## Abstract

A new model is developed for the effects of the Kerr optical nonlinearity in a microring (racetrack) resonator coupled to input and output waveguides, which takes into account the nonlinearity in the couplers as well as the microring sections. It is shown how the nonlinear microring add-drop filter can be used as a self-adjusting all-optical beam splitter to extend the dynamic range of photodetectors and protect optoelectronic circuitry from high input powers.

© 2008 Optical Society of America

## 1. Introduction: the nonlinear microring add-drop filter

The microring resonator sandwiched between two waveguides—the add-drop filter—has become one of the most important components in integrated optics. It continues to be widely used in a wide variety of semiconductor, polymeric and glass-based material platforms for signal routing, filtering, modulation, lasing, etc. Here, we develop a new model of the effects of the optical Kerr nonlinearity (intensity dependent refractive index) in this device.

Although the Kerr effect in the microring add-drop filter has been studied [1–5], this paper is the first to consider the fully nonlinear case: not only is the microring in the nonlinear regime, but so are the couplers between the microring and the waveguides. The generalization is of practical merit since microring filters in integrated optics are usually not fabricated as circular rings, but instead as racetracks [6], and often with such small diameters that the couplers comprise a significant part of the overall length.

Based on our new model, we propose a new application of the microring add-drop filter: we show that it can be used as an all-optical self-limiting add-drop filter to protect optical receiver circuitry from intense light by limiting the amount of light sent to port “D” in Fig. 1(a). It can also be used to achieve linear detection over a wide dynamic range by a self-adjusted splitting of input light among one or several photodiodes as shown in Fig. 1(b), depending on whether the input intensity is low or high.

The apparent simplicity of this problemis deceptive: The fully-nonlinear microring add-drop filter cannot be solved using matrices alone and requires an iterative algorithm using, at each step, a numerical solution of two coupled nonlinear differential equations. For simplicity, the constituent waveguides are assumed to support only a single transverse mode, and the light is assumed to be linearly polarized.

## 2. Calculation of the transfer function

The transmission can be calculated separately within each of the four sections labeled in Fig. 1, of which two sections represent couplers, *C*
_{1} and *C*
_{2}, and the other two sections are half-rings, labeled *R*
_{1} and *R*
_{2}.

#### 2.1. Coupler sections:

For the coupler sections, the perturbative effects of the Kerr nonlinearity can be accounted for using Jensen’s method [7]. The evolution of the field along the propagation direction is described by a coupled system of nonlinear Schrödinger equations,

where α and β2 are the linear and two-photon absorption coefficients, *M*
_{a,b} and *κ*
_{ab,ba} are the usual self-coupling and cross-coupling coefficients of a linear directional coupler, e.g., see [8, Eq. (13.8.4–5)], and *Q*
_{3} and *Q*
_{4} are the corresponding nonlinear corrections [7, Eqs. (11)–(12)].

An analytic solution to Eqs. (1a)–(1b) is not known. If absorption can be ignored, an analytic solution can be written down in terms of elliptic functions, as found by Jensen [7]. However, in the regime we are interested in, the role of absorption—including two-photon absorption—cannot be ignored. Here, the solution is found numerically by using the split-step Fourierdomain beam-propagation algorithm, which evaluates the nonlinear and dispersive operators separately (the latter in Fourier space) over each infinitesimal step in the propagation direction.

#### 2.2. Half-ring sections:

If absorption can be ignored, it is simple to calculate the propagation along a waveguide. The transfer function *b*
_{2}/*b*
_{1} is obtained as ${e}^{i\left(k{L}_{\mathrm{hr}}+{\varphi}_{\mathrm{hr}}\right)}$ where *ϕ*
_{hr}=*k*
_{0}
*n*
_{2}|*b*
_{1}|^{2} is the nonlinear phase shift accumulated over the length of propagation *L*
_{hr}, which is the length of the half-ring portions of the racetrack resonator (from *b*
_{1} to *b*
_{2} in Fig. 2).

A closed form expression can also be obtained in the presence of two-photon absorption. The propagation in each half-ring (*R*
_{1} and *R*
_{2}) is obtained from the following equation [9]:

$${\varphi}_{\mathrm{hr}}\equiv \frac{1}{2K}\mathrm{log}\left[1+2{k}_{0}{n}_{2}K{\mid {b}_{1}\mid}^{2}{L}_{\mathrm{hr}-\mathrm{eff}}\right],\phantom{\rule{.2em}{0ex}}{L}_{\mathrm{hr}-\mathrm{eff}}\equiv \frac{\left[1-\mathrm{exp}\left(-\alpha {L}_{\mathrm{hr}}\right)\right]}{\alpha},$$

$$\mathrm{where}\phantom{\rule{.2em}{0ex}}K\equiv \frac{{\beta}_{2}}{2{k}_{0}{n}_{2}},\phantom{\rule{.2em}{0ex}}k\equiv \frac{2\pi n}{\lambda},\phantom{\rule{.2em}{0ex}}{k}_{0}\equiv \frac{2\pi}{\lambda}.$$

