## Long range surface plasmons on asymmetric suspended thin film structures for biosensing applications |

Optics Express, Vol. 18, Issue 18, pp. 19009-19019 (2010)

http://dx.doi.org/10.1364/OE.18.019009

Acrobat PDF (1061 KB)

### Abstract

We show that long-range surface plasmons (LRSPs) are supported in a physically asymmetric thin film structure, consisting of a low refractive index medium on a metal slab, supported by a high refractive index dielectric layer (membrane) over air, as a suspended waveguide. For design purposes, an analytic formulation is derived in 1D yielding a transcendental equation that ensures symmetry of the transverse fields of the LRSP within the metal slab by constraining its thicknesses and that of the membrane. Results from the formulation are in quantitative agreement with transfer matrix calculations for a candidate slab waveguide consisting of an H_{2}O-Au-SiO_{2}-air structure. Biosensor-relevant figures of merit are compared for the asymmetric and symmetric structures, and it is found that the asymmetric structure actually improves performance, despite higher losses. The finite difference method is also used to analyse metal stripes providing 2D confinement on the structure, and additional constraints for non-radiative LRSP guiding thereon are discussed. These results are promising for sensors that operate with an aqueous solution that would otherwise require a low refractive index-matched substrate for the LRSP.

© 2010 OSA

## 1. Introduction

1. W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature **424**(6950), 824–830 (2003). [CrossRef] [PubMed]

2. B. Liedberg, C. Nylander, and I. Lundstrom, “Surface plasmon resonance for gas detection and biosensing,” Sens. Actuators **4**, 299–304 (1983). [CrossRef]

10. J. Homola, “Surface plasmon resonance sensors for detection of chemical and biological species,” Chem. Rev. **108**(2), 462–493 (2008). [CrossRef] [PubMed]

11. P. Berini, “Plasmon-polariton waves guided by thin lossy metal films of finite width: Bound modes of symmetric structures,” Phys. Rev. B **61**(15), 10484–10503 (2000). [CrossRef]

23. A. V. Krasavin and A. V. Zayats, “Three-dimensional numerical modeling of photonic integration with dielectric-loaded SPP waveguides,” Phys. Rev. B **78**(4), 045425 (2008). [CrossRef]

24. P. Berini, “Long-range surface plasmon-polaritons,” Adv. Opt. Phot. **1**(3), 484–588 (2009). [CrossRef]

8. P. Berini, “Bulk and surface sensitivities of surface plasmon waveguides,” N. J. Phys. **10**(10), 105010 (2008). [CrossRef]

24. P. Berini, “Long-range surface plasmon-polaritons,” Adv. Opt. Phot. **1**(3), 484–588 (2009). [CrossRef]

*s*and

_{b}*ss*modes respectively [24

_{b}^{0}24. P. Berini, “Long-range surface plasmon-polaritons,” Adv. Opt. Phot. **1**(3), 484–588 (2009). [CrossRef]

25. D. Sarid, “Long-range surface-plasma waves on very thin metal-films,” Phys. Rev. Lett. **47**(26), 1927–1930 (1981). [CrossRef]

30. I. Breukelaar, R. Charbonneau, and P. Berini, “Long-Range Surface Plasmon-Polariton Mode Cutoff and Radiation in Embedded Strip Waveguides,” J. Appl. Phys. **100**(4), 043104 (2006). [CrossRef]

31. A. W. Wark, H. J. Lee, and R. M. Corn, “Long-range surface plasmon resonance imaging for bioaffinity sensors,” Anal. Chem. **77**(13), 3904–3907 (2005). [CrossRef] [PubMed]

34. R. Daviau, E. Lisicka-Skrzek, R. N. Tait, and P. Berini, “Broadside excitation of surface plasmon waveguides on Cytop,” Appl. Phys. Lett. **94**(9), 091114 (2009). [CrossRef]

35. R. Slavík and J. Homola, “Optical multilayers for LED-based surface plasmon resonance sensors,” Appl. Opt. **45**(16), 3752–3759 (2006). [CrossRef] [PubMed]

_{2}O

_{5}bi-layer system was used to match (effectively) the index of the aqueous environment on the other side of an Au slab. A second approach involves combining a finite 1D photonic crystal, which can support Bloch surface waves in the bandgap, with an Au slab (and the bounding medium on the other side) such that a symmetric LRSP field distribution is achieved throughout the structure [36

