## Inverse design of a three-dimensional nanophotonic resonator |

Optics Express, Vol. 19, Issue 11, pp. 10563-10570 (2011)

http://dx.doi.org/10.1364/OE.19.010563

Acrobat PDF (1126 KB)

### Abstract

The inverse design of a three-dimensional nanophotonic resonator is presented. The design methodology is computationally fast (10 minutes on a standard desktop workstation) and utilizes a 2.5-dimensional approximation of the full three-dimensional structure. As an example, we employ the proposed method to design a resonator which exhibits a mode volume of 0.32(*λ/n*)^{3} and a quality factor of 7063.

© 2011 OSA

## 1. Motivation

1. D. A. B. Miller, “Rationale and challenges for optical interconnects to electronic chips,” Proc. IEEE **88**, 728–749 (2000). [CrossRef]

## 2. 2.5-dimensional approximation

2. J. Lu and J. Vuckovic, “Inverse design of nanophotonic structures using complementary convex optimization,” Opt. Express **18**, 3793–3804 (2010). [CrossRef] [PubMed]

^{7}× 10

^{7}) and ill-conditioned matrix, namely that given by the time-harmonic Maxwell’s equation in three dimensions, which for the electric (

*E*) field is

*E*-field inside such a holey fiber is expressed as where

*β*is the wave-vector along the fiber axis. Solving for Eq. (1) now requires solving for a much smaller (∼ 10

^{5}× 10

^{5}) matrix, which can be accomplished using standard linear algebra packages. This simulation technique is thus very similar to a two-dimensional finite-difference frequency-domain solver.

*β*, which we obtain by fitting a sinusoid to the out-of-plane decay in the three-dimensional FDTD solution, our approximation captures most of the characteristics of the three-dimensional field at the center plane of the slab.

*t*, since the in-plane values of

*ε*remain the same. To estimate

*t*for the full three-dimensional device, we treat the resonant mode as a guided mode in a slab waveguide surrounded by a cladding of

*n*= 1. The expression for

*t*is then the thickness of the slab for the lowest-order TE mode of a slab waveguide with effective refractive index

*n*

_{eff}[3]: Here,

*α*is the wave-vector which determines the evanescent decay of the slab waveguide mode into air, while

*k*is an approximated in-plane k-vector. To determine

_{x}*n*

_{eff}, the effective refractive index, we use the following approximation, In the above formulation and for the remainder of the manuscript, the norm (|| · ||) refers to the 2-norm over the entire grid and is equivalent to the spatial integral (∫ | · |

^{2}

*dr*)

^{1/2}.

## 3. Inverse design method

### 3.1. Original formulation

*ε*and

*E*are the permittivity and electric vector field respectively, defined within the 2.5-dimensional approximation and discretized along a standard Yee grid [4

4. K. Yee, “Numerical solution of initial boundary value problems involving maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. Mag. **14**, 302–307 (1966). [CrossRef]

*ε*was constrained to be isotropic by only varying

*ε*in each cell, and then determining

_{z}*ε*and

_{x}*ε*via spatial interpolation. FT is the two-dimensional Fourier transform.

_{y}*ε*and

*E*must satisfy, that is, the wave equation for the 2.5-dimensional fiber mode and the transversality condition. Lastly, Eq. (12) is a binary constraint on epsilon, denoting that we only want the structure to be composed of air or silicon.

*ε*and

*E*. To make matters worse, the binary constraint in Eq. (12) generally results in a problem which is NP-hard [5]. For these reasons, our strategy is to employ an alternating directions scheme [6

6. S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein are preparing a manuscript to be called, “Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers,” http://www.stanford.edu/~boyd/papers/distr_opt_stat_learning_admm.html.

### 3.2. Alternating directions: field optimization sub-problem

*E*while holding

*ε*constant, which is a quadratic problem with linear equality constraints, and is easily solved using a standard factor-solve method for sparse matrices [7].

*E*–

*μω*

^{2}

*εE*||) as the

*physics residual*.

