## Increased surface plasmon resonance sensitivity with the use of double Fourier harmonic gratings

Optics Express, Vol. 16, Issue 16, pp. 11691-11702 (2008)

http://dx.doi.org/10.1364/OE.16.011691

Acrobat PDF (264 KB)

### Abstract

A biomolecular sensor consisting of a thin metallic grating deposited on a glass prism is studied in the formalism of poles and zeros of the scattering matrix. Surface plasmon resonance is used to increase the sensitivity of the device with respect to a variation of the refractive index of the substrate. It is shown that a direct coupling between counter propagating surface plasmons using double-harmonic Fourier gratings leads to an enhancement of the sensitivity. The result of the stronger coupling is the transfer of the working point from the lower to the upper edge of the band gap in the dispersion diagram.

© 2008 Optical Society of America

## 1. Introduction

1. E. Kretschmann, “Determination of optical constants of metals by excitation of surface Plasmons,” Z. Phys. **241**, 313- (1971). [CrossRef]

3. J. J. Cowan and E. T. Arakawa, “Dispersion of surface plasmons in dielectric-metal coatings on concave diffraction gratings,” Z. Phys. **235**, 97- (1970). [CrossRef]

4. D. Maystre and R. C. McPhedran, “A detailed theoretical study of the anomalies of a sinusoidal diffraction grating,” Optica Acta **21**, 413–421 (1974). [CrossRef]

10. F. Pigeon, I. F. Salakhutdinov, and A. V. Tishchenko, “Identity of long-range surface plasmons along asymetric structures and their potential for refractometric sensors,” J. Appl. Phys. **90**, 852–859 (2001) [CrossRef]

10. F. Pigeon, I. F. Salakhutdinov, and A. V. Tishchenko, “Identity of long-range surface plasmons along asymetric structures and their potential for refractometric sensors,” J. Appl. Phys. **90**, 852–859 (2001) [CrossRef]

11. C. J. Alleyne, A. G. Kirk, R. C. McPhedran, N-A. P. Nicorovici, and D. Maystre, “Enhanced SPR sensitivity using periodic metallic structures,” Opt. Express **15**, 8163–8169 (2007). [CrossRef] [PubMed]

13. M. Breidne and D. Maystre, “A systematic numerical study of Fourier gratings”, J. Opt. **13**, 71–79 (1982). [CrossRef]

## 2. Single harmonic Fourier grating

_{1}=1.33), the superstrate is glass (n

_{3}=1.5). The thickness of the metallic layer is taken to be 40 nm. The period of the grating is 0.3 μm. It is illuminated from the glass by a plane wave in Transverse Magnetic polarization (TM), with angle of incidence θ. In this way, it is possible to excite from the 0th order a surface plasmon resonance at the water-metal interface when the condition

*α*

_{i}≡ n

_{3}sinθ =

*α*

_{spp1-2}is satisfied, where

*α*

_{spp1-2}is the normalized propagating constant of the surface plasmon. This surface plasmon will propagate in the Ox direction. It is possible to excite simultaneously from the −1st order a surface plasmon propagating along both metallic interfaces in the −Ox direction when the two equalities: n

_{3}sinθ −

*λ*/d = −

*α*

_{spp1-2}and n

_{3}sin θ −

*λ*/d = −

*α*

_{spp3-2}are satisfied. These surface plasmons propagate on the lower and upper metallic interfaces respectively. The surface plasmon

*α*

_{spp3-2}propagating on the interface between prism and metal will not be modified by a slight variation of the refractive index n

_{1}of the water and is not of interest in this study.

14. J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J.Opt. **11**, 235–241 (1980). [CrossRef]

15. L. Li, “Oblique-coordinate-system-based Chandezon method for modeling one-dimensionally periodic, multilayer, inhomogeneous, anisotropic gratings,” J. Opt. Soc. Am. **16**, 2521–2531 (1999). [CrossRef]

*α*) I = D, where I and D represent column vectors made of the Fourier components of the x and z components of the incident (I) and diffracted (D) electric and magnetic fields. The complex solutions

*α*

_{p}denoting poles are the solutions of the homogeneous problem:

*α*

_{z}represents a zero of the 0th reflected order:

*α*

_{spp}satisfies Eq. (1) and is identified with

*α*

_{p}. When considering perfectly conducting metals,

*α*

_{z}=

*α*

_{p}

^{*}[10

10. F. Pigeon, I. F. Salakhutdinov, and A. V. Tishchenko, “Identity of long-range surface plasmons along asymetric structures and their potential for refractometric sensors,” J. Appl. Phys. **90**, 852–859 (2001) [CrossRef]

**90**, 852–859 (2001) [CrossRef]

*α*

_{i}= n

_{3}sinθ. If the imaginary part of the zero is zero, then the reflected efficiency will be equal to zero.