36. V. N. Konopsky and E. V. Alieva, “Long-range propagation of plasmon polaritons in a thin metal film on a one-dimensional photonic crystal surface,” Phys. Rev. Lett. **97**(25), 253904 (2006). [CrossRef]

37. P. Berini, R. Charbonneau, and N. Lahoud, “Long-range surface plasmons on ultrathin membranes,” Nano Lett. **7**(5), 1376–1380 (2007). [CrossRef] [PubMed]

38. P. Berini, R. Charbonneau, and N. Lahoud, “Long-range surface plasmons along membrane-supported metal stripes,” IEEE J. Sel. Top. Quantum Electron. **14**(6), 1479–1495 (2008). [CrossRef]

8. P. Berini, “Bulk and surface sensitivities of surface plasmon waveguides,” N. J. Phys. **10**(10), 105010 (2008). [CrossRef]

39. P. Berini, “Figures of merit for surface plasmon waveguides,” Opt. Express **14**(26), 13030–13042 (2006). [CrossRef] [PubMed]

_{2}O - Au - SiO

_{2}- air suspended structure, which has a potential advantage over the membrane waveguide of Refs. [37

37. P. Berini, R. Charbonneau, and N. Lahoud, “Long-range surface plasmons on ultrathin membranes,” Nano Lett. **7**(5), 1376–1380 (2007). [CrossRef] [PubMed]

38. P. Berini, R. Charbonneau, and N. Lahoud, “Long-range surface plasmons along membrane-supported metal stripes,” IEEE J. Sel. Top. Quantum Electron. **14**(6), 1479–1495 (2008). [CrossRef]

## 2. LRSP along an IMII slab structure (1D)

### 2.1 Geometry of the IMII slab waveguide

*x*and

*y*directions. The LRSP is assumed to propagate along the +

*x*-direction at a vacuum wavelength of

*λ*=1310 nm according to e

_{0}^{+j}

*(e*

^{βx}^{-j}

*time dependence assumed). The H*

^{ωt}_{2}O region is semi-infinite and its relative permittivity is (1.3159 +

*j*1.639 × 10

^{−5})

^{2}. The second layer is a Au slab, which has a relative permittivity of −86.08 +

*j*8.322 and a thickness

*t*. Beneath the Au slab is a supporting dielectric membrane of thickness

*d*, such as SiO

_{2}having a relative permittivity of 2.0932. The bottom semi-infinite layer is air, having a relative permittivity of 1. (The values of relative permittivity originate from Ref. [38

38. P. Berini, R. Charbonneau, and N. Lahoud, “Long-range surface plasmons along membrane-supported metal stripes,” IEEE J. Sel. Top. Quantum Electron. **14**(6), 1479–1495 (2008). [CrossRef]

_{2}layer is thick enough to be mechanically stable and deposited without too much compressive strain. It can be fabricated by etching through an underlying Si substrate (not shown) to release the SiO

_{2}layer, thereby creating a free-standing SiO

_{2}membrane supported around its perimeter [37

37. P. Berini, R. Charbonneau, and N. Lahoud, “Long-range surface plasmons on ultrathin membranes,” Nano Lett. **7**(5), 1376–1380 (2007). [CrossRef] [PubMed]

_{2}O interface is high, based on results obtained for similar structures [8

8. P. Berini, “Bulk and surface sensitivities of surface plasmon waveguides,” N. J. Phys. **10**(10), 105010 (2008). [CrossRef]

### 2.2 Theoretical model of the IMII: restored LRSP symmetry in the metal

*d*that restores the symmetry of the transverse LRSP mode fields within the metal slab. Practically, this is achieved by enforcing that the transverse magnetic field (and transverse electric field) be identical along its boundaries, as sketched in Fig. 1(b). Although field symmetry over the full cross-section is not achieved, symmetry within the metal slab is achieved, thus capturing an IMII configuration that supports a LRSP; this LRSP shall henceforth be referred to as the “

*symmetry-constrained LRSP*”.

