## Sensitivity analysis and optimization of sub-wavelength optical gratings using adjoints |

Optics Express, Vol. 22, Issue 11, pp. 12971-12981 (2014)

http://dx.doi.org/10.1364/OE.22.012971

Acrobat PDF (1332 KB)

### Abstract

Numerical optimization of photonic devices is often limited by a large design space the finite-differences gradient method requires as many electric field computations as there are design parameters. Adjoint-based optimization can deliver the same gradients with only two electric field computations. Here, we derive the relevant adjoint formalism and illustrate its application for a waveguide slab, and for the design of optical sub-wavelength gratings.

© 2014 Optical Society of America

## 1. Introduction

*primal solve*, one has to solve the

*adjoint problem*once. The adjoint solution acts as a kind of influence function regarding the respective target function and avoids to solve the primal problem for each parameter variation. This is the principle idea behind

*adjoint-based optimization and sensitivity analysis*. This approach is well-known to several fields of application as control theory, mesh adaptation, error estimation or propagation [1

1. J. L. Lions, *Optimal Control of Systems Governed by Partial Differential Equations* (Springer, 1971). [CrossRef]

5. J. Reuther, A. Jameson, J. J. Alonso, M. J. Remlinger, and D. Saunders, “Constrained multipoint aerodynamic shape optimization using an adjoint formulation and parallel computers, part 1,” Journal of Aircraft **36**(1), 51–60 (1999). [CrossRef]

6. J. Reuther, A. Jameson, J. J. Alonso, M. J. Remlinger, and D. Saunders, “Constrained multipoint aerodynamic shape optimization using an adjoint formulation and parallel computers, part 2,” Journal of Aircraft **36**(1), 61–74 (1999). [CrossRef]

7. Y. seek Chung, Changyul-Cheon, I.-H. Park, and S.-Y. Hahn, “Optimal shape design of microwave device using fdtd and design sensitivity analysis,” Microwave Theory and Techniques, IEEE Transactions on **48**, 2289–2296 (2000). [CrossRef]

10. N. K. Nikolova, Y. Li, Y. Li, and M. H. Bakr, “Sensitivity analysis of scattering parameters with electromagnetic time-domain simulators,” Microwave Theory and Techniques, IEEE Transactions on **54**, 1598–1610 (2006). [CrossRef]

11. G. Veronis, R. W. Dutton, and S. Fan, “Method for sensitivity analysis of photonic crystal devices,” Optics Letters **29**, 2288–2290 (2004). [CrossRef] [PubMed]

15. O. D. Miller, C. W. Hsu, M. T. H. Reid, W. Qiu, B. G. DeLacy, J. D. Joannopoulos, M. Soljačić, and S. G. Johnson, “Fundamental limits to extinction by metallic nanoparticles,” Physical Review Letters **112**, 123903 (2014). [CrossRef] [PubMed]

16. D. Fattal, J. Li, Z. Peng, M. Fiorentino, and R. G. Beausoleil, “Flat dielectric grating reflectors with focusing abilities,” Nature Photonics **4**, 466–470 (2010). [CrossRef]

17. V. Liu, D. Miller, and S. Fan, “Highly tailored computational electromagnetics methods for nanophotonic design and discovery,” Proceedings of the IEEE **101**, 484–493 (2013). [CrossRef]

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

## 2. Problem description

*T*(

**x**,

**p**) which depends on the physical state of the system

**x**, as well as a number of design parameters

**p**. For optical gratings, for example, the physical state of the system can be described conveniently by the electric field (or sometimes just a single component thereof) at each point of the computational grid. The vector of design parameters

**p**contains, for example, geometric parameters describing the layout of our grating. In order to determine the state of our system

**x**(

**p**), we solve a given vector equation

**R**(

**x**,

**p**) =

**0**, such as Maxwell’s equations.

*p*, this derivative is provided that the design parameters are mutually independent. What makes the evaluation of the above equation computationally costly is the the term

_{k}**x**on the design parameters

**p**is

*implicit*via the condition that

**R**(

**x**,

**p**) =

**0**. Otherwise, we would be able to analytically compute the relevant derivatives with respect to our design parameters and our optimization problem would be trivial.