*α*

_{p}corresponding to the two counter propagating surface plasmons at the water-metal interface as a function of h

_{1}for the refractive indices n

_{1}= 1.33 and 1.34 (water and water infused with a biomolecule). The two counter propagating surface plasmons excited by the 0th and the −1st orders present two different values of

*α*

_{p}, in full and dashed lines in Fig. 2, since two different equations must be satisfied, respectively:

*α*

_{spp1-2}increases.

*α*

_{p}must be decreased to fulfill Eq. (3) and must be increased to fulfill Eq. (4). As a result, the full lines and dashed lines in Fig. 2 that represent respectively the propagation constant of the surface plasmons excited by the −1st and the 0th orders approach each other when increasing the groove depth.

16. D. Maystre and R. Petit, “Brewster incidence for metallic gratings,” Opt. Commun. **17**, 196–200 (1976). [CrossRef]

17. M. C. Hutley and D. Maystre, “The total absorption of light by a diffraction grating,” Opt. Commun. **19**, 431–436 (1976) [CrossRef]

_{1}of the substrate. The difference between real parts in Fig. 3(a) has to be maximized, while the imaginary part of

*α*

_{z}has to be minimized. When increasing h

_{1}, the coupling between the two counter propagating surface plasmons increases. As a result, the constants of propagation tend toward the same value, with an increase of the imaginary part.

_{1}= 1.34 stops to increase further when increasing h

_{1}, in other words when the imaginary part becomes important for n

_{1}= 1.34 and still negligible for n

_{1}=1.33, which happens around h

_{1}= 20 nm. Figure 4 shows the difference between the blue and red dashed lines of Fig. 3a as a function of h

_{1}, together with the imaginary parts of

*α*

_{z}. The sensitivity of the propagation constant of the surface plasmon with the refractive index n

_{1}is maximal for h

_{1}= 0.019 μm. However, the imaginary part of

*α*

_{z}with n

_{1}= 1.34 is too high so that the minimum of reflectivity will not be pronounced, and the width at mid-height will be too large. As a consequence, h

_{1}= 0.017 μm is preferable for the plot of the reflectivity as a function of the angle of incidence (Fig. 5). The two minima of reflectivity obtained with n

_{1}=1.33 and 1.34 are separated by Δθ = 6.25° (using the leftmost dip for n = 1.33 and the rightmost dip for n = 1.34).

## 3. Double Fourier harmonic grating

_{2}. The profile represents a function with equation:

_{1}and h

_{2}has been computed in order to obtain the largest variation of the absorption as a function of θ associated with a very low reflectivity, close to zero. As a result of this, the values h

_{1}= 35 nm, h

_{2}= 59 nm, t = 40 nm and φ = π/2 have been selected. In order to understand why the second modulation permits an enhanced SPR sensitivity, the poles are plotted as a function of the groove depth h defined by the groove geometry (Fig. 6):

_{1}is then at a maximum.

_{1}is maximal and imaginary parts are minimized. As a consequence, in the plot of the reflectivity of the device as a function of the angle of incidence with h=35 nm (Fig. 9), the minima are well pronounced and very close to zero, and they are very well separated, with a variation of Δθ = 10° with Δn = 0.01.

_{min}corresponding to the minimum reflectivity. This can be made by using a diverging incident beam and measuring a response using CCD camera. The sensitivity varies between 0.7 and 1.4 degree per 0.001 Δn, as observed in Fig. 10(b).

18. M. D. Malinsky, K. L. Kelly, G. C. Schatz, and R. P. Van Duyne, “Chain Length Dependence and Sensing Capabilities of the Localized Surface Plasmon Resonance of Silver Nanoparticles Chemically Modified with Alkanethiol Self-Assembled Monolayers,” J. Am. Chem. Soc. **123**, 1471–1482 (2001). [CrossRef]

18. M. D. Malinsky, K. L. Kelly, G. C. Schatz, and R. P. Van Duyne, “Chain Length Dependence and Sensing Capabilities of the Localized Surface Plasmon Resonance of Silver Nanoparticles Chemically Modified with Alkanethiol Self-Assembled Monolayers,” J. Am. Chem. Soc. **123**, 1471–1482 (2001). [CrossRef]