*εE*

^{2}||) as the

*design objective*. Note that although the denominator in the original formulation has been removed in order to make the term convex, the present formulation is still equivalent because we fix the

*E*-field in the center of the structure (Eq. (14)), where we want the maximum to occur. Also note that the constraint in Eq. (14) is crucial in order to avoid the trivial solution

*E*= 0.

*η*coefficient in Eq. (13) allows us to trade-off between the physics residual and the design objective. Thus, in order to achieve a small mode volume, the initial value of

*η*is large and is subsequently exponentially reduced once per iteration at all points in the grid in order to bring the physics residual to 0. Furthermore,

*η*can also be given a non-constant spatial weighting in order to reduce in-plane losses; this strategy was implemented in the results presented here.

### 3.3. Alternating directions: structure optimization sub-problem

*ε*while holding

*E*constant, which is a quadratic problem with linear inequality constraints and is solved using CVX [8

8. M. Grant and S. Boyd, CVX: Matlab software for disciplined convex programming, version 1.21. http://cvxr.com/cvx, January 2011.

*ε*is relaxed to allow for any value in the range from

*ε*

_{air}to

*ε*

_{silicon}. The sub-problem is now relatively straightforward to solve, although it is very unlikely that one obtains a completely binary structure. In practice various digitization schemes can be implemented [9

9. D. Englund, I. Fushman, and J. Vuckovic, “General recipe for designing photonic crystal cavities,” Opt. Express **13**, 5961–5975 (2005). [CrossRef] [PubMed]

## 4. Result

*ω*= 0.16

*c/*Δ

*x*, and the out-of-plane wave vector,

*β*= 0.24(Δ

*x*)

^{−1}, are chosen by the user. Here, Δ

*x*is the grid spacing and

*c*is the speed of light in vacuum. Lastly, an initial dielectric structure consisting entirely of silion,

*ε*=

*ε*

_{silicon}everywhere on a 160 × 160 grid, is used as a starting point—although other initial structures (e. g. air everywhere or random

*ε*) yield nearly identical structures in this case.

*E*is updated to be the solution of Eq. (13)–(16) and is then plugged into Eq. (17)–(18), the solution of which becomes the updated value of

*ε*. The subsequent iteration then uses the updated

*ε*variable to re-optimize

*E*once again. And the algorithm continues to proceed in this way until the physics residual (||∇ × ∇ ×

*E – μω*

^{2}

*εE*||) is reduced to an acceptable value.

*ε*and

*E*(

*E*shown only), respectively, after 75 iterations of our inverse design method. The entire inverse design process takes only 10 minutes on a standard desktop computer for the 160 × 160 grid. An animation of the values of

_{y}*ε*and

*E*at each iteration of the algorithm is included in the supplementary material.

_{y}*η*(Eq. (13)), since as

*η*decreases, increasing emphasis is placed on minimizing the physics residual in the field sub-problem.

## 5. Verification

### 5.1. Accuracy

*x*. However, a small sampling of thicknesses in that vicinity resulted in a more accurate thickness of 8.5Δ

*x*, which matched the resonant frequency of the target field and 2.5-dimensional fiber mode.

*E*component of the target field, 2.5-dimensional fiber mode and the full three-dimensional FDTD field side-by-side. Figure 5 plots the horizontal cross sections of the magnitudes of the fields on a logarithmic scale. We see that the target field and fiber mode match very well, while there is significantly more discrepancy between the target field and the full three-dimensional field. This is expected with the use of our approximation; however, the error is still relatively small (below 1% of the maximum field strength) and confined mostly to the edges of the structure.

_{y}### 5.2. Performance

*λ/n*)

^{3}and the total quality factor was 7063. The radiative (out-of-plane) quality factor was 8808 and the in-plane quality factor was 35648.

*E*field is shown because it is the dominant field component and contributes most heavily to the out-of-plane leakage [10

_{y}10. J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, “Optimization of Q-factor in photonic crystal microcavities,” IEEE J. Quantum Electron. **38**, 850–856 (2002). [CrossRef]

*E*component dominates because the symmetry of the structure dictates that it will contain a non-zero DC component (central component in the light cone), unlike the

_{y}*E*and

_{x}*E*components.