*p*is a small variation on the

_{k}*k*-th design variable, and

**p**

_{Δpk}denotes the vector of design parameters where Δ

*p*was added to the

_{k}*k*-th component only. As we can see from the above finite-difference equation, this approach demands for one reference computation of

**R**(

**x**,

**p**) =

**0**to compute

**x**(

**p**), and then one additional

**R**(

**x**,

**p**

_{Δpk}) =

**0**to find the necessary

**x**(

**p**

_{Δpk}) for each component of the gradient. To compute the full gradient with respect to

*n*design variables, we would therefore have to solve the vector equation

**R**=

**0**a total of

*n*+ 1 times. For costly

**R**and large numbers of design parameters, this often exceeds the available resources of computational power and time.

## 3. Derivation of adjoint approach

**R**(

**x**,

**p**) =

**0**, and one solution of the adjoint equation. As long as the latter is is of similar numerical complexity as the direct equation

**R**, the adjoint-based optimization is likely to reduce the computational cost of

**p**, and any corresponding state

**x**, we know that the governing vector equation

**R**vanishes, Since this condition has to hold for

*any*set of parameters

**p**with corresponding

**x**, its Taylor expansion also has to vanish. Intuitively speaking, in the solution space, where we only consider design parameters and their corresponding states, the derivative of the state function

**R**with respect to a design parameter

*p*also vanishes, We can now multiply

_{k}**v**and subtract the product from

**v**such that the prefactor of

*n*+ 1 solutions of our state function to find the gradient

**R**=

**0**once to find our

**x**(

**p**). Then, we will solve our adjoint equation to find a convenient

**v**. And these two computations then straightforwardly allow us to determine the derivative of our target function with respect to our design parameters,

## 4. Optical systems

*et al.*[9

9. N. K. Nikolova, H. W. Tam, and M. H. Bakr, “Sensitivity analysis with the fdtd method on structured grids,” Microwave Theory and Techniques, IEEE Transactions on **52**, 1207–1216 (2004). [CrossRef]

**J**denotes the density of current sources. The constitutive relations are where

*μ*denotes the permeability tensor,

*ε*the permittivity tensor, and

*σ*the specific conductivity tensor. These equations can be combined into

*σ*vanishes. Furthermore, we write our electric field (and source) as

**E**(

*t*) = Re(

**Ē**e

^{−iωt}), where

**Ē**∈ ℂ. Note that the permittivity

*ε*describes the layout of our optical grating and therefore directly depends on our design parameters. We rewrite Eq (11) as where we introduced To compute an electric field

**Ē**that satisfies Eq. (11), we can use any convenient Maxwell solver, such as any FDTD [19

19. R. Courant, K. Friedrichs, and H. Lewy, “Über die partiellen Differenzengleichungen der mathematischen Physik,” Mathematische Annalen **100**, 32–74 (1928). [CrossRef]

20. K. Yee, “Numerical solution of initial boundary value problems involving maxwell’s equations in isotropic media,” Antennas and Propagation, IEEE Transactions on **14**, 302–307 (1966). [CrossRef]

*T*is the electric field strength in a specific region Ω where

**Ē**

_{0}=

**Ē**within the region Ω and zero outside, and

*N*is some (irrelevant) normalization constant introduced for mathematical rigor, only. With this notation, we also introduce the additional hypothesis that

**Ē**varies predominantly in norm (rather than in phase), as we vary

**p**. This will allow us to approximate the derivative of our target function with respect to the electric field as

*ε*

^{†}=

*ε*and

*μ*

^{†}=

*μ*we have

*ℳ*

^{†}=

*ℳ*. Second, we use

**Ē**computed as the numerical solution of Eq. (12) in Eq. (7) and find

**v̄**. Third, we rewrite this equation in a way that shows its correspondence to the direct problem Eq. (12) and conclude that

**v̄**can be interpreted as an electric field that was created by a particular set of sources. The adjoint sources term,

**Ē**, and of the adjoint problem,

**v̄**. Our target function (14) does not depend on the design parameters

**p**. Hence,

*ε*. Therefore,

*ε*with respect to each of the design parameters

*p*. However, we certainly know the current dielectric layout of the device we are trying to optimize. Hence, we can compute

_{k}**Ē**, and the adjoint solution

**v̄**.

## 5. Example 1: waveguide slab

*h*, described by the cover material index

*n*, the guiding film index

_{c}*n*, and the substrate index

_{f}*n*. We would like to know what modes are supported by this waveguide slab for a given wavelength

_{s}*λ*. The vacuum wave vector is

*β*. Therefore, solving Eq. (17) means finding one or all

*β*for which this equation holds.