## 4. Dispersion diagrams

*ω*-k diagrams, which show the formation of an

*ω*-gap and its link with the poles of the scattering matrix, discussed in the previous sections. Figure 12a presents the map of the intensity of the 0th reflected order as a function of k

_{x}and

*ω*/c (in μm

^{−1}) for the single-harmonic grating with h

_{1}= 20 nm (h

_{2}= 0), with c standing for the speed of light in vacuum. One can observe the existence of an

*ω*-gap between 7.3 and 7.8 μm

^{−1}. For simplicity, we have neglected the dispersion of the media. The values of the intensity for k

_{x}> 1.5

*ω*/c can be larger than unity, because this region corresponds to angles of incidence larger than 90°, taking into account that the refractive index of the cladding is equal to 1.5. The working wavelength has been fixed in the previous sections to 0.85 μm, and is presented by the horizontal dashed purple line. It lies at the lower boundary of the forbidden gap that is formed due to the interaction between the plasmons, propagating in the opposite direction on the substrate-metal interface. The real parts of their constants of propagation are superposed on the figure (blue thick lines). The corresponding imaginary parts are presented in Fig. 12b. When the real parts of the constants of propagation approach each other (with increasing

*ω*), the interaction leads to a sharp increase of the imaginary part, leading to a formation of the band gap. Further increase of

*ω*leads to a separation of the propagation constants, a decrease of the imaginary part, and a creation of the upper propagation region.

^{−1}is accompanied by a slight curvature of the real parts and is due to the interaction between the plasmon surface waves propagating on the upper and lower surfaces of the metallic layer.

^{−1}in

*ω*/c. Second, the imaginary parts of the propagation constants are larger than in the single-harmonic case. As a result, the working point is shifted from the lower to the upper boundary of the forbidden gap. As observed in Fig. 13a, the upper boundary is flatter than the lower one, so that one can expect stronger sensitivity with respect to the substrate refractive index, which is the case, as shown previously.

## 5. Conclusion

## Acknowledgments

## References and links

1. | E. Kretschmann, “Determination of optical constants of metals by excitation of surface Plasmons,” Z. Phys. |

2. | R. W. Wood, “On a remarkable case of uneven distribution of light in a diffraction grating spectrum,” Philos. Mag. |

3. | J. J. Cowan and E. T. Arakawa, “Dispersion of surface plasmons in dielectric-metal coatings on concave diffraction gratings,” Z. Phys. |

4. | D. Maystre and R. C. McPhedran, “A detailed theoretical study of the anomalies of a sinusoidal diffraction grating,” Optica Acta |

5. | M. Neviere, “The homogeneous problem,” in |

6. | H. Raether, |

7. | M. J. Jory, P. S. Vukusic, and J. R. Sambles, “Development of a prototype gas sensor using surface plasmon resonance on gratings,” Sens. Actuators B |

8. | U. Schroter and D. Heitmann, “Grating couplers for surface plasmons excited on thin metal films in the Kretschmann-Raether configuration,” Phys. Rev. B |

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

10. | F. Pigeon, I. F. Salakhutdinov, and A. V. Tishchenko, “Identity of long-range surface plasmons along asymetric structures and their potential for refractometric sensors,” J. Appl. Phys. |

11. | C. J. Alleyne, A. G. Kirk, R. C. McPhedran, N-A. P. Nicorovici, and D. Maystre, “Enhanced SPR sensitivity using periodic metallic structures,” Opt. Express |

12. | D. Maystre, “General study of grating anomalies from electromagnetic surface modes”, in |

13. | M. Breidne and D. Maystre, “A systematic numerical study of Fourier gratings”, J. Opt. |

14. | J. Chandezon, D. Maystre, and G. Raoult, “A new theoretical method for diffraction gratings and its numerical application,” J.Opt. |

15. | L. Li, “Oblique-coordinate-system-based Chandezon method for modeling one-dimensionally periodic, multilayer, inhomogeneous, anisotropic gratings,” J. Opt. Soc. Am. |

16. | D. Maystre and R. Petit, “Brewster incidence for metallic gratings,” Opt. Commun. |

17. | M. C. Hutley and D. Maystre, “The total absorption of light by a diffraction grating,” Opt. Commun. |

