal conversion between different spatial modes with the same polarization. Given the prominence of non-reciprocal polarization conversion effects in standard magneto-optics, it should certainly be of interest to seek to achieve similar polarization effects with electro-optic dynamic modulation as well.
In this paper, we show that non-reciprocal polarization conversion can be achieved in dynamically modulated systems. Different from the recently proposed opto-mechanically induced Faraday effect [18] and optically driven Faraday effect using nonlinear media [19], both of which used Fabry-Perót resonators that support degenerate modes in free-space, our design works for on-chip waveguide which supports quasi-TE and TM modes and has a broad bandwidth. The dynamic modulated polarization optical isolator we proposed in this paper is a direct mimic of the waveguide magneto-optic isolator. Instead of using magneto-optical materials that have asymmetric permittivity tensor and need external magnetic field to break the time-reversal symmetry, we employ the time modulation to achieve the polarization rotation and non-reciprocity. The isolator based on dynamic modulation should be amenable to on-chip implementation using electro-optic materials [20–23].
This paper is organized as follows. In Section 2, we introduce the set up of a rectangular waveguide isolator design based on the concept of direct interband photonic transition. In Section 3, we derive a vectorial coupled mode theory formalism for the setup in Section 2. In Section 4, we provide a numerical demonstration using three-dimensional finite-element-method simulations and compare the results with those from the vectorial coupled mode theory. We conclude in Section 5.
2. Non-magnetic polarization isolator design
Figure 1(a) shows the configuration of the polarization isolator based on spatio-temporal dynamic modulation. The device consists of rectangular dielectric waveguides. It includes two modulated regions with different modulation phases, separated by a passive interferometric region where the width of the waveguide is different from the modulated regions. The rectangular waveguide supports the fundamental TE mode (Fig. 2(b)) and the lowest-order TM mode (Fig. 2(c)) which has a higher frequency at the same propagation constant. Figure 2(a) shows the band structures of the waveguide, where the solid lines are the band structure in the modulated regions and the dash lines are the band structure in the interferometric region.
Fig. 1. (a) Optical isolator design. The blue region represents a static waveguide. The dark shaded regions are modulated regions. Only one quadrant of the waveguide is uniformly modulated for each modulated region. (b) In the forward propagation, an optical wave in the TE mode is injected from the left side, and it remains in the same TE mode after passing through the device. (c) In the backward propagation, an optical wave in the TE mode is injected from the right side. After passing through the device, it is fully converted into the TM mode.
Fig. 2. (a) Band structures of the TE/TM mode. The dielectric waveguide has a static relative permittivity $\epsilon _r = 12.25$. The width and height of the rectangular waveguide is $d_x = 0.6~\mu$m and $d_y = 0.3~\mu$m respectively. The interferometric region has the same height but a different width $d_c = 0.7~\mu$m. The solid lines represent the waveguide band structure in the modulated regions and the dash lines represent the band structure of the interferometric region. (b) Electric field distribution of the fundamental TE mode in the $x-y$ plane. (c) Electric field distribution of the lowest-order TM mode in the $x-y$ plane.
In the modulated regions, the relative permittivity is modulated sinusoidally in time:
where $\epsilon _r(x,y,z)$ is the static relative permittivity of the chosen materials, $\delta (x,y)$ is the modulation strength, $\Omega = \omega _2 - \omega _1$ is the modulation frequency and $\phi$ is the modulation phase. $\omega _1$ and $\omega _2$ are the angular frequencies for the fundamental TE mode and the lowest-order TM mode, respectively. The modulation profile $\delta (x,y)$ is uniform along the $z$ direction for each modulated region. By choosing appropriate modulation profiles, such as applying modulation on only one quadrant of the waveguide, a direct photonic transition between TE and TM modes with the same propagation constant can be induced. Here the transition is direct because the modulation profile of Eq. (1) has no $z$-dependency.Optical isolation can be achieved based on the concept of direct photonic transition as discussed above [15]. As shown in Fig. 1(a), the two modulation sections have different modulation phases, denoted as $\phi _l$ and $\phi _r$ respectively. Non-reciprocal phase response between forward and backward propagations appears as long as $\phi _l \neq \phi _r$ [15]. In the interferometric region, since the waveguide has a different width from that of the modulated regions, the TE/TM modes have different propagation constants as compared with that of the modulated regions. Thus, the two polarizations will accumulate a phase difference $\Delta \phi$ as light propagates through the interferometric region. As illustrated in Fig. 1(b), consider the forward process where a TE mode transmits through the structure while remaining at the TE mode. This process involves two pathways. In the first pathway, the field undergoes a photonic transition in the first modulated region with the associated phase $\pi /2 + \phi _l$, where the extra phase $\pi /2$ can be seen from an explicit calculation and in fact is analogous to the $\pi /2$ phase difference between reflected and transmitted light in a beam splitter [15]. The field then propagates through the interference region, and then undergoes a photonic transition in the second modulated region with the associated phase $\pi /2 -\phi _r$. In the second pathway, the field transmits through the entire structure without undergoing a photonic transition. The phase difference between the two pathways is then $\pi + \phi _l - \phi _r + \Delta \phi$. With the choice $\phi _l-\phi _r = \pi /2$, and $\Delta \phi = \pi /2$, these two pathways interfere constructively. In contrast, in the backward direction, the phase difference between the same two pathways is $\pi + \phi _r - \phi _l + \Delta \phi$. Notice that the accumulated phase associated with the modulation phases changes sign since the direction of the photonic transition reverses. Consequently, the two pathways interfere destructively in the backward direction. As a result of such interference, for a TE input light, in the forward direction the transmitted light stays at as a TE mode, whereas in the backward direction the transmitted light converts to a TM mode, resulting in a non-reciprocal intensity response. With a passive polarization filter [24,25], the device can achieve complete optical isolation.