*β*changes as we modify, for example, the height of the slab, we would usually have to solve Eq. (17) for different values of

*h*. Worse, even, if we were to modify the indices

*n*,

_{c}*n*, and

_{f}*n*, too, we would have to solve that same equation over and over again to derive how

_{s}*β*depends on them. Nowadays computers can solve this transcendental equation really easily, but for the sake of explaining the adjoint method, pretend that this is an arduous task.

*R*(

*β*,

**p**) depends on the longitudinal wave vector, as well as a number of design parameters

**p**= (

*h*,

*n*,

_{c}*n*,

_{f}*n*)

_{s}*, and vanishes for all solutions*

^{T}*β*(

**p**). For the sake of simplicity, we shall define our target function

*T*to be simply the longitudinal wave vector,

*v*such that In this case,

*β*changes as we vary the height of the guiding film,

*p*=

_{k}*h*, and we therefore need to compute and finally, following Eq. (28), As we can see in Fig. 1, the adjoint method leads to the correct gradient. However, is far more efficient for computing multi-variable optimizations than finite-differences gradients.

## 6. Example 2: grating coupler

21. D. Taillaert, W. Bogaerts, P. Bienstman, T. Krauss, P. van Daele, I. Moerman, S. Verstuyft, K. De Mesel, and R. Baets, “An out-of-plane grating coupler for efficient butt-coupling between compact planar waveguides and single-mode fibers,” Quantum Electronics, IEEE Journal of **38**, 949–955 (2002). [CrossRef]

22. G. Roelkens, D. V. Thourhout, and R. Baets, “High efficiency silicon-on-insulator grating coupler based on a poly-silicon overlay,” Optics Express **14**, 11622–11630 (2006). [CrossRef] [PubMed]

*p*and duty cycle

*d*of the grating grooves. Our simulations are assuming a waveguide thickness of 0.22

*μ*m, a groove-depth of 0.08

*μ*m, a wavelength of 1.55

*μ*m.

- compute the direct electromagnetic field for specific design parameters (12),
- compute the adjoint field for the same design parameters (15),
- combine the direct and adjoint solutions to (trivially) compute the gradient (16),
- update the design parameters for steepest ascent,
- repeat steps 1–4 until convergence.

23. A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: A flexible free-software package for electromagnetic simulations by the FDTD method,” Computer Physics Communications **181**, 687–702 (2010). [CrossRef]

**Ē**) before and after optimization. The coupling of the incident light from the top to inside the waveguide (inside the black square) has improved from 0.39 to 0.88.

*n*design parameters, the advantage of adjoint-based optimization would be even more striking: direct computation of the finite-differences gradient would require at least

*n*+ 1 field computations per gradient, whereas the adjoint method still only requires two.

## 7. Conclusion

**Ē**and the corresponding adjoint field

**v̄**are the only non-trivial computations required to calculate the gradient, needed to perform gradient ascent (or gradient descent) on the design parameters. We believe this method will prove itself immensely valuable in the design of various grating structures, including a host of non-periodic diffractive structures.

## Acknowledgments

## References and links

1. | J. L. Lions, |

2. | R. Becker and R. Rannacher, “An optimal control approach to error control and mesh adaption in finite element methods,” Acta Numerica |

3. | K. Eriksson, D. Estep, P. Hansbo, and C. Johnson, “Introduction to adaptive methods for differential equations,” Acta Numerica |

4. | D. E. Rumelhart, G. E. Hinton, and R. J. Williams, “Learning internal representations by error propagation,” Parallel Data Processing |

5. | J. Reuther, A. Jameson, J. J. Alonso, M. J. Remlinger, and D. Saunders, “Constrained multipoint aerodynamic shape optimization using an adjoint formulation and parallel computers, part 1,” Journal of Aircraft |

6. | J. Reuther, A. Jameson, J. J. Alonso, M. J. Remlinger, and D. Saunders, “Constrained multipoint aerodynamic shape optimization using an adjoint formulation and parallel computers, part 2,” Journal of Aircraft |

7. | Y. seek Chung, Changyul-Cheon, I.-H. Park, and S.-Y. Hahn, “Optimal shape design of microwave device using fdtd and design sensitivity analysis,” Microwave Theory and Techniques, IEEE Transactions on |

8. | N. Georgieva, S. Glavic, M. Bakr, and J. Bandler, “Feasible adjoint sensitivity technique for em design optimization,” Microwave Theory and Techniques, IEEE Transactions on |

