## Efficient 3D sensitivity analysis of surface plasmon waveguide structures

Optics Express, Vol. 16, Issue 21, pp. 16371-16381 (2008)

http://dx.doi.org/10.1364/OE.16.016371

Acrobat PDF (157 KB)

### Abstract

We present a novel analytical approach for efficient sensitivity analysis of surface plasmon polaritons (SPPs) waveguide-based structures using the beam propagation method (BPM). Our approach exploits the adjoint variable technique to extract the response sensitivities with respect to all the design parameters regardless of their number. No extra BPM simulations are required. The accuracy of the results are comparable to those obtained using the expensive central finite difference approximations applied at the response level. Our approach is successfully applied to different SPPs structures for different applications.

© 2008 Optical Society of America

## 1. Introduction

2. P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of symmetric structures,” Phys. Rev. B, Condens. Matter. **61**, 10484–10503 (2000). [CrossRef]

10. J. Shibayama, S. Takagi, T. Yamazaki, J. Yamauchi, and H. Nakano, “Numerical analysis of waveguide-based surface Plasmon resonance sensor with adsorbed layer using two- and three-dimensional beam-propagation methods,” IEICE Trans. Electron.E **90-C**, 95–100 (2007). [CrossRef]

6. N. N. Feng, M. L. Brongersma, and L. D Negro, “Metal-Dielectric slot-waveguide structures for the propagation of surface plasmon polaritons at 1.55 µm,” IEEE J.Quantum Electron. **43**, 479–485 (2007). [CrossRef]

7. J. Homola, S. S. Yee, and G. Gauglitz, “Surface plasmon resonance sensors: review,” Sens. Actuators B **54**, 3–15 (1999). [CrossRef]

9. J. Homola, “Present and future of surface plasmon resonance biosensors,” Anal. Bioanal. Chem. **377**, 528–539 (2003). [CrossRef] [PubMed]

2. P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of symmetric structures,” Phys. Rev. B, Condens. Matter. **61**, 10484–10503 (2000). [CrossRef]

3. G. Veronis and S. Fan, “Modes of Subwavelength Plasmonic Slot Waveguides,” J. Lightwave Technol. **25**, 2511–2521 (2007). [CrossRef]

5. I. Breukelaar, R. Charbonneau, and P. Berini, “Long-range surface plasmon-polariton mode cutoff and radiation in embedded strip waveguides,” J. Appl. Phys. **100**, 043104-1-043104-9 (2006). [CrossRef]

10. J. Shibayama, S. Takagi, T. Yamazaki, J. Yamauchi, and H. Nakano, “Numerical analysis of waveguide-based surface Plasmon resonance sensor with adsorbed layer using two- and three-dimensional beam-propagation methods,” IEICE Trans. Electron.E **90-C**, 95–100 (2007). [CrossRef]

*N*design parameters

*N*extra simulations are need to extract the sensitivity information if the forward finite difference approximation is used.

10. J. Shibayama, S. Takagi, T. Yamazaki, J. Yamauchi, and H. Nakano, “Numerical analysis of waveguide-based surface Plasmon resonance sensor with adsorbed layer using two- and three-dimensional beam-propagation methods,” IEICE Trans. Electron.E **90-C**, 95–100 (2007). [CrossRef]

11. J. Shibayama, S. Takagi, T. Yamazaki, J. Yamauchi, and H. Nakano, “Eigenmode analysis of a light-guiding metal line loaded on a dielectric substrate using the imaginary-distance beam-propagation method,” J. Lightwave Technol. **23**, 1533–1539 (2005). [CrossRef]

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

13. M. A. Swillam, M. H. Bakr, and X. Li, “Accurate sensitivity analysis of photonic devices exploiting the finite-difference time-domain central adjoint variable method,” Appl. Opt. **46**, 1492–1499.(2007). [CrossRef] [PubMed]

14. M. A. Swillam, M. H. Bakr, and X. Li, “Efficient adjoint sensitivity analysis exploiting the FD-BPM,” J. Lightwave Technol. **25**, 1861–1869 (2007). [CrossRef]

15. M. A. Swillam, M. H. Bakr, and X. Li, “Full vectorial 3D sensitivity analysis and design optimization using BPM,” J. Lightwave Technol. **26**, 528–536 (2008). [CrossRef]

14. M. A. Swillam, M. H. Bakr, and X. Li, “Efficient adjoint sensitivity analysis exploiting the FD-BPM,” J. Lightwave Technol. **25**, 1861–1869 (2007). [CrossRef]