3. Vectorial coupled mode theory
To select the modulation profile ($\delta (x,y)$ in Eq. (1)) and quantitatively analyze the required dimensions of each region, we derive a vectorial coupled mode theory formalism for the three-dimensional waveguide under dynamic modulation described by Eq. (1).
In the modulated region, the electric and magnetic fields satisfy Maxwell’s equations:
We plot out the $E_z$ field distributions of the fundamental TE and TM modes in Figs. 3(a) and (b), respectively. The overlap of the $E_z$ fields of these two modes exhibit a quadrupolar shape, as shown in Fig. 3(c). Thus, in order to maximize the coupling strength, according to the quadrupolar shape of the field overlap, the ideal modulation profile has a positive sign in the first and third quadrants and a negative sign in the second and fourth quadrants, as shown in Fig. 3(d). In practice, as long as the modulation is asymmetric about the $x$-plane and $y$-plane of the waveguide, there will be non-zero coupling strength. For silicon modulators, modulation profile that has the required asymmetry can be achieved by controlling the profile of the p-n junction [27].
Fig. 3. (a) The $z$-component electric field distribution for TE mode. The dimensions and permittivity of the square waveguide are the same as those in Fig. 2. (b) The $z$-component electric field distribution for TM mode. (c) The mode overlap between the $z$-component electric field for the TE and TM modes. (d) An ideal modulation profile based on the mode overlap between the TE and TM modes. The first and third quadrants have the opposite sign to the second and fourth quadrants in order.
4. Numerical demonstration
In this section, we numerically demonstrate the rectangular waveguide isolator designed by a first-principle three-dimensional finite-element-method (FEM) simulation with multi-frequency components [28,29] and compare the simulation results with the vectorial coupled mode theory.
The rectangular dielectric waveguide we use in the simulation has static relative permittivity $\epsilon _r = 12.25$ with width $d_x = 0.6~\mu$m and height $d_y = 0.3~\mu$m and is surrounded by air. The width of the passive interferometric region is $d_c = 0.7~\mu$m. The band structures and the modal profiles of the waveguide have been plotted in Fig. 2. The input TE signal has frequency $\omega _1 = 2\pi \times 193.1$ THz and the corresponding TM mode has frequency $\omega _2 = 2\pi \times 213.5$ THz. The direct interband transition happens when the modulation frequency $\Omega = \omega _2 - \omega _1 = 2\pi \times 20.4$ THz. The modulation profile is chosen to be quadrupolar in order to minimize the modulation length and has an amplitude of $\delta = 0.1\epsilon _r$. The modulation phases of the two modulation sections are $\phi _l = 0$ and $\phi _r = -\pi /2$ respectively. In order to achieve equal amplitudes for both modes after passing through the first modulator, the modulation length $L_m$ is chosen to be $8.19~\mu$m according to Eqs. (5) and (6). The length of the interferometric region is $L_c = 13.95~\mu$m such that the TE and TM modes accumulate a $\pi /2$ phase difference.
To simulate the effects of dynamic modulation, we implement the multi-frequency formalism as discussed in [28], using the COMSOL multiphysics software which is based on finite-element method (FEM). This formalism allows us to simulate the effects of a harmonic modulation of the dielectric function, entirely from first-principles, with no uncontrolled approximations. In the simulation, we only need to consider three frequency components $\omega _1, \omega _1\pm \Omega$. The coupling to other modulation side bands at frequencies $\omega +n \Omega$, where $|n|>1$, are negligible since there are no corresponding guided modes at these modulation side bands.
Figure 4 shows the photon flux in the TE and TM modes as a function of propagation distance, when an optical wave in the TE mode is injected either in the forward direction from the left (Fig. 4(a)), or in the backward direction from the right (Fig. 4(b)). In the forward direction (Fig. 4(a)), the wave is partially converted to the TM mode in the first modulated region. It then propagates as a linear superposition of TE and TM modes through the interferometeric region. The photon flux in the TE and TM modes is unchanged in this region since it is unmodulated. Finally, the wave passes through the second modulated region, and is completely converted back to the TE mode. The insects in Fig. 4(a) show the electric field distribution on the input and output cross-section of the waveguides. Both distributions are in a pure TE mode, indicating the lack of modal conversion in the forward direction. In contrast, in the backward direction (Fig. 4(b)), a TE mode injected from the right is completely converted to the TM mode upon passing through the structure. The contrast between Fig. 4(a) and Fig. 4(b) demonstrates a strong non-reciprocal response from this structure.