*H*, for a TM mode:

_{i·y}*E*is obtained from:

_{i·x}*A*,

*B*,

*C*,

*D*,

*F*,

*G*are the amplitude coefficients for

*H*and

_{i·y}*E*at the boundaries. Subscripts 1-4 indicate the H

_{i·x}_{2}O, Au, SiO

_{2}and air layers, respectively, and

*k*= (

_{i}*β*

^{2}-

*k*

_{0}^{2}

*ε*)

_{i}^{1/2}where

*k*represents the vacuum wavenumber,

_{0}*ε*is the complex relative permittivity of the

_{i}*i*

^{th}layer and

*β*is the complex TM mode wavenumber. Then by applying the boundary conditions (tangential fields must match) to Eqs. (1) and (2), we obtain the

*β*-

*d*-

*t*transcendental equation for TM modes on this structure:where

*ε*,

_{i}*β*} = 0) mode cut-off occurs when

*β*=

*k*

_{0}ε_{1}^{1/2}(mode wavenumber equals the wavenumber of plane waves in H

_{2}O). Substituting this condition into Eq. (3) yields the following equation which constrains

*d*and

*t*at the cut-off point of the TM modes:As mentioned above, the transverse magnetic field of the LRSP must be identical along the boundaries of the metal slab if the field is to be symmetric therein; this implies

*B*=

*C*. Adding this constraint to the boundary conditions (tangential fields must match), and applying them to Eqs. (1) and (2) yields:and:

*B*=

*C*yields two transcendental equations, instead of one (Eq. (3)) as is usually the case; the system is overspecified with this constraint, and to remain consistent, the Eq. (5) and (6) must be simultaneously satisfied to provide the symmetry-constrained LRSP solution.Eq. (6) is generated from the field solutions in the H

_{2}O and Au layers only (

*H*,

_{1y}*H*,

_{2y}*E*,

_{1x}*E*), and is the same as for the LRSP of the symmetric IMI [26

_{2x}26. J. J. Burke, G. I. Stegeman, and T. Tamir, “Surface-polariton-like waves guided by thin, lossy metal films,” Phys. Rev. B Condens. Matter **33**(8), 5186–5201 (1986). [CrossRef] [PubMed]

*ε*,

_{i}*β*} = 0). A solution strategy then consists of first solving Eq. (6) using a root-finding algorithm to determine

*β*for a value of

*t*, and then inserting both values into Eq. (5) to determine

*d*directly. Equations (5) and (6) are thus useful for generating

*t*,

*d*pairs for which the LRSP is symmetry-constrained. Equation (3) always holds (including losses) so the complex mode wavenumber (including attenuation) can then be readily determined for a particular design.

*t*and

*d*satisfying Eq. (5) for the LRSP in the IMII considered (H

_{2}O - Au - SiO

_{2}- air at 1310 nm) in the lossless case, and so corresponds to designs for which the LRSP is symmetry-constrained. Note that the thinner the Au slab, the thicker the SiO

_{2}layer must be to compensate for the index discrepancy between H

_{2}O and air. For example, for a 50 nm thick Au slab, a 342 nm thick SiO

_{2}layer is needed, but for a 20 nm thick Au slab, a 382 nm thick SiO

_{2}layer is needed. The other curve in Fig. 2(a) corresponds to cut-off configurations obtained using Eq. (4), below which there is no purely bound LRSP. The symmetry-constrained designs are above cut-off but as the metal slab thickness decreases, cut-off is approached.

*n*) of the symmetry-constrained LRSP computed using Eq. (5), which increases with Au thickness as expected, approaching limiting values when the Au thickness is larger than ~100 nm. When the Au layer is thinner, the effective index is close to the refractive index of H

_{eff}_{2}O.

### 2.3 Computations using the transfer matrix method

40. C. Chen, P. Berini, D. Feng, S. Tanev, and V. Tzolov, “Efficient and accurate numerical analysis of multilayer planar optical waveguides in lossy anisotropic media,” Opt. Express **7**(8), 260–272 (2000). [CrossRef] [PubMed]

*d*= 341.7 nm and its effective index and attenuation (TMM) are 1.324633 and 31.39 dB/mm, respectively. When the Au slab is 20 nm thick, the symmetry-constrained LRSP occurs at

*d*= 381.5 nm and its effective index and attenuation (TMM) are 1.3182884 and 3.26 dB/mm, respectively.