_{z}## 6. Conclusion

*N*-port couplers, multiplexers, and grating couplers.

## Acknowledgments

## References and links

1. | D. A. B. Miller, “Rationale and challenges for optical interconnects to electronic chips,” Proc. IEEE |

2. | J. Lu and J. Vuckovic, “Inverse design of nanophotonic structures using complementary convex optimization,” Opt. Express |

3. | U. Inan and A. Inan, |

4. | K. Yee, “Numerical solution of initial boundary value problems involving maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. Mag. |

5. | S. Boyd and L. Vandenberghe, |

6. | S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein are preparing a manuscript to be called, “Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers,” http://www.stanford.edu/~boyd/papers/distr_opt_stat_learning_admm.html. |

7. | Y. Chen, T. A. Davis, W. W. Hager, and S. Rajamanickam, “Algorithm 887: CHOLMOD, supernodal sparse Cholesky factorization and update/downdate,” ACM Trans. Math. Software |

8. | M. Grant and S. Boyd, CVX: Matlab software for disciplined convex programming, version 1.21. http://cvxr.com/cvx, January 2011. |

9. | D. Englund, I. Fushman, and J. Vuckovic, “General recipe for designing photonic crystal cavities,” Opt. Express |

10. | J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, “Optimization of Q-factor in photonic crystal microcavities,” IEEE J. Quantum Electron. |

**OCIS Codes**

(230.5750) Optical devices : Resonators

(230.5298) Optical devices : Photonic crystals

**ToC Category:**

Optical Devices

**History**

Original Manuscript: March 22, 2011

Revised Manuscript: May 6, 2011

Manuscript Accepted: May 7, 2011

Published: May 13, 2011

**Citation**

Jesse Lu, Stephen Boyd, and Jelena Vučković, "Inverse design of a three-dimensional nanophotonic resonator," Opt. Express **19**, 10563-10570 (2011)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-11-10563

Sort: Year | Journal | Reset

### References

- D. A. B. Miller, “Rationale and challenges for optical interconnects to electronic chips,” Proc. IEEE 88, 728–749 (2000). [CrossRef]
- J. Lu and J. Vuckovic, “Inverse design of nanophotonic structures using complementary convex optimization,” Opt. Express 18, 3793–3804 (2010). [CrossRef] [PubMed]
- U. Inan and A. Inan, Electromagnetic Waves (Prentice Hall, 2000), p. 296.
- K. Yee, “Numerical solution of initial boundary value problems involving maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. Mag. 14, 302–307 (1966). [CrossRef]
- S. Boyd and L. Vandenberghe, Convex Optimization (Cambridge University Press, 2004).
- S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein are preparing a manuscript to be called, “Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers,” http://www.stanford.edu/~boyd/papers/distr_opt_stat_learning_admm.html .
- Y. Chen, T. A. Davis, W. W. Hager, and S. Rajamanickam, “Algorithm 887: CHOLMOD, supernodal sparse Cholesky factorization and update/downdate,” ACM Trans. Math. Software 35(3), 2 (2009).
- M. Grant and S. Boyd, CVX: Matlab software for disciplined convex programming , version 1.21. http://cvxr.com/cvx , January 2011.
- D. Englund, I. Fushman, and J. Vuckovic, “General recipe for designing photonic crystal cavities,” Opt. Express 13, 5961–5975 (2005). [CrossRef] [PubMed]
- J. Vuckovic, M. Loncar, H. Mabuchi, and A. Scherer, “Optimization of Q-factor in photonic crystal microcavities,” IEEE J. Quantum Electron. 38, 850–856 (2002). [CrossRef]

## Cited By |
Alert me when this paper is cited |

OSA is able to provide readers links to articles that cite this paper by participating in CrossRef's Cited-By Linking service. CrossRef includes content from more than 3000 publishers and societies. In addition to listing OSA journal articles that cite this paper, citing articles from other participating publishers will also be listed.

« Previous Article | Next Article »

OSA is a member of CrossRef.