9. | N. K. Nikolova, H. W. Tam, and M. H. Bakr, “Sensitivity analysis with the fdtd method on structured grids,” Microwave Theory and Techniques, IEEE Transactions on |

10. | N. K. Nikolova, Y. Li, Y. Li, and M. H. Bakr, “Sensitivity analysis of scattering parameters with electromagnetic time-domain simulators,” Microwave Theory and Techniques, IEEE Transactions on |

11. | G. Veronis, R. W. Dutton, and S. Fan, “Method for sensitivity analysis of photonic crystal devices,” Optics Letters |

12. | Y. Jiao, S. Fan, and D. A. B. Miller, “Photonic crystal device sensitivity analysis with wannierbasis gradients,” Optics Letters |

13. | P. Seliger, M. Mahvash, C. Wang, and A. F. J. Levi, “Optimization of aperiodic dielectric structures,” Journal of Applied Physics |

14. | C. M. Lalau-Keraly, S. Bhargava, O. D. Miller, and E. Yablonovitch, “Adjoint shape optimization applied to electromagnetic design,” Optics Express |

15. | O. D. Miller, C. W. Hsu, M. T. H. Reid, W. Qiu, B. G. DeLacy, J. D. Joannopoulos, M. Soljačić, and S. G. Johnson, “Fundamental limits to extinction by metallic nanoparticles,” Physical Review Letters |

16. | D. Fattal, J. Li, Z. Peng, M. Fiorentino, and R. G. Beausoleil, “Flat dielectric grating reflectors with focusing abilities,” Nature Photonics |

17. | V. Liu, D. Miller, and S. Fan, “Highly tailored computational electromagnetics methods for nanophotonic design and discovery,” Proceedings of the IEEE |

18. | J. Lu and Vučković, “Inverse design of nanophotonic structures using complementary convex optimization,” Optics Express |

19. | R. Courant, K. Friedrichs, and H. Lewy, “Über die partiellen Differenzengleichungen der mathematischen Physik,” Mathematische Annalen |

20. | K. Yee, “Numerical solution of initial boundary value problems involving maxwell’s equations in isotropic media,” Antennas and Propagation, IEEE Transactions on |

21. | D. Taillaert, W. Bogaerts, P. Bienstman, T. Krauss, P. van Daele, I. Moerman, S. Verstuyft, K. De Mesel, and R. Baets, “An out-of-plane grating coupler for efficient butt-coupling between compact planar waveguides and single-mode fibers,” Quantum Electronics, IEEE Journal of |

22. | G. Roelkens, D. V. Thourhout, and R. Baets, “High efficiency silicon-on-insulator grating coupler based on a poly-silicon overlay,” Optics Express |

23. | A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: A flexible free-software package for electromagnetic simulations by the FDTD method,” Computer Physics Communications |

**OCIS Codes**

(050.0050) Diffraction and gratings : Diffraction and gratings

(220.4830) Optical design and fabrication : Systems design

(230.0230) Optical devices : Optical devices

(350.4238) Other areas of optics : Nanophotonics and photonic crystals

(310.6805) Thin films : Theory and design

**ToC Category:**

Diffraction and Gratings

**History**

Original Manuscript: March 31, 2014

Revised Manuscript: May 1, 2014

Manuscript Accepted: May 12, 2014

Published: May 21, 2014

**Citation**

Armand C. R. Niederberger, David A. Fattal, Nicolas R. Gauger, Shanhui Fan, and Raymond G. Beausoleil, "Sensitivity analysis and optimization of sub-wavelength optical gratings using adjoints," Opt. Express **22**, 12971-12981 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-11-12971