_{2}O interface for increasing

*d*, and to the Au/SiO

_{2}interface for decreasing

*d*. The LRSP tends towards cut-off with decreasing

*d*, explaining the decreasing effective index.

*n*/∂

_{eff}*n*) and surface (∂

_{c}*n*/∂

_{eff}*a*) sensitivities of the LRSP computed using the TMM.

*n*denotes the refractive index of the carrier fluid, in this case H

_{c}_{2}O, and

*a*denotes the thickness of a thin biochemical adlayer of refractive index 1.5 placed on the metal slab at the Au/H

_{2}O interface [8

**10**(10), 105010 (2008). [CrossRef]

*d*follow the trends of the mode fields described in the preceding paragraph. In the case of the

*t*= 20 nm thick Au slab, the bulk sensitivity increases as

*d*decreases below the symmetry-constrained design because the mode nears cut-off, and as it does its fields extend deeply into the higher index medium which is H

_{2}O. The sensitivities are larger for the

*t*= 50 nm slab because of the stronger mode confinement.

*G*and the

*M*figure of merit [8

_{2}**10**(10), 105010 (2008). [CrossRef]

39. P. Berini, “Figures of merit for surface plasmon waveguides,” Opt. Express **14**(26), 13030–13042 (2006). [CrossRef] [PubMed]

_{2}membrane thicknesses, calculated using the TMM.

*G*is the ratio of the surface sensitivity to the normalised attenuation (

*k*), and is particularly relevant for surface sensing [8

_{eff}**10**(10), 105010 (2008). [CrossRef]

*M*measures the confinement-to-loss ratio, where confinement is defined as the mode’s distance from the light-line [39

_{2}39. P. Berini, “Figures of merit for surface plasmon waveguides,” Opt. Express **14**(26), 13030–13042 (2006). [CrossRef] [PubMed]

*G*and

*M*are maximised at the same membrane thickness, which is fairly close to the symmetry-constrained design. For example, when the Au slab thickness is

_{2}*t*= 50 nm,

*G*and

*M*peak at

_{2}*d*= 350 nm (the symmetry-constrained membrane thickness in this case is

*d*= 341.7 nm). Thus the symmetry-constrained structure generates good designs for sensing applications.

*H*fields, showing that they are identical for

_{y}*z*> 0,

*i.e.*, within the metal slab and in the H

_{2}O region. The field below the metal slab in the case of the IMII is more confined due to the SiO

_{2}and Air layers. Table 1 summarises the modal quantities for

*t*= 20 and 50 nm. The effective indices are similar, though not identical, but the attenuation of the LRSP in the IMII is significantly larger. Although the field distribution in the metal is nearly identical to that of the corresponding IMI (Fig. 3(d)), the mode overlap with the metal is stronger because of the increased confinement in the SiO

_{2}and air regions, leading to the larger attenuation. The overlap with the biochemical adlayer is also larger for the LRSP on the IMII (due to the same reason) leading to a larger surface sensitivity. Despite the higher losses of the IMII structure, the figure of merit

*G*is actually improved for the IMII structures as compared to the IMI structures, which favours their use for biosensor applications. (Only one adlayer was assumed for the IMI in the sensitivity calculations, which is more practical for microfluidic implementation.)

_{2}O layer of finite thickness and an additional air top-layer, we also calculated the effective index (

*n*) with different H

_{eff}_{2}O layer thicknesses for the 50 nm gold layer symmetry-constrained configuration. For a distance of one free-space wavelength (1310 nm), the real part of the effective index of the mode shows only 0.4% change as compared to the semi-infinite H

_{2}O layer (shown in Table 1). The percentage change then drops by an order of magnitude for each additional wavelength in thickness.

### 2.4 Comparison with TE and TM dielectric waveguide modes

**14**(6), 1479–1495 (2008). [CrossRef]

_{2}O - Au - SiO

_{2}- air at 1310 nm), and of the TE

_{0}and TM

_{0}modes in the corresponding dielectric slab waveguide. When the thickness of the membrane (

*d*) is less than 390 nm, the LRSP has a larger effective index than the TE

_{0}and TM

_{0}modes. From Fig. 2(a), this range of thickness (

*d*< 390 nm) corresponds to an Au slab thickness (

*t*) that is greater than 18 nm. It is therefore expected that the corresponding metal stripe waveguide will support a purely bound (non-radiative) symmetry-constrained LRSP for

*d*< 390 nm and

*t*> 18 nm (approximately). Note that this additional constraint leads to a lower bound on the Au thickness, and thus a lower bound on the achievable attenuation.