#### 2.3. Iterative algorithm:

Note that the solution to Eqs. (1a)–(1b), i.e., the output of a nonlinear directional coupler requires a knowledge of the input field amplitudes in *both* waveguides since an increase or decrease in the intensity in either waveguide alters the refractive indices and coupling integrals. Mathematically, the division of the input, *a*
_{0}, between the two outputs *a*
_{1} and *b*
_{1} depends on *b*
_{0}, and vice versa. Therefore, we need to know both *a*
_{0} (which is the external input to the filter) and *b*
_{0} (which is the circulating power in the microring) in order to determine *a*
_{1} and *b*
_{1}. However, *b*
_{1} is also related to *b*
_{0} by the the sequence of operations shown in Fig. 2.

To analyze the nonlinear microring (racetrack) add-drop filter, we need to augment Jensen’s theory with an iterative calculation procedure: For a given value of *a*
_{0}, we assume a value of *b*
_{0} (a first guess is 1/*|κ|* times larger than *a*
_{0}) and calculate *a*
_{1} and *b*
_{1} using the split-step method to solve Eqs. (1a)–(1b). The transfer function of the second coupler *C*
_{2} is calculated like *C*
_{1}, except with the assumption that there is no input at the add port indicated in Fig. 1. Finally, the transfer function of the second half-ring is calculated, similar to Eq. (2). The final result, *b*
^{′}
_{0}, is different from the value of *b*
_{0} assumed initially. Therefore, an iterative algorithm is used to converge to the correct guess for *b*
_{0}, and thus obtain the correct nonlinear transfer functions. Under some circumstances, the system can become chaotic even if the coupler is assumed to be linear [10–12].

## 3. Calculation results and discussion

In Figs. 3–5, it is assumed that the length of each directional coupler *L*
_{cplr} = 50 *µ*m, the half-circumference of the microring *L*
_{hr} = 22.5 *µ*m, the linear and nonlinear absorption coefficients are *α* = 0.1/cm and *β* = 10^{-9} cm/W, the linear refractive index *n*
_{0} = 3.47, the nonlinear refractive index coefficient *n*
_{2} = 1.4 × 10^{-13} cm^{2}/W at a wavelength *λ* = 1550 nm, and the waveguide cross-section = 0.3 *µ*m × 0.3 *µ*m. The center-to-center waveguide separation is 480 nm, so that in the linear regime, the coupling coefficient |*κ*
*L*
_{cplr}| = 0.2.

Fig. 3 shows the drop and the through port transfer functions for a wavelength that is on resonance with the microring at low intensities. At low power levels, i.e., the linear regime, the drop fraction is high, i.e. most of the input power transfers to the drop port [13]. The 3-dB bandwidth around the resonance wavelength is 7 GHz. As the input intensity *a*
_{1} is increased, the intensity in the microring *b*
_{0} increases even more, which leads to dephasing of the directional coupler, and a smaller coupling coefficient between the input waveguide and the microring. (Such behavior was demonstrated first in a directional coupler [14,15], and then in a microring coupled to a single waveguide [16].) Correspondingly, the fraction of the input that is extracted by the second waveguide at the drop port also decreases.

This self-limiting behavior of the drop ratio can protect photonic circuitry located at the drop port from intense light in the input waveguide [e.g., > 1 MW/cm^{2} in Fig. 3]. Qualitatively, whereas for low input intensities, the microring sends most of the light to the drop port, for high intensities, most of the light is transmitted along the waveguide itself, and only a small fraction is sent to the drop port. As shown in Fig. 1(b), the light that remains in the input waveguide can be detected by subsequent receiver sections also configured in the same way, i.e., “capped” by a microring front-end.

Fig. 4 quantifies the gain in receiver dynamic range. At low intensities, the first detector receives all the input power (blue line), with a small reduction from the input (black line) because of linear absorption. As the intensity increases beyond 0.1 MW/cm^{2} (corresponding to 2.5 mW power in the waveguide), the gap between the *D*
_{1} drop (blue squares) and the black line (input) starts to widen. The power reaching *D*
_{1} is now capped and most of the input power reaches the through port. Two-photon absorption is not a limiting factor for this range of input powers, but comes into play at higher powers, as shown in Fig. 3.

When the intensity is as high as 100 MW/cm^{2} (90 mW power), the intensity at port *D*
_{1} is less than 1 MW/cm^{2} (1 mW power). An additional 89 mW of power reaches the through port for detection by *D*
_{2}, *D*
_{3}, etc. which is indicated by the shaded area in Fig. 4. This represents a gain in dynamic range of 19.5 dB, increasing to nearly 40 dB at the right edge of Fig. 4. Note that the gain in dynamic range arises purely because of a self-adjusted splitting of the input power in the optical domain. There may be additional performance gains in the opto-electronic conversion at the detectors themselves, since no individual detector would “see” more than a milliwatt of optical power.