Fig. 4. The comparisons between the vectorial coupled mode theory and FEM numerical simulation. The circles in plots (a) and (b) represent the simulation results and the solid lines are from the vectorial coupled mode theory. (a) The photon flux of TE and TM modes as a function of propagation distance for the forward propagation. (b) The photon flux of TE and TM modes as a function of propagation distance for the backward propagation.
To quantify the performance of the device, we define the insertion loss and the isolation ratio as
where the $T_{f\mathrm {TE}}$ and $T_{b\mathrm {TE}}$ are the normalized output photon flux number of TE mode for the forward and backward propagations, respectively. The insertion loss and isolation ratio calculated based on the numerical simulation photon flux results are 0.05 dB and 29.95 dB, respectively.In Fig. 4, we also compare the results from the first-principle simulations (open circles), with those from the vectorial coupled mode theory (solid lines). The vectorial compuled mode theory results agree with the first-principle simulations. We have thus provide a validation of the vectorial coupled mode theory. In the simulation, we have used a large modulation frequency and modulation strength in order to reduce the simulation time. Below, we provide a design for more realistic modulation parameters using the vectorial coupled mode theory.
The state-of-the-art integrated lithium niobate (LiNbO$_3$) electro-optic modulators can achieve a modulation frequency over $100$ GHz [22,23]. In order to bring the frequencies of TE and TM modes close to each other, we need a nearly square waveguide. The LiNbO$_3$ waveguide’s width is designed to be $d_x = 605$ nm and the height is $d_y = 600$ nm with refractive index $2.2$ and surrounded by silicon dioxide layer with refractive index $1.45$. With these parameters, the fundamental TE and TM mode has a frequency of $\omega _1 = 2\pi \times 193.410$ THz and $\omega _2 = 2\pi \times 193.512$ THz, respectively, with a frequency difference of $102$ GHz. The modulation is applied to one quarter of the waveguide with modulation strength $\delta /\epsilon _r = 10^{-3}$. The length required for each modulated section is $L_m = 11.58$ mm according to our coupled mode theory calculation. The passive interference region has width $d_c = 700$ nm and the corresponding length is $L_c = 19.04~\mu$m. In Fig. 5, we perform a vectorial coupled mode analysis for the LiNbO$_3$ modulator based design according to Eqs. (5) and (6). For the forward propagation, the TE mode is completely transmitted as TE mode, while in the backward direction, the TE mode fully converts to TM mode. Due to the nearly parallel bands of the TE and TM modes inside the waveguide, the device has a broad bandwidth. The vectorial coupled mode theory calculation indicates that the isolation ratio for the above device is above $30$ dB over a frequency bandwidth of $5$ THz near the operating frequency $\omega _1$.
Fig. 5. The vectorial coupled mode theory analysis for the LiNbO$_3$ modulator based design. (a) The photon flux of TE and TM modes as a function of propagation distance for the forward propagation. (b) The photon flux of TE and TM modes as a function of propagation distance for the backward propagation.
Next, we conduct a tolerance analysis by changing the width of the waveguide. Suppose we change the width of the waveguide by 1 nm. The TE/TM mode separation changes to 123 GHz for 606 nm width and 82 GHz for 604 nm width, as shown in Fig. 6(a). To accommodate such a change, one only needs to adjust the modulation frequency accordingly. We perform a vectorial coupled mode analysis for the width perturbed devices without changing other dimensions. As shown in Fig. 6(b), the device still has isolation ratio above 30 dB in a broad frequency range.
Fig. 6. (a) Band structures for devices with different width perturbation. (b) The isolation ratio for devices under width perturbation. The target design has the width of 605 nm, working at the input frequency around 193.410 THz. The figure shows the isolation ratio if the fabrication deviates from the target design by 1 nm.
The insertion loss of the proposed device will be affected by the static loss introduced by the optical modulator and the hybrid integration [23] but the isolation ratio will not change [30]. The footprint and energy consumption of our proposed device can benefit from the developments of high-efficiency optical modulators design [31]. In addition, one possible way to shrink the size and reduce the energy consumption of the modulator is to use the ring resonator design [13,32], which however sacrifices the operation bandwidth.
We end the discussion here with a brief comment. On-chip optical isolators based on magneto-optical effects typically operate in the Voigt geometry [6–9]. In these isolators, optical isolation is usually achieved only for one polarization. On the other hand, quasi-phase matching [10–12] has been demonstrated to overcome birefringence in the Faraday geometry to achieve isolation for both polarizations. Our construction here is analogous to the Faraday geometry in magneto-optical devices and can provide isolation for both polarizations simultaneously.
5. Conclusions
In conclusion, we have proposed an integrated photonic device that can achieve polarization isolation without using magneto-optical effect. The polarization optical isolator based on dynamic modulation can achieve complete isolation with low loss, high isolation ratio, and broad bandwidth, which provides an alternative platform for integrated polarization-rotation optical isolator design.
Funding
Air Force Office of Scientific Research (FA9550-18-1-0379).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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