 
Modeling Kerr nonlinear devicesMain Researcher: Bjorn Maes
Nonlinear materials are actively researched for designing alloptical devices. The Kerr effect is a promising nonlinearity as it means a nearinstantaneous intensity dependent refractive index change. Because of the weakness of this effect strong field confinements are necessary, which are now possible in advanced structures such as photonic crystals or photonic wires. To simulate these devices efficient and accurate new methods are needed.
We developed an extension to the linear mode expansion method. In this way we inherit many of the benefits of the original method, such as advanced boundary conditions, rigorousness, small computational domain, etc.
Our method consists of an iteration of linear calculations, using CAMFR. The nonlinear material is divided in a grid and each small rectangle is assigned its refractive index during iteration.
Example of a spatial grid.
Starting from the linear index distribution, a guess or a previously calculated solution, we perform a linear eigenmode simulation. This gives us the intensities in the rectangles with which we can update the indexes using the Kerr constitutive relation of the material. If the new index distribution is approximately equal to the old one, we have converged to a solution of the full nonlinear problem.
We have implemented this algorithm for finite and periodic infinite structures. Calculations are especially fast for structures with small nonlinear sections, as the linear parts only have to be simulated once.
A nonlinear photonic crystal switch (Soljacic et al, 2002).
As mode expansion is a frequency domain technique we immediately obtain steadystate solutions. For this purpose time domain methods such as FiniteDifference TimeDomain (FDTD) require the simulation of long pulses.
We employ modes propagating in both directions, thus reflection and interference effects can be observed. This feedback in combination with nonlinearity can give rise to bistability, which means two possible outputs for one input. Our algorithm is able to track both branches of such bistable solution curves. FDTD needs a strong input excitation pulse before the main pulse to scan the upper branch, which complicates matters.
Bistable solution of the photonic crystal switch.
Other people involved: PhD thesises Publications International Journals

M. Fiers, T. Van Vaerenbergh, K. Caluwaerts, D. Vande Ginste, B. Schrauwen, J. Dambre, P. Bienstman,
Timedomain and frequencydomain modeling of nonlinear optical components on circuitlevel using a nodebased approach, Journal of the Optical Society of America B, 29(5), p.896–900 (2012) .

A. Omari, I. Moreels, F. Masia, W. Langbein, P. Borri, D. Van Thourhout, P. Kockaert, Z. Hens,
Role of interband and photoinduced absorption in the nonlinear refraction and absorption of resonantly excited PbS quantum dots around 1550 nm, Physical Review B, 85(115318), (2012) .

Zhiyong Xu, B. Maes, Xunya Jiang, John D. Joannopoulos, Lluis Torner, Marin Soljacic,
Nonlinear photonic crystals near the supercollimation point, Optics Letters, 33(15), p.1762 (2008) .

B. Maes, P. Bienstman, R. Baets, Bobo Hu, Phillip Sewell, Trevor Benson,
Modeling comparison of secondharmonic generation, Optical and Quantum Electronics, 40(1), p.1322 (2008) .

P. Vandersteegen, B. Maes, P. Bienstman, R. Baets,
Using the complex Jacobi method to simulate Kerr nonlinear photonic components, Optical and Quantum Electronics, 38(13), p.3544 (2006) .

G. Van der Sande, B. Maes, P. Bienstman, J. Danckaert, R. Baets, I. Veretennicoff,
Nonlinear lattice model for spatially guided solitons in nonlinear photonic crystals, Optics Express, 13(5), p.15441554 (2005) .

B. Maes, P. Bienstman, R. Baets,
Bloch modes and selflocalized waveguides in nonlinear photonic crystals, J. Opt. Soc. Am. B, 22(3), p.613619 (2005) .
International Conferences

A. Omari, I. Moreels, F. Masia, W. Langbein, D. Van Thourhout, P. Kockaert, Z. Hens,
Nonlinear optical properties due to inter and intraband transitions in PbS quantum dots, 7th International Conference on Quantum Dots, United States, (2012) .

B. Kuyken, S. Clemmen, S. Selvaraja, W. Bogaerts, S. Massar, R. Baets, G. Roelkens,
Self phase modulation in highly nonlinear hydrogenated amorphous silicon, Photonics Society Annual Meeting, United States, (2010) .

B. Maes, K. Huybrechts, G. Morthier, P. Bienstman, R. Baets,
Switching with Coupled Photonic Crystal Cavities, SIAM Conference on Nonlinear Waves and Coherent Structures (invited), Italy, (2008) .

P. Bienstman, P. Vandersteegen, B. Maes, R. Baets,
Modeling methods for highindex contrast linear and nonlinear nanophotonics, NUSOD 2006 (Numerical Simulation of Optoelectronics Devices) (invited), Singapore, (2006) .

P. Vandersteegen, P. Bienstman, R. Baets,
Extensions of the Complex Jacobi Iteration to simulate Photonic Wavelength Scale Components, European Conference on Computational Fluid Dynamics (ECCOMAS CFD 2006), Netherlands, (2006) .

P. Vandersteegen, P. Bienstman, R. Baets, A. Dewandre, M. Haelterman,
Simulations of Kerr based non linear optical components with the Complex Jacobi iteration, ICTON (COSTP11 training school), p.We.P.14 (2006) .

P. Vandersteegen, B. Maes, P. Bienstman, R. Baets,
Simulating nonlinear third order effects with the adapted complex Jacobi iteration method, 2005 IEEE/LEOS Symposium Benelux Chapter Proceedings, Belgium, p.193196 (2005) .

G. Van der Sande, B. Maes, P. Bienstman, J. Danckaert, R. Baets, I. Veretennicoff,
Nonlinear lattice model for selflocalized waveguides in nonlinear photonic crystals, SPIE International Congress on Optics and Optoelectronics, Poland, p.594922 (2005).

P. Bienstman, B. Maes, P. Vandersteegen, R. Baets,
Modelling of nonlinear nanophotonic devices, OWTNM (invited), Australia, p.32 (2005) .

B. Maes, G. Van der Sande, P. Bienstman, J. Danckaert, R. Baets, I. Veretennicoff,
Selflocalized Waveguides in Nonlinear Photonic Crystals, IPRA, United States, p.ITuB2 (2005) .

P. Vandersteegen, P. Bienstman, R. Baets,
Extending the Complex Jacobi Iteration method to simulate Kerr nonlinear effects, OWTNM 2005, France, (2005) .

B. Maes, P. Bienstman, R. Baets,
Bloch modes and selflocalized waveguides in nonlinear photonic crystals, IEEE/Leos Benelux Annual Symposium 2003, Netherlands, p.233236 (2003).

B. Maes, P. Bienstman, R. Baets,
Rigorous modelling of nonlinear photonic components with mode expansion and spatial index discretisation, IPR, United States, p.105107 (2003).

B. Maes, P. Bienstman, R. Baets,
Rigorous modelling of nonlinear wavelengthscale structures with mode expansion and spatial index discretisation, OWTNM 2003 (Optical Waveguide Theory and Numerical Modelling), 11, Czech Republic, p.98 (2003).

B. Maes,
Modelling of nonlinear effects in a mode expansion context, Workshop 2D photonic crystals, Switzerland, p.I05 (2002).
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