## Investigation of LPG-SPR sensors using the finite element method and eigenmode expansion method |

Optics Express, Vol. 21, Issue 12, pp. 13875-13895 (2013)

http://dx.doi.org/10.1364/OE.21.013875

Acrobat PDF (10744 KB)

### Abstract

As compared to the well-known traditional couple-mode theory, in this study, we proposed a visual, graphical, and simple numerical simulation method for long-period fiber-grating surface-plasmon-resonance (LPG-SPR) sensors. This method combines the finite element method and the eigenmode expansion method. The finite element method was used to solve for the guided modes in fiber structures, including the surface plasmon wave. The eigenmode expansion method was used to calculate the power transfer phenomenon of the guided modes in the fiber structure. This study provides a detailed explanation of the key reasons why the periodic structure of long-period fiber-grating (LPG) can achieve significantly superior results for our method compared to those obtained using other numerical methods, such as the finite-difference time-domain and beam propagation methods. All existing numerical simulation methods focus on large-sized periodic components; only the method established in this study has 3D design and analysis capabilities. In addition, unlike the offset phenomenon of the design wavelength λ_{D} and the maximum transmission wavelength λ_{max} of the traditional coupled-mode theory, the method established in this study has rapid scanning LPG period capabilities. Therefore, during the initial component design process, only the operating wavelength must be set to ensure that the maximum transmission wavelength of the final product is accurate to the original setup, for example, λ = 1550 nm. We verified that the LPG-SPR sensor designed in this study provides a resolution of ~-45 dB and a sensitivity of ~27000 nm/RIU (refractive index unit). The objective of this study was to use the combination of these two numerical simulation methods in conjunction with a rigorous, simple, and complete design process to provide a graphical and simplistic simulation technique that reduces the learning time and professional threshold required for research and applications of LPG-SPR sensors.

© 2013 OSA

## 1. Introduction

1. J. Homola, “Optical fiber sensor based on surface plasmon excitation,” Sens. Actuators B Chem. **29**(1-3), 401–405 (1995). [CrossRef]

4. Ó. Esteban, R. Alonso, M. C. Navarrete, and A. González-Cano, “Surface plasmon excitation in fiber-optical sensors: a novel theoretical approach,” J. Lightwave Technol. **20**(3), 448–453 (2002). [CrossRef]

5. S. Patskovsky, A. V. Kabashin, M. Meunier, and J. H. Luong, “Silicon-based surface plasmon resonance sensing with two surface plasmon polariton modes,” Appl. Opt. **42**(34), 6905–6909 (2003). [CrossRef] [PubMed]

8. S. Maruo, O. Nakamura, and S. Kawata, “Evanescent-wave holography by use of surface-plasmon resonance,” Appl. Opt. **36**(11), 2343–2346 (1997). [CrossRef] [PubMed]

### 1.1 Defining the LPG-SPR sensor structure and solving for guided modes

9. Y. J. He, Y. L. Lo, and J. F. Huang, “Optical-fiber surface-plasmon-resonance sensor employing long-period fiber grating in multiplexing,” J. Opt. Soc. Am. B **23**(5), 801–811 (2006). [CrossRef]

### 1.2 Defining the mathematical model of fiber grating and explaining the equation of the unconjugated form of the coupled-mode

9. Y. J. He, Y. L. Lo, and J. F. Huang, “Optical-fiber surface-plasmon-resonance sensor employing long-period fiber grating in multiplexing,” J. Opt. Soc. Am. B **23**(5), 801–811 (2006). [CrossRef]

### 1.3 Defining the bar and cross-transmission power spectrum

_{11}after traveling a certain distance (z) and when incidence occurs (z = 0). In other words, for incident mode HE

_{11}, after traveling a certain distance (z), if the power can be coupled to other modes, the power ratio of the modes and the incidence (z = 0) HE

_{11}denote the cross transmission power. The power spectrum diagram of LPG-SPR sensors can be obtained by using these two parameters and configuring the wavelengths in diagram form.

