Photonic analog computing, in which differential and integral operations are performed using light, enables analog signals to be computed optically without the need for analog-to-digital data conversion (ADC), digital-to-analog conversion (DAC), or optical-electrical-optical (OEO) conversion. The differentiator (DIFF) and the differential equation (DE) solver constitute the two fundamental systems in analog computing. Photonic differentiators have wide application in numerous areas, including pulse characterization, ultra-fast signal generation, and ultra-high-speed coding. Optical DE (ODE) systems are useful in describing behavior across many fields of science and engineering, including temperature diffusion processes, optomechanical systems, and the response of different resistor-capacitor photonic circuits.

Many approaches have been implemented in the design of DIFFs and ODE solver systems. Photonic DIFFs have been realized using semiconductor optical amplifiers (SOAs)^{1, 2} and highly nonlinear fibers,^{3} and ODE solvers have been implemented using an optical feedback loop.^{4} These systems are, however, bulky, complex, and inefficient. Computing based on integrated silicon photonics may enable the realization of compact ultra-high-speed and ultra-wide-band signal processors with low power consumption.

Photonic analog computing can be regarded as a linear system. The spectra of silicon integrated waveguides can therefore be tailored to meet the requirements of analog computing. We have developed several DIFFs and ODE solvers on this basis, using an integrated silicon Mach-Zehnder Interferometer (MZI) and a microring resonator (MRR), respectively: see Figure 1(a) and (b). To develop our *N*th-order photonic DIFF, we employed a linear system with a frequency response that varies with a frequency powered by *N*. Because the MZI structure has a cosine frequency response, showing an approximately linear frequency near the null point,^{5} it is capable of acting as the first-order DIFF. To achieve *N*th-order DIFF, the MZI unit can be cascaded *N* times with the resonant frequencies aligned.

**Figure 1. **(a) Our photonic differentiator (DIFF) system, based on a silicon Mach-Zehnder interferometer (MZI). (b) Our optical differential equation (ODE) solver system, based on a silicon microring resonator (MRR). Both systems mathematically modify the input signals. k: Constant with a unit of 1/ps.

With this in mind, we have designed and fabricated cascaded MZIs on a commercial silicon-on-insulator (SOI) wafer: see Figure 2(a). MZI units that operate on varying orders were used to implement different-order DIFFs. We then injected a Gaussian pulse train—see Figure 2(b)—into the MZI unit. The output temporal waveforms of the first-order, second-order, and third-order DIFFs show that the measured differentiated pulses fit well with the simulated pulses: see Figure 2(c–e).

**Figure 2. **(a) Microscope image of our on-chip MZIs. Insets: Photos of MZI structures operating on different orders (first order: MZI-1, second order: MZI-2, and third order: MZI-3). (b) Input pulse. Temporal waveforms for (c) first-, (d) second-, and (e) third-order DIFFs. a.u.: Arbitrary units.

When a fractional order number (*N*) is used, the DIFF operates as a fractional-order DIFF. In mathematics, fractional-order DIFFs can be used for calculations in fractal geometry, chaos circuits, system encryption, signal singularity detection, and so on. They can be considered a generalization of integer-order DIFFs, with the potential to accomplish what integer-order DIFFs cannot. Asymmetrical power between the two arms of the MZI unit leads to a frequency response phase shift of less than π. However, the loss of one arm can be tuned by carrier modulation using an external power supply, resulting in a variable phase jump of the MZI response. This enables the MZI unit to be used in the implementation of fractional-order DIFF.

We fabricated the MZI unit with a PN junction at a commercial semiconductor foundry: see Figure 3(a–d).^{6} A Gaussian-like pulse train was employed as input—see Figure 3(e)—and the output differentiated waveforms were measured under varying applied voltages: see Figure 3(f–j). The simulated waveforms of the fractional-order DIFFs show perfect accordance with the experimental results, with fractional orders of *N* = 0.83, 0.85, 0.98, 1.00, and 1.03, respectively. Additionally, we fabricated a silicon MRR, allowing us to demonstrate high-order and fractional-order DIFF in a different device.^{7} Due to its generally much smaller bandwidth, the MRR is better suited for processing narrow-bandwidth analog signals.