15. M. A. Swillam, M. H. Bakr, and X. Li, “Full vectorial 3D sensitivity analysis and design optimization using BPM,” J. Lightwave Technol. **26**, 528–536 (2008). [CrossRef]

## 2. 3D ADI BPM

16. Y. Hsueh, M. Yang, and H. Chang,“Three-dimensional noniterative full-vectorial beam propagation method based on the alternating direction implicit method,” J. Lightwave Technol. **19**, 2389–2397 (1999). [CrossRef]

*Ψ**is the slowly varying envelope of the transverse electric field component*

_{y}

*E**. Similarly, the BPM equation for the envelope of the field component in the*

_{y}*x*direction

**Ψ**

*can be obtained [16*

_{x}16. Y. Hsueh, M. Yang, and H. Chang,“Three-dimensional noniterative full-vectorial beam propagation method based on the alternating direction implicit method,” J. Lightwave Technol. **19**, 2389–2397 (1999). [CrossRef]

*k*is the free space wave number,

*n*is the refractive index, and

*n*is the reference refractive index.

_{o}

*Ψ*^{l+1}

_{x,y}at the propagation step

*z*

_{l+1}=(

*l*+1)Δ

*z*as follows [16

16. Y. Hsueh, M. Yang, and H. Chang,“Three-dimensional noniterative full-vectorial beam propagation method based on the alternating direction implicit method,” J. Lightwave Technol. **19**, 2389–2397 (1999). [CrossRef]

**is the identity matrix. In Eq. (3),**

*I*

*B**and*

_{x}

*B**represent the*

_{y}*x*-dependent and

*y*-dependent components of the operator

*P**given in Eq. (1), respectively. They are given by*

_{yy}

*P**xx*can be separated to

*x*-dependent and

*y*-dependent components for the TE case. The solution of the system of equations given in Eq. (2) is performed by solving two tridiagonal system of equations at each propagation step [16

**19**, 2389–2397 (1999). [CrossRef]

## 3. Sensitivity analysis using adjoint variable method (AVM)

*f*with respect to the design parameter

*p*is given by [15

_{i}15. M. A. Swillam, M. H. Bakr, and X. Li, “Full vectorial 3D sensitivity analysis and design optimization using BPM,” J. Lightwave Technol. **26**, 528–536 (2008). [CrossRef]

*i*=2…,

*N*is the index of the design parameters

*p*. These design parameters may be dimensions of discontinuities or constitutive parameters. In Eq. (6), the objective function is assumed to depend on the field states at different propagation steps

_{i}*z*i.e.

_{k}*l*∈

_{k}*I*∀

*k*, where

*I*is the index set of the associated field states. By differentiating (2) with respect to the ith design parameter

*p*, we get

_{i}**26**, 528–536 (2008). [CrossRef]

14. M. A. Swillam, M. H. Bakr, and X. Li, “Efficient adjoint sensitivity analysis exploiting the FD-BPM,” J. Lightwave Technol. **25**, 1861–1869 (2007). [CrossRef]

**26**, 528–536 (2008). [CrossRef]

*Q*grid points due to the change in the design parameter

*p*. Following a similar approach to that in [17

_{i}17. P. A. W. Basl, M. H. Bakr, and N. K. Nikolova, “Efficient estimation of sensitivities in TLM with dielectric discontinuities,” IEEE Microwave Wirel. Compon. Lett. **15**, 89–91 (2005). [CrossRef]

*q*th grid point

*∂*/

**R***∂n*is analytically calculated. This derivative is sparse with non zero elements at the elements corresponding to the grid point

_{q}*q*. This derivative

*∂*/

**R***∂n*can be calculated in terms of

_{q}*∂*/

**B**_{x}*∂n*and

_{q}*∂*/

**B**_{y}*∂n*which can be calculated analytically. For example, in the case of finite difference discritization of the operators given in Eq.(5), the matrix

_{q}*can be written as*

**B**x*x*is the spatial grid size in the

*x*direction.

*with respect to the refractive index of grid point*

**B**x*q*we obtain

*is the identity matrix. The term*

**I***∂*/

**B**_{y}*∂n*can be calculated using a similar approach. It follows that the derivative

_{q}*∂*/

**R***∂n*can be calculated analytically in an efficient manner. It is calculated only once for all the design parameters. This approach reduces the computational cost of obtaining the derivative of the system matrices especially for structures with large number of design parameters.