## 2. The finite element method

11. D. Sun, J. Manges, Xingchao Yuan, and Z. Cendes, “Spurious modes in finite-element methods,” IEEE Antennas Propag. Mag. **37**(5), 12–24 (1995). [CrossRef]

- A. Use the equivalent functional for solving the minimum value problem to replace the PDE with boundary conditions.
- B. Partition the geometric domain to be solved into several sufficiently small blocks. These blocks are known as elements, such as triangular or quadrilateral elements. The objects connecting elements are called nodes, and the unknown variable of the nodes is the desired answer.
- C. One appropriate interpolation function can be used by each true function to be solved in each element for approximation. The easiest interpolation function to conduct is the linear interpolation function. Thus, the polynomial function formed by nodes can be employed for approximation of the true function to be solved in each element. This polynomial coefficient is known as the shape function.
- D. Substitute this polynomial function into the functional to determine a minimum value and obtain a set of simultaneous equations.
- E. Enter the boundary conditions into the simultaneous equations, and solve these equations using the Gaussian elimination method. Thus, the answer to all nodes in the domain to be solved can be obtained. After the answer to each node is obtained, each true function to be solved within the element can be further approximated using the polynomial function of the node. In other words, all unknown variables or values of the entire geometric domain to be solved can be obtained.

^{−4}as a guideline for all cutting resolutions.

_{11}) with the effective refractive index

_{11}) entered the LPG domain, the mode coupling phenomenon only occurred between HE

_{11}and the cladding modes of azimuthal order

_{11}) and 19 cladding modes, were solved during the preliminary process. We used the same naming convention commonly employed in fiber optics to name the 20 guided modes obtained, excluding the core mode. We then numbered the 19 cladding modes based on their effective refractive index value, where the greater the n

_{eff}value was, the smaller the

_{11}and its 2D and 3D power distribution diagrams, as shown in Fig. 4 and Fig. 5, where its effective refractive index was

^{−4}.

9. Y. J. He, Y. L. Lo, and J. F. Huang, “Optical-fiber surface-plasmon-resonance sensor employing long-period fiber grating in multiplexing,” J. Opt. Soc. Am. B **23**(5), 801–811 (2006). [CrossRef]

**23**(5), 801–811 (2006). [CrossRef]

## 3. The eigenmode expansion method

_{k-1}) and block object k (Bk

_{k}) is called junction k-1 (J

_{k-1}), which is presented in Segment (1) of Fig. 9 and Fig. 10. The diagram clearly shows that by employing the same cutting mechanism, the block objects obtained from cutting segment object (m) and segment object (n) are identical completely. In other words, under the same cutting procedure, completing the cutting computation for a segment object equates to completing the cutting computation for an entire LPG-SPR senor periodic object. In this study, we cut a segment object into 300 uniform block objects. The eigenmode expansion method targeted one segment object from the LPG-SPR sensor periodic object and the segment was cut into 300 uniform block objects. In each block object, the guided modes that may exist were recalculated. Next, power conversions between block objects were performed using the Fourier series expansion method. The steps in this procedure can also be performed to complete the power transfers of the entire LPG-SPR sensor [12

12. C. H. Herry and Y. Shani, “Analysis of mode propagation in optical waveguide devices by Fourier expansion,” IEEE J. Quantum Electron. **27**(3), 523–530 (1991). [CrossRef]

14. D. F. G. Gallagher and T. P. Felici, “Eigenmode expansion methods for simulation of optical propagation in photonics-Pros and cons,” Proc. SPIE **4987**, 69–82 (2003). [CrossRef]

_{k-1}can be obtained, as shown in Eq. (10) and Eq. (11). After the propagated electromagnetic fields for a uniform block object Bk

_{k-1}are obtained, power must be precisely transferred from Block object Bk

_{k-1}to Block object Bk

_{k}. Here, we used the scattering matrix to transform the forward and backward power propagation of two adjacent block objects, as shown in Eq. (12). In Fig. 12,

_{k-1}and the uniform block object Bk

_{k}, and

^{−3}. Otherwise, the traditional unconjugated form of coupled-mode theory would exhibit significant errors. Compared to the traditional unconjugated form of coupled-mode theory, the eigenmode expansion method employed in this study can detailed cut one segment into 300 uniform block objects and re-solve for the guided modes of each uniform block object. In other words, this method is not disadvantaged by requiring that the induced refractive index charge is not excessive. In addition, unless otherwise specified, all the induced refractive index changes in this study used

^{−4}. Figure 13 shows the eigenmode expansion method that employed 20 guided modes (i.e.,

^{−4}criterion. Therefore, unless otherwise specified, the standard number of all guided modes used in this study was 20, and the standard number of block objects used in this study was 300.