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### References

- J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations (Springer, 1971). [CrossRef]
- R. Becker, R. Rannacher, “An optimal control approach to error control and mesh adaption in finite element methods,” Acta Numerica 10, 1–102 (2001). [CrossRef]
- K. Eriksson, D. Estep, P. Hansbo, C. Johnson, “Introduction to adaptive methods for differential equations,” Acta Numerica 4, 105–158 (1995). [CrossRef]
- D. E. Rumelhart, G. E. Hinton, R. J. Williams, “Learning internal representations by error propagation,” Parallel Data Processing 1, 318–362 (1986).
- J. Reuther, A. Jameson, J. J. Alonso, M. J. Remlinger, D. Saunders, “Constrained multipoint aerodynamic shape optimization using an adjoint formulation and parallel computers, part 1,” Journal of Aircraft 36(1), 51–60 (1999). [CrossRef]
- J. Reuther, A. Jameson, J. J. Alonso, M. J. Remlinger, D. Saunders, “Constrained multipoint aerodynamic shape optimization using an adjoint formulation and parallel computers, part 2,” Journal of Aircraft 36(1), 61–74 (1999). [CrossRef]
- Y. seek Chung, Changyul-Cheon, I.-H. Park, S.-Y. Hahn, “Optimal shape design of microwave device using fdtd and design sensitivity analysis,” Microwave Theory and Techniques, IEEE Transactions on 48, 2289–2296 (2000). [CrossRef]
- N. Georgieva, S. Glavic, M. Bakr, J. Bandler, “Feasible adjoint sensitivity technique for em design optimization,” Microwave Theory and Techniques, IEEE Transactions on 50, 2751–2758 (2002). [CrossRef]
- N. K. Nikolova, H. W. Tam, M. H. Bakr, “Sensitivity analysis with the fdtd method on structured grids,” Microwave Theory and Techniques, IEEE Transactions on 52, 1207–1216 (2004). [CrossRef]
- N. K. Nikolova, Y. Li, Y. Li, M. H. Bakr, “Sensitivity analysis of scattering parameters with electromagnetic time-domain simulators,” Microwave Theory and Techniques, IEEE Transactions on 54, 1598–1610 (2006). [CrossRef]
- G. Veronis, R. W. Dutton, S. Fan, “Method for sensitivity analysis of photonic crystal devices,” Optics Letters 29, 2288–2290 (2004). [CrossRef] [PubMed]
- Y. Jiao, S. Fan, D. A. B. Miller, “Photonic crystal device sensitivity analysis with wannierbasis gradients,” Optics Letters 30, 302–304 (2005). [CrossRef]
- P. Seliger, M. Mahvash, C. Wang, A. F. J. Levi, “Optimization of aperiodic dielectric structures,” Journal of Applied Physics 100, 034310 (2006). [CrossRef]
- C. M. Lalau-Keraly, S. Bhargava, O. D. Miller, E. Yablonovitch, “Adjoint shape optimization applied to electromagnetic design,” Optics Express 21, 21693–21701 (2013). [CrossRef] [PubMed]
- O. D. Miller, C. W. Hsu, M. T. H. Reid, W. Qiu, B. G. DeLacy, J. D. Joannopoulos, M. Soljačić, S. G. Johnson, “Fundamental limits to extinction by metallic nanoparticles,” Physical Review Letters 112, 123903 (2014). [CrossRef] [PubMed]
- D. Fattal, J. Li, Z. Peng, M. Fiorentino, R. G. Beausoleil, “Flat dielectric grating reflectors with focusing abilities,” Nature Photonics 4, 466–470 (2010). [CrossRef]
- V. Liu, D. Miller, S. Fan, “Highly tailored computational electromagnetics methods for nanophotonic design and discovery,” Proceedings of the IEEE 101, 484–493 (2013). [CrossRef]
- J. Lu, Vučković, “Inverse design of nanophotonic structures using complementary convex optimization,” Optics Express 18, 3793–3804 (2010). [CrossRef] [PubMed]
- R. Courant, K. Friedrichs, H. Lewy, “Über die partiellen Differenzengleichungen der mathematischen Physik,” Mathematische Annalen 100, 32–74 (1928). [CrossRef]
- K. Yee, “Numerical solution of initial boundary value problems involving maxwell’s equations in isotropic media,” Antennas and Propagation, IEEE Transactions on 14, 302–307 (1966). [CrossRef]
- D. Taillaert, W. Bogaerts, P. Bienstman, T. Krauss, P. van Daele, I. Moerman, S. Verstuyft, K. De Mesel, R. Baets, “An out-of-plane grating coupler for efficient butt-coupling between compact planar waveguides and single-mode fibers,” Quantum Electronics, IEEE Journal of 38, 949–955 (2002). [CrossRef]
- G. Roelkens, D. V. Thourhout, R. Baets, “High efficiency silicon-on-insulator grating coupler based on a poly-silicon overlay,” Optics Express 14, 11622–11630 (2006). [CrossRef] [PubMed]
- A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, S. G. Johnson, “MEEP: A flexible free-software package for electromagnetic simulations by the FDTD method,” Computer Physics Communications 181, 687–702 (2010). [CrossRef]

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