Finally, Fig. 5 compares the calculated response at the drop port predicted by the fully-nonlinear model with the simpler conventional model, in which it is assumed that the ring is nonlinear, but the coupler is linear. Ignoring the nonlinear behavior of the coupler overestimates the switching intensity threshold by 30 dB.

## 4. Conclusion

This paper discusses how to calculate the transfer function of the nonlinear microring (racetrack) add-drop filter taking into account the Kerr nonlinearity in both the microring and the coupler regions, using an iterative algorithm. A new application of the microring add-drop filter has been proposed for protecting receiver circuitry from intense light, and maintaining the linearity of photoreceivers by automatically limiting the drop fraction at any stage.

## Acknowledgment

The authors are grateful to the National Science Foundation for support (ECCS-0642603).

## References and links

**1. **J. E. Heebner and R. W. Boyd, “Enhanced all-optical switching by use of a nonlinear fiber ring resonator,” Opt. Lett. **24**, 847–849 (1999). [CrossRef]

**2. **V. Van, T. Ibrahim, K. Ritter, P. Absil, F. Johnson, R. Grover, J. Goldhar, and P.-T. Ho, “All-optical nonlinear switching in GaAs-AlGaAs microring resonators,” IEEE Photon. Tech. Lett. **14**, 74–76 (2002). [CrossRef]

**3. **V. Van, T. Ibrahim, P. Absil, F. Johnson, and P.-T. Ho, “Optical signal processing using nonlinear semiconductor microring resonators,” IEEE J. Sel. Top. Quantum Electron. **8**, 705–713 (2002). [CrossRef]

**4. **S. Pereira, P. Chak, and J. E. Sipe, “All-optical AND gate by use of a Kerr nonlinear microresonator structure,” Opt. Lett. **28**, 444–446 (2003). [CrossRef] [PubMed]

**5. **J. E. Heebner and R. W. Boyd, “Strong Dispersive and Nonlinear Optical Properties of Microresonator-Modified Optical Waveguides,” in *Laser Resonators and Beam Control VI*, A. V. Kudryashov and A. H. Paxton, eds., Proc. SPIE Vol.4969. pp. 185–194 (2003). [CrossRef]

**6. **V. Van, P. P. Absil, J. V. Hryniewicz, and P. T. Ho, “Propagation loss in single mode GaAs-AlGaAs microring resonators: measurement and model,” J. Lightwave Technol. **19**, 1734–1739 (2001). [CrossRef]

**7. **S. M. Jensen, “The nonlinear directional coupler,” IEEE Trans. Microwave Theory Tech. **MTT-30**, 1568–1571 (1982). [CrossRef]

**8. **A. Yariv, *Optical Electronics in Modern Communications*, 5th ed. (Oxford, New York, 1997).

**9. **S. Blair, J. E. Heebner, and R. W. Boyd, “Beyond the absorption-limited nonlinear phase shift with microring resonators,” Opt. Lett. **27**, 357–359 (2002). [CrossRef]

**10. **H. Nakatsuka, S. Asaka, H. Itoh, K. Ikeda, and M. Matsuoka, “Observation of Bifurcation to Chaos in an All-Optical Bistable System,” Phys. Rev. Lett. **50**, 109–112 (1983). [CrossRef]

**11. **K. Ikeda and M. Mizuno, “Frustrated Instabilities in Nonlinear Optical Resonators,” Phys. Rev. Lett. **53**, 1340–1343 (1984). [CrossRef]

**12. **B. Crosignani, B. Daino, P. D. Porto, and S. Wabnitz, “Optical multistability in a fiber-optic passive-loop resonator,” Opt. Commun. **59**, 309–312 (1986). [CrossRef]

**13. **A. Yariv, “Universal relations for coupling of optical power between microresonators and dielectric waveguides,” Electron. Lett. **36**, 321–322 (2000). [CrossRef]

**14. **P. Li Kam Wa, J. E. Stitch, J. J. Mason, J. S. Roberts, and P. N. Robson, “All optical multiple quantum well waveguide switch,” Electron. Lett. **21**, 26–28 (1985). [CrossRef]

**15. **A. Villeneuve, C. C. Yang, P. G. J., G. I. Wigley, J. S. Stegeman, C. N. Aitchison, and Ironside, “Ultrafast all-optical switching in semiconductor nonlinear directional couplers at half the band gap,” Appl. Phys. Lett. **61**, 147–149 (1992). [CrossRef]

**16. **A. D. Bristow, R. Iyer, J. S. Aitchison, H. M. van Driel, and A. L. Smirl, “Switchable Al[sub x]Ga[sub 1 - x]As all-optical delay line at 1.55 mu m,” Appl. Phys. Lett. **90**, 101112 (2007). [CrossRef]