_{q}*n*represents the real refractive index of the material and

_{r}*n*represents the losses of this material. Using the theory of complex analysis, the derivative of the system matrix with respect to the refractive index for the complex case is given by:

_{im}*and*

**R**^{k}_{r}*are the real part and the imaginary part of the system matrix*

**R**^{k}_{m}*. Here, the matrix*

**R**^{k}*is assumed to be analytical. In addition, the derivatives of the system matrices with respect to the imaginary part of the refractive index can be efficiently calculated using the Cauchy-Riemann formula as follow [18]:*

**R**^{k}## 4. Numerical examples

11. J. Shibayama, S. Takagi, T. Yamazaki, J. Yamauchi, and H. Nakano, “Eigenmode analysis of a light-guiding metal line loaded on a dielectric substrate using the imaginary-distance beam-propagation method,” J. Lightwave Technol. **23**, 1533–1539 (2005). [CrossRef]

11. J. Shibayama, S. Takagi, T. Yamazaki, J. Yamauchi, and H. Nakano, “Eigenmode analysis of a light-guiding metal line loaded on a dielectric substrate using the imaginary-distance beam-propagation method,” J. Lightwave Technol. **23**, 1533–1539 (2005). [CrossRef]

**23**, 1533–1539 (2005). [CrossRef]

**23**, 1533–1539 (2005). [CrossRef]

19. FEMLAB, 2.3 ed. COMSOL AB, Sweden, 2002. http://www.comsol.com

### 4.1 A metal loaded on a channel dielectric waveguide

*n*=0.18-

_{m}*j*10.2 at a wavelength of 1.55 µm. The refractive indexes of silicon (Si) and the insulator (SiO2) are

*n*=3.46, and

_{s}*n*=1.46, respectively. The thickness of the metal layer and the silicon layer are

_{i}*t*=0.10 µm, and

_{m}*t*=0.50 µm, respectively. The width of the metal and silicon layers are taken as

_{s}*W*=

_{m}*W*=0.5 µm. The propagation length of this waveguide structure is given as

_{s}*Lp*=1/(2Im(

*β*)), where

*β*is the complex propagation constant of the fundamental TM mode of this structure. This length is calculated using the imaginary distance 3D ADI BPM to be 32.0 µm. The vector of design parameters of this structure is

**=[**

*p**t*

_{s}*t*

_{m}*W*

_{s}*W*]

_{m}*. The sensitivity of the propagation length and the effective index of the fundamental mode are calculated using the proposed AVM. The results are compared with the sensitivity information obtained using the CFD approach applied directly at the response level. A very good agreement is obtained between our approach and the CFD as shown in Figs. 4-6. The CFD requires 8 additional simulations. Our approach, however, requires no additional simulations. In general, to obtain an optimal response, the sensitivity should be ideally zero. Thus our target is to minimize the sensitivity to enhance the response. As clear from Fig. 4, the sensitivity of the propagation length is decreasing with the increase of the thickness of the silicon layer*

^{T}*t*. Thus increasing the silicon layer thickness reduces the loss of the fundamental mode. On the other hand, Fig. 5 shows that the sensitivity of propagation length approaching zero when the width of the metal layer

_{s}*W*approaches 0.5 µm which is the same value for the width of the silicon layer

_{m}*W*. The same behavior can be seen in Fig. 6, where the sensitivity of the propagation length decreases when

_{s}*W*approaches the nominal value of

_{s}*W*which is 0.5 µm. Figs. 5 and 6 indicate that in order to reduce the losses and hence increase the propagation length, the width of both the silicon layer and the metal layer should be similar or ideally the same. The width mismatch between the two layers forces the field lines at the edges of the metals to pass through the air before reaching the silicon region. This results in an additional discontinuity. Thus we may conclude that the presence of the additional discontinuity due to width mismatch increases the losses of the whole structure.

_{m}### 4.2 A compact 1×3 power splitter

*n*=0.18-

_{m}*j*10.2 deposited on an insulator (SiO

_{2}) with refractive index

*n*=1.459. The various widths of the metal sections define the regions of single mode and multimode operation. The design of this structure is based on the multimode interference in the multimode waveguide by utilizing the self imaging principle [20

_{s}20. L. B. Soldano and E. C. Pennings, “Optical multi-mode interference devices based on self-imaging: principles and applications,” J. Lightwave Technol. **13**, 615–627 (1995). [CrossRef]

*W*is taken to be 15.0 µm. The width of the metal in the single waveguides

_{m}*W*is taken to be 3.0 µm to ensure single mode operation at 1.55 µm [11

_{s}**23**, 1533–1539 (2005). [CrossRef]

*t*is taken to be 0.2 µm. The thickness of the metal layer of the single mode region

_{m}*t*is also taken to be 0.2 µm The sensitivity of the coupling coefficient of the power splitter with respect to the design parameters of the structure