**Figure 3. **(a) Micrographs of the MZI, (b) coupling grating, (c) multimode interference (MMI), and (d) an integrated p-i-n diode. (e) An input Gaussian-like pulse and (f–j) output fractional-order DIFF waveforms with order numbers N = 0.83, N =0.85, N = 0.98, N =1.00, and N = 1.03, respectively. Sim: Simulated. Exp: Experimental.

We also fabricated an ODE solver. We found the frequency response at the drop port of an add-drop MRR to inherently match with that of the first-order ODE solver. Furthermore, changing the *Q*-factor of the MRR enables the ODE constant-coefficient to be tuned.^{8} Figure 4(a) and (b) shows microscope images of the MRR. We were able to control the ring waveguide using electrodes, enabling the *Q*-factor of the ring to be changed via applied voltage. A super-Gaussian pulse was chosen as the input signal: see Figure 4(c). Using the measured output from 0 and 1.3V, the constant-coefficient was determined to be 0.038/ps and 0.082/ps, respectively: see Figure 4(c) and (d). These calculated waveforms accord well with the measured ones. Our ODE solver is therefore very accurate within a certain bandwidth, restrained by the free spectral range of the MRR.

In summary, we have designed silicon integrated devices (MZI and MRR) to implement photonic DIFF and ODE solvers. These analog photonic integrated circuits show great promise for application in future high-speed, low-cost, and compact optical computing systems. In future work, we plan to incorporate these basic computing units into more complex computing modules.

**Figure 4. **(a) Microscope image of the fabricated MRR and (b) closer image of the ring region. (c) The super-Gaussian input waveform. (d, e) Measured output waveforms with different voltages applied. Yellow solid line: Measured waveforms. Black dash line: Fitted waveforms. Red attunement line: Calculated ideal outputs.

Jianji Dong, Ting Yang, Aoling Zheng, Xinliang Zhang

Huazhong University of Science and Technology

Wuhan, China

Jianji Dong received his PhD in 2008 and is currently a professor in Wuhan National Lab for Optoelectronics. His research interests include photonic digital/analog computing, optical vortex manipulation, and microwave photonics. He has published more than 90 peer-reviewed journal papers.

References:

1. J. Xu, X. Zhang, J. Dong, D. Liu, D. Huang, High-speed all-optical differentiator based on a semiconductor optical amplifier and an optical filter, *Opt. Lett.* 32(13), p. 1872-1874, 2007.

2. J. Xu, X. Zhang, J. Dong, D. Liu, D. Huang, All-optical differentiator based on cross-gain modulation in semiconductor optical amplifier, *Opt. Lett.* 32(20), p. 3029-3031, 2007.

3. P. Velanas, A. Bogris, A. Argyris, D. Syvridis, High-speed all-optical first- and second-order differentiators based on cross-phase modulation in fibers, *J. Lightw. Technol.* 26(18), p. 3269-3276, 2008.

4. S. Tan, Z. Wu, L. Lei, S. Hu, J. Dong, X. Zhang, All-optical computation system for solving differential equations based on optical intensity differentiator, *Opt. Express* 21(6), p. 7008-7013, 2013.

5. J. Dong, A. Zheng, D. Gao, L. Lei, D. Huang, X. Zhang, Compact, flexible, and versatile photonic differentiator using silicon Mach-Zehnder interferometers, *Opt. Express* 21(6), p. 7014-7024, 2013.

6. A. Zheng, T. Yang, X. Xiao, Q. Yang, X. Zhang, J. Dong, Tunable fractional-order differentiator using an electrically tuned silicon-on-isolator Mach-Zehnder interferometer, *Opt. Express* 22(15), p. 18232-18237, 2014.

7. J. Dong, A. Zheng, D. Gao, S. Liao, L. Lei, D. Huang, X. Zhang, High-order photonic differentiator employing on-chip cascaded microring resonators, *Opt. Lett.* 38(5), p. 628-630, 2013.

8. T. Yang, J. Dong, L. Lu, L. Zhou, A. Zheng, X. Zhang, J. Chen, All-optical differential equation solver with constant-coefficient tunable based on a single microring resonator, *Sci. Rep.* 4, p. 5581, 2014.