_{3}N

_{4}as the membrane are also computed, as shown in Fig. 4(b). In this case, the effective index of the TE

_{0}mode is larger than that of the symmetry-constrained LRSP over the entire range of membrane thickness, suggesting that Si

_{3}N

_{4}is not a good choice for the corresponding metal stripe waveguide.

## 3. LRSP along an IMII stripe structure (2D)

### 3.1 Geometry of the IMII stripe waveguide

*w*provides additional confinement in the

*y*dimension, and propagation occurs along the stripe in the

*x*dimension. The same material parameters and operating wavelength (as in Subsection 2.1) are retained and the width of the Au stripe is selected as being

*w*= 5 μm. Due to the geometrical complexity of the structure, theoretical and TMM approaches are not suitable for analysis, so the finite-difference method (FDM) implemented in commercial software [41

41. Lumerical Solutions, Inc., http://www.lumerical.com.

### 3.2 Finite difference results

*d*,

*t*) (also observed in [38

**14**(6), 1479–1495 (2008). [CrossRef]

*d*) varies (also observed in [38

**14**(6), 1479–1495 (2008). [CrossRef]

*d*~330 and 380 nm in the case of the 50 and 20 nm thick Au stripes, respectively (as shown in Fig. 7 ), similarly to the slab waveguide (Fig. 4). One reason for the similarity is the large width selected for the stripe (

*w*= 5 μm) which reduces the interaction of the LRSP fields with the stripe corners. Another reason is the relatively small contrast between the selected membrane material (SiO

_{2}) and H

_{2}O. Indeed assuming Cytop [34

34. R. Daviau, E. Lisicka-Skrzek, R. N. Tait, and P. Berini, “Broadside excitation of surface plasmon waveguides on Cytop,” Appl. Phys. Lett. **94**(9), 091114 (2009). [CrossRef]

_{2}O) renders the slab and stripe results essentially indistinguishable. Also, the lower the index of the membrane the larger the design space for the stripe waveguide because radiation into the TE

_{0}and TM

_{0}modes of the dielectric slab (on either side of the stripe) occurs over a smaller range of membrane thickness.

*H*) of the LRSP over the cross-section of a stripe IMII with symmetry-constrained thicknesses as computed by the FDM. For a Au stripe thickness of

_{y}*t*= 50 nm, the symmetry-constrained LRSP occurs for a membrane thickness of

*d*= 330 nm, and for

*t*= 20 nm, then

*d*= 380 nm. The

*H*field along the top boundary of the Au stripe is identical to that along its bottom boundary as observed.

_{y}## 5. Summary and concluding remarks

_{2}O-Au-SiO

_{2}-air suspended structure. We found that the symmetry-constrained LRSP exhibits fairly low attenuation at a particular metal slab thickness. Furthermore, its attenuation and confinement decreased with decreasing metal thickness, following the conventional IMI. We also found that the thinner the metal slab, the thicker the dielectric layer needs to be to satisfy the symmetry constraint because the LRSP extends further into the dielectric for a thinner metal slab. As expected, the LRSP in this IMII waveguide has a high surface sensitivity as well as high figures of merit (

*G*for surface sensing and

*M*for waveguiding) supporting its potential for sensing applications.

_{2}_{0}and TM

_{0}modes supported by the dielectric slabs on either side of the stripe to ensure no lateral radiation into these modes. This constraint places upper bounds on the thickness and the refractive index of the membrane, and on how thin the stripe may be, thereby also placing a lower bound on the LRSP attenuation. Despite these constraints, practical stripe designs have been found for the candidate structure (

*w*= 5 μm,

*t*= 20 nm,

*d*= 380 nm, yielding a symmetry-constrained LRSP attenuation of ~3.27 dB/mm at

*λ*= 1310 nm).