_{s}**=[**

*p**W*

_{m}*t*

_{m}*L*

_{m}*t*]

_{s}*is studied using our AVM technique. The coupling coefficient is defined as*

^{T}

*Φ**and*

_{N}**are the normalized modal field of the output waveguides and the normalized field distribution at the output waveguides, respectively. The length of the multimode section**

*Ψ**L*is calculated to be 78.2 µm. This length, which is calculated using the 3D ADI BPM, maximizes the power coupling coefficient at the output waveguides. A good agreement is obtained between the sensitivity information obtained using our AVM approach and the sensitivity obtained using the time consuming CFD approach. In order to illustrate the accuracy of our approach the normalized sensitivity of the power coupling with respect to width of the multimode region

_{m}*W*is shown in Fig. 8. The normalized sensitivity of the power coupling sensitivity is also calculated with respect to the thickness of the metal in the multimode region

_{m}*t*as shown in Fig 9. As shown from this figure, the sensitivity of the coupling power approaches zero when the thickness reaches the value of 0.2 µm. Hence, the power coupling is maximum and the loss is minimized once the thickness exceeds this value. This result agrees with the result obtained in [11

_{m}**23**, 1533–1539 (2005). [CrossRef]

## 5. Conclusion

## References and links

1. | S. A. Maier, |

2. | P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of symmetric structures,” Phys. Rev. B, Condens. Matter. |

3. | G. Veronis and S. Fan, “Modes of Subwavelength Plasmonic Slot Waveguides,” J. Lightwave Technol. |

4. | W. L. Barnes, A. Dereux, and T. W. Ebbesen “Surface plasmon subwavelength optics,” Nature. |

5. | I. Breukelaar, R. Charbonneau, and P. Berini, “Long-range surface plasmon-polariton mode cutoff and radiation in embedded strip waveguides,” J. Appl. Phys. |

6. | N. N. Feng, M. L. Brongersma, and L. D Negro, “Metal-Dielectric slot-waveguide structures for the propagation of surface plasmon polaritons at 1.55 µm,” IEEE J.Quantum Electron. |

7. | J. Homola, S. S. Yee, and G. Gauglitz, “Surface plasmon resonance sensors: review,” Sens. Actuators B |

8. | R. D. Harris and J. S. Wilkinson, “Waveguide surface plasmon resonance sensors,” Sens. Actuators B |

9. | J. Homola, “Present and future of surface plasmon resonance biosensors,” Anal. Bioanal. Chem. |

10. | J. Shibayama, S. Takagi, T. Yamazaki, J. Yamauchi, and H. Nakano, “Numerical analysis of waveguide-based surface Plasmon resonance sensor with adsorbed layer using two- and three-dimensional beam-propagation methods,” IEICE Trans. Electron.E |

11. | J. Shibayama, S. Takagi, T. Yamazaki, J. Yamauchi, and H. Nakano, “Eigenmode analysis of a light-guiding metal line loaded on a dielectric substrate using the imaginary-distance beam-propagation method,” J. Lightwave Technol. |

12. | G. Veronis, R. W. Dutton, and S. Fan, “Method for sensitivity analysis of photonic crystal devices,” Opt. Lett. |

13. | M. A. Swillam, M. H. Bakr, and X. Li, “Accurate sensitivity analysis of photonic devices exploiting the finite-difference time-domain central adjoint variable method,” Appl. Opt. |

14. | M. A. Swillam, M. H. Bakr, and X. Li, “Efficient adjoint sensitivity analysis exploiting the FD-BPM,” J. Lightwave Technol. |

15. | M. A. Swillam, M. H. Bakr, and X. Li, “Full vectorial 3D sensitivity analysis and design optimization using BPM,” J. Lightwave Technol. |

16. | Y. Hsueh, M. Yang, and H. Chang,“Three-dimensional noniterative full-vectorial beam propagation method based on the alternating direction implicit method,” J. Lightwave Technol. |

17. | P. A. W. Basl, M. H. Bakr, and N. K. Nikolova, “Efficient estimation of sensitivities in TLM with dielectric discontinuities,” IEEE Microwave Wirel. Compon. Lett. |

18. | J. W. Brown and R. V. Churchil, |

19. | FEMLAB, 2.3 ed. COMSOL AB, Sweden, 2002. http://www.comsol.com |

20. | L. B. Soldano and E. C. Pennings, “Optical multi-mode interference devices based on self-imaging: principles and applications,” J. Lightwave Technol. |