18. | M. D. Malinsky, K. L. Kelly, G. C. Schatz, and R. P. Van Duyne, “Chain Length Dependence and Sensing Capabilities of the Localized Surface Plasmon Resonance of Silver Nanoparticles Chemically Modified with Alkanethiol Self-Assembled Monolayers,” J. Am. Chem. Soc. |

**OCIS Codes**

(050.1950) Diffraction and gratings : Diffraction gratings

(130.6010) Integrated optics : Sensors

(240.6680) Optics at surfaces : Surface plasmons

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: May 20, 2008

Revised Manuscript: July 14, 2008

Manuscript Accepted: July 16, 2008

Published: July 21, 2008

**Virtual Issues**

Vol. 3, Iss. 9 *Virtual Journal for Biomedical Optics*

**Citation**

Nicolas Bonod, Evgeny Popov, and Ross C. McPhedran, "Increased surface plasmon resonance sensitivity with the use of double Fourier harmonic gratings," Opt. Express **16**, 11691-11702 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-16-11691

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### References

- E. Kretschmann, "Determination of optical constants of metals by excitation of surface Plasmons," Z. Phys. 241, 313- (1971). [CrossRef]
- R. W. Wood, "On a remarkable case of uneven distribution of light in a diffraction grating spectrum," Philos. Mag. 4, 396-402 (1902)
- J. J. Cowan and E. T. Arakawa, "Dispersion of surface plasmons in dielectric-metal coatings on concave diffraction gratings," Z. Phys. 235, 97- (1970). [CrossRef]
- D. Maystre and R. C. McPhedran, "A detailed theoretical study of the anomalies of a sinusoidal diffraction grating," Optica Acta 21, 413-421 (1974). [CrossRef]
- M. Neviere, "The homogeneous problem," in Electromagnetic theory of gratings, R. Petit ed. (Springer-Verlag, 1980), ch.5
- H. Raether, Surface Plasmons on Smooth and Rough Surfaces and on Gratings (Springer-Verlag, Berlin, 1988)
- M. J. Jory, P. S. Vukusic, and J. R. Sambles, "Development of a prototype gas sensor using surface plasmon resonance on gratings," Sens. Actuators B 17, 203-209 (1994). [CrossRef]
- U. Schroter and D. Heitmann, "Grating couplers for surface plasmons excited on thin metal films in the Kretschmann-Raether configuration," Phys. Rev. B 60, 4992-4999 (1999). [CrossRef]
- J. Homola, S. S. Yee and G. Gauglitz, "Surface plasmon resonance sensors: review," Sens. Act. B 54, 3-15 (1999) [CrossRef]
- F. Pigeon, I. F. Salakhutdinov, A. V. Tishchenko, "Identity of long-range surface plasmons along asymetric structures and their potential for refractometric sensors," J. Appl. Phys. 90, 852-859 (2001) [CrossRef]
- C. J. Alleyne, A.G. Kirk, R. C. McPhedran, N-A. P. Nicorovici and D. Maystre, "Enhanced SPR sensitivity using periodic metallic structures," Opt. Express 15, 8163-8169 (2007). [CrossRef] [PubMed]
- D. Maystre, "General study of grating anomalies from electromagnetic surface modes", in Electromagnetic Surface Modes, A.D. Boardman, ed. (John Wiley, 1982), Chap. 17.
- M. Breidne and D. Maystre, "A systematic numerical study of Fourier gratings", J. Opt. 13, 71-79 (1982). [CrossRef]
- J. Chandezon, D. Maystre, and G. Raoult, "A new theoretical method for diffraction gratings and its numerical application," J.Opt. 11, 235-241 (1980). [CrossRef]
- L. Li, "Oblique-coordinate-system-based Chandezon method for modeling one-dimensionally periodic, multilayer, inhomogeneous, anisotropic gratings," J. Opt. Soc. Am. 16, 2521-2531 (1999). [CrossRef]
- D. Maystre and R. Petit, "Brewster incidence for metallic gratings," Opt. Commun. 17, 196-200 (1976). [CrossRef]
- M. C. Hutley and D. Maystre, "The total absorption of light by a diffraction grating," Opt. Commun. 19, 431-436 (1976) [CrossRef]
- M. D. Malinsky, K. L. Kelly, G. C. Schatz, and R. P. Van Duyne, "Chain Length Dependence and Sensing Capabilities of the Localized Surface Plasmon Resonance of Silver Nanoparticles Chemically Modified with Alkanethiol Self-Assembled Monolayers," J. Am. Chem. Soc. 123, 1471-1482 (2001). [CrossRef]

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