_{0}## Acknowledgements

## References and links

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

2. | B. Liedberg, C. Nylander, and I. Lundstrom, “Surface plasmon resonance for gas detection and biosensing,” Sens. Actuators |

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

4. | F.-C. Chien and S.-J. Chen, “A sensitivity comparison of optical biosensors based on four different surface plasmon resonance modes,” Biosens. Bioelectron. |

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7. | K. A. Tetz, L. Pang, and Y. Fainman, “High-resolution surface plasmon resonance sensor based on linewidth-optimized nanohole array transmittance,” Opt. Lett. |

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23. | A. V. Krasavin and A. V. Zayats, “Three-dimensional numerical modeling of photonic integration with dielectric-loaded SPP waveguides,” Phys. Rev. B |

24. | P. Berini, “Long-range surface plasmon-polaritons,” Adv. Opt. Phot. |

25. | D. Sarid, “Long-range surface-plasma waves on very thin metal-films,” Phys. Rev. Lett. |

26. | J. J. Burke, G. I. Stegeman, and T. Tamir, “Surface-polariton-like waves guided by thin, lossy metal films,” Phys. Rev. B Condens. Matter |

27. | L. Wendler and R. Haupt, “Long-range surface plasmon-polaritons in asymmetric layer structures,” J. Appl. Phys. |

28. | F. Yang, J. R. Sambles, and G. W. Bradberry, “Long-range surface modes supported by thin films,” Phys. Rev. B Condens. Matter |

29. | P. Berini, “Plasmon-Polariton Waves Guided by Thin Lossy Metal Films of Finite Width: Bound Modes of Asymmetric Structures,” Phys. Rev. B |

30. | I. Breukelaar, R. Charbonneau, and P. Berini, “Long-Range Surface Plasmon-Polariton Mode Cutoff and Radiation in Embedded Strip Waveguides,” J. Appl. Phys. |

31. | A. W. Wark, H. J. Lee, and R. M. Corn, “Long-range surface plasmon resonance imaging for bioaffinity sensors,” Anal. Chem. |

32. | R. Slavík and J. Homola, “Ultrahigh resolution long range surface plasmon-based sensor,” Sens. Actuators B Chem. |

33. | J. Dostálek, A. Kasry, and W. Knoll, “Long Range Surface Plasmons for Observation of Biomolecular Binding Events at Metallic Surfaces,” Plasmonics |

34. | R. Daviau, E. Lisicka-Skrzek, R. N. Tait, and P. Berini, “Broadside excitation of surface plasmon waveguides on Cytop,” Appl. Phys. Lett. |

35. | R. Slavík and J. Homola, “Optical multilayers for LED-based surface plasmon resonance sensors,” Appl. Opt. |

36. | V. N. Konopsky and E. V. Alieva, “Long-range propagation of plasmon polaritons in a thin metal film on a one-dimensional photonic crystal surface,” Phys. Rev. Lett. |

37. | P. Berini, R. Charbonneau, and N. Lahoud, “Long-range surface plasmons on ultrathin membranes,” Nano Lett. |

38. | P. Berini, R. Charbonneau, and N. Lahoud, “Long-range surface plasmons along membrane-supported metal stripes,” IEEE J. Sel. Top. Quantum Electron. |

39. | P. Berini, “Figures of merit for surface plasmon waveguides,” Opt. Express |

40. | C. Chen, P. Berini, D. Feng, S. Tanev, and V. Tzolov, “Efficient and accurate numerical analysis of multilayer planar optical waveguides in lossy anisotropic media,” Opt. Express |

41. | Lumerical Solutions, Inc., http://www.lumerical.com. |

**OCIS Codes**

(130.2790) Integrated optics : Guided waves

(240.6680) Optics at surfaces : Surface plasmons

(280.1415) Remote sensing and sensors : Biological sensing and sensors

**ToC Category:**

Optics at Surfaces

**History**

Original Manuscript: July 23, 2010

Revised Manuscript: August 18, 2010

Manuscript Accepted: August 18, 2010

Published: August 20, 2010

**Virtual Issues**

Vol. 5, Iss. 13 *Virtual Journal for Biomedical Optics*

**Citation**

Qiao Min, Chengkun Chen, Pierre Berini, and Reuven Gordon, "Long range surface plasmons on asymmetric suspended thin film structures for biosensing applications," Opt. Express **18**, 19009-19019 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-18-19009

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### References

- W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003). [CrossRef] [PubMed]
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