## 4. Design and analysis of LPG-SPR sensors

_{11}) to facilitate full coupling with SPW (ν = 9) of the same propagation direction. A diagram of the relationship between transmission power and LPG periods is shown in Fig. 16. A diagram of the relationship between the transmission power and LPG period number for the core mode (HE

_{11}) fully coupled to SPW (ν = 9) of the same propagation direction is shown in Fig. 17. Combining Fig. 16 and Fig. 17, we found that for period Λ = 47.57399 μm, and when the number of periods was N = 29, the core mode (HE

_{11}) was fully coupled to the SPW (SPW, ν = 9) of the same propagation direction. Based on these two parameters, we inputted the core mode (HE

_{11}) from the left end of Fig. 15 to observe the power propagation at the Y = 0 plane or the X-Z plane. The results are shown in Fig. 18. Evidently, under LPG disturbance, the core mode power was fully coupled to the SPW (ν = 9) of the same propagation direction during the propagation process. In addition, to verify the accuracy of mode coupling, we obtained the 2D and 3D power distributions of various z-axis positions shown in Fig. 18.

^{−4}. Obviously, the propagation distance and power loss conditions of this example must also be examined. As shown in Fig. 26, the power loss can satisfy the less than 10

^{−4}requirement specified in Step 3 of the design process.

15. T. Erdogan, “Cladding-mode resonances in short and long period fiber grating filters,” J. Opt. Soc. Am. A **14**(8), 1760–1773 (1997). [CrossRef]

16. T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. **15**(8), 1277–1294 (1997). [CrossRef]

15. T. Erdogan, “Cladding-mode resonances in short and long period fiber grating filters,” J. Opt. Soc. Am. A **14**(8), 1760–1773 (1997). [CrossRef]

16. T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. **15**(8), 1277–1294 (1997). [CrossRef]

_{11}) fully coupled to the SPW (ν = 9) of the same propagation direction were Λ = 47.57399 μm and N = 29. We calculated the spectrum graph, and the results are shown in Fig. 27. From this figure, we found that the maximum transmission wavelength

_{4}= 1.4, the sensor had a resolution of approximately −45 dB. Based on Fig. 28, we corresponded the resonance wavelength to the analyte refractive index n

_{4}; the results are shown in Fig. 29. Using Fig. 29, we can easily calculate that the sensor had a sensitivity of approximately 27000 nm/RIU (refractive index unit). The analyte refractive index changes and resonance wavelength shift directions can be reasonably explained using the approximate Eq. (13) shown below.For the Eq. (13) in the spectral graph, the LPG period (

## 5. Conclusion

## Acknowledgment

## References and links

1. | J. Homola, “Optical fiber sensor based on surface plasmon excitation,” Sens. Actuators B Chem. |

2. | R. Slavík, J. Homola, and J. Čtyroký, “Single-mode optical fiber surface plasmon resonance sensor,” Sens. Actuators B Chem. |

3. | S.-M. Tseng, K.-Y. Hsu, H.-S. Wei, and K.-F. Chen, “Analysis and experiment of thin metal-clad fiber polarizer with index overlay,” IEEE Photon. Technol. Lett. |

4. | Ó. Esteban, R. Alonso, M. C. Navarrete, and A. González-Cano, “Surface plasmon excitation in fiber-optical sensors: a novel theoretical approach,” J. Lightwave Technol. |

5. | S. Patskovsky, A. V. Kabashin, M. Meunier, and J. H. Luong, “Silicon-based surface plasmon resonance sensing with two surface plasmon polariton modes,” Appl. Opt. |

6. | S. Patskovsky, A. V. Kabashin, M. Meunier, and J. H. Luong, “Properties and sensing characteristics of surface-plasmon resonance in infrared light,” J. Opt. Soc. Am. A |

7. | A. J. C. Tubb, F. P. Payne, R. B. Millington, and C. R. Lowe, “Single-mode optical fibre surface plasma wave chemical sensor,” Sens. Actuators B Chem. |

8. | S. Maruo, O. Nakamura, and S. Kawata, “Evanescent-wave holography by use of surface-plasmon resonance,” Appl. Opt. |

9. | Y. J. He, Y. L. Lo, and J. F. Huang, “Optical-fiber surface-plasmon-resonance sensor employing long-period fiber grating in multiplexing,” J. Opt. Soc. Am. B |

10. | E. D. Palik, |

11. | D. Sun, J. Manges, Xingchao Yuan, and Z. Cendes, “Spurious modes in finite-element methods,” IEEE Antennas Propag. Mag. |