**OCIS Codes**

(000.4430) General : Numerical approximation and analysis

(230.1360) Optical devices : Beam splitters

(240.6680) Optics at surfaces : Surface plasmons

(250.5403) Optoelectronics : Plasmonics

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: July 30, 2008

Revised Manuscript: September 16, 2008

Manuscript Accepted: September 17, 2008

Published: September 29, 2008

**Citation**

Mohamed A. Swillam, Mohamed H. Bakr, and Xun Li, "Efficient 3D sensitivity analysis of surface plasmon waveguide structures," Opt. Express **16**, 16371-16381 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-21-16371

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### References

- S. A. Maier, Plasmonics: Fundamentals and Applications, (Springe, 2007).
- P. Berini, "Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of symmetric structures," Phys. Rev. B, Condens. Matter. 61, 10484-10503 (2000). [CrossRef]
- G. Veronis and S. Fan, "Modes of Subwavelength Plasmonic Slot Waveguides,"J. Lightwave Technol. 25, 2511-2521 (2007). [CrossRef]
- W. L. Barnes, A. Dereux, and T. W. Ebbesen "Surface plasmon subwavelength optics," Nature. 424, 824-830 (2003). [CrossRef] [PubMed]
- I. Breukelaar, R. Charbonneau, and P. Berini, " Long-range surface plasmon-polariton mode cutoff and radiation in embedded strip waveguides," J. Appl. Phys. 100, 043104-1-043104-9 (2006). [CrossRef]
- N. N. Feng, M. L. Brongersma, and L. D Negro, "Metal-Dielectric slot-waveguide structures for the propagation of surface plasmon polaritons at 1.55 μm," IEEE J.Quantum Electron. 43, 479 - 485 (2007). [CrossRef]
- J. Homola, S. S. Yee, and G. Gauglitz, "Surface plasmon resonance sensors: review," Sens. Actuators B 54, 3-15 (1999). [CrossRef]
- R. D. Harris and J. S. Wilkinson, "Waveguide surface plasmon resonance sensors," Sens. Actuators B 29, 261-267 (1995). [CrossRef]
- J. Homola, "Present and future of surface plasmon resonance biosensors," Anal. Bioanal. Chem. 377, 528-539 (2003). [CrossRef] [PubMed]
- J. Shibayama, S. Takagi, T. Yamazaki, J. Yamauchi, and H. Nakano, "Numerical analysis of waveguide-based surface Plasmon resonance sensor with adsorbed layer using two- and three-dimensional beam-propagation methods," IEICE Trans. Electron. E90-C, 95-100 (2007). [CrossRef]
- J. Shibayama, S. Takagi, T. Yamazaki, J. Yamauchi, and H. Nakano, "Eigenmode analysis of a light-guiding metal line loaded on a dielectric substrate using the imaginary-distance beam-propagation method," J. Lightwave Technol. 23, 1533- 1539 (2005). [CrossRef]
- G. Veronis, R. W. Dutton, and S. Fan, "Method for sensitivity analysis of photonic crystal devices," Opt. Lett. 29, 2288-2290 (2004). [CrossRef] [PubMed]
- M. A. Swillam, M. H. Bakr, and X. Li, "Accurate sensitivity analysis of photonic devices exploiting the finite-difference time-domain central adjoint variable method," Appl. Opt. 46, 1492-1499.(2007). [CrossRef] [PubMed]
- M. A. Swillam, M. H. Bakr, and X. Li, "Efficient adjoint sensitivity analysis exploiting the FD-BPM," J. Lightwave Technol. 25, 1861 - 1869 (2007). [CrossRef]
- M. A. Swillam, M. H. Bakr, and X. Li, "Full vectorial 3D sensitivity analysis and design optimization using BPM," J. Lightwave Technol. 26, 528-536 (2008). [CrossRef]
- Y. Hsueh, M. Yang, and H. Chang," Three-dimensional noniterative full-vectorial beam propagation method based on the alternating direction implicit method," J. Lightwave Technol. 19, 2389-2397 (1999). [CrossRef]
- P. A. W. Basl, M. H. Bakr, and N. K. Nikolova, "Efficient estimation of sensitivities in TLM with dielectric discontinuities," IEEE Microwave Wirel. Compon. Lett. 15, 89-91 (2005). [CrossRef]
- J. W. Brown and R. V. Churchil, Complex Variables and Applications (McGraw-Hill, 2003).
- FEMLAB, 2.3 ed. COMSOL AB, Sweden, 2002. http://www.comsol.com
- L. B. Soldano and E. C. Pennings, "Optical multi-mode interference devices based on self-imaging: principles and applications," J. Lightwave Technol. 13, 615-627 (1995). [CrossRef]

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