12. | C. H. Herry and Y. Shani, “Analysis of mode propagation in optical waveguide devices by Fourier expansion,” IEEE J. Quantum Electron. |

13. | G. Sztefka and H. P. Nolting, “Bidirectional eigenmode Propagation for Large refractive index steps,” IEEE Photon. Technol. Lett. |

14. | D. F. G. Gallagher and T. P. Felici, “Eigenmode expansion methods for simulation of optical propagation in photonics-Pros and cons,” Proc. SPIE |

15. | T. Erdogan, “Cladding-mode resonances in short and long period fiber grating filters,” J. Opt. Soc. Am. A |

16. | T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. |

**OCIS Codes**

(240.6680) Optics at surfaces : Surface plasmons

(240.6690) Optics at surfaces : Surface waves

(350.2770) Other areas of optics : Gratings

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: November 2, 2012

Revised Manuscript: January 10, 2013

Manuscript Accepted: January 15, 2013

Published: June 3, 2013

**Citation**

Yue Jing He, "Investigation of LPG-SPR sensors using the finite element method and eigenmode expansion method," Opt. Express **21**, 13875-13895 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-12-13875

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### References

- J. Homola, “Optical fiber sensor based on surface plasmon excitation,” Sens. Actuators B Chem.29(1-3), 401–405 (1995). [CrossRef]
- R. Slavík, J. Homola, and J. Čtyroký, “Single-mode optical fiber surface plasmon resonance sensor,” Sens. Actuators B Chem.54(1-2), 74–79 (1999). [CrossRef]
- S.-M. Tseng, K.-Y. Hsu, H.-S. Wei, and K.-F. Chen, “Analysis and experiment of thin metal-clad fiber polarizer with index overlay,” IEEE Photon. Technol. Lett.9(5), 628–630 (1997). [CrossRef]
- Ó. Esteban, R. Alonso, M. C. Navarrete, and A. González-Cano, “Surface plasmon excitation in fiber-optical sensors: a novel theoretical approach,” J. Lightwave Technol.20(3), 448–453 (2002). [CrossRef]
- S. Patskovsky, A. V. Kabashin, M. Meunier, and J. H. Luong, “Silicon-based surface plasmon resonance sensing with two surface plasmon polariton modes,” Appl. Opt.42(34), 6905–6909 (2003). [CrossRef] [PubMed]
- S. Patskovsky, A. V. Kabashin, M. Meunier, and J. H. Luong, “Properties and sensing characteristics of surface-plasmon resonance in infrared light,” J. Opt. Soc. Am. A20(8), 1644–1650 (2003). [CrossRef] [PubMed]
- A. J. C. Tubb, F. P. Payne, R. B. Millington, and C. R. Lowe, “Single-mode optical fibre surface plasma wave chemical sensor,” Sens. Actuators B Chem.41(1-3), 71–79 (1997). [CrossRef]
- S. Maruo, O. Nakamura, and S. Kawata, “Evanescent-wave holography by use of surface-plasmon resonance,” Appl. Opt.36(11), 2343–2346 (1997). [CrossRef] [PubMed]
- Y. J. He, Y. L. Lo, and J. F. Huang, “Optical-fiber surface-plasmon-resonance sensor employing long-period fiber grating in multiplexing,” J. Opt. Soc. Am. B23(5), 801–811 (2006). [CrossRef]
- E. D. Palik, Handbook of Optical Constants of Solids (Academic Press, 1985).
- D. Sun, J. Manges, Xingchao Yuan, and Z. Cendes, “Spurious modes in finite-element methods,” IEEE Antennas Propag. Mag.37(5), 12–24 (1995). [CrossRef]
- C. H. Herry and Y. Shani, “Analysis of mode propagation in optical waveguide devices by Fourier expansion,” IEEE J. Quantum Electron.27(3), 523–530 (1991). [CrossRef]
- G. Sztefka and H. P. Nolting, “Bidirectional eigenmode Propagation for Large refractive index steps,” IEEE Photon. Technol. Lett.5(5), 554–557 (1993). [CrossRef]
- D. F. G. Gallagher and T. P. Felici, “Eigenmode expansion methods for simulation of optical propagation in photonics-Pros and cons,” Proc. SPIE4987, 69–82 (2003). [CrossRef]
- T. Erdogan, “Cladding-mode resonances in short and long period fiber grating filters,” J. Opt. Soc. Am. A14(8), 1760–1773 (1997). [CrossRef]
- T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol.15(8), 1277–1294 (1997). [CrossRef]

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