Abstract

A numerical study is presented of surface plasmon waves excitation in a metal film applied to the cladding of a standard bent single-mode optical fiber. It was shown that by adjusting the bend radius and metal film thickness one can achieve effective coupling between the fiber fundamental mode and symmetric surface plasmon mode through the intermediary of whispering gallery modes supported by the cladding of the bent fiber. This effect is demonstrated to allow for refractometric measurement both in the wavelength and intensity-modulated regimes with a resolution of up to 10−8 RIU. Usage of standard noise reduction techniques for intensity-modulated optical signals promises further increase in accuracy.

© 2014 Optical Society of America

1. Introduction

One of the major trends in the development of biosensing technologies nowadays deals with refractometric sensors based on surface plasmon resonance (SPR) [14]. Thanks to their extreme sensitivity and the ability to dispense with fluorescent markers, SPR-sensors are becoming a universally accepted tool to study biomolecular interactions and finding increasing application in the detection of chemical and biological analytes [1, 3, 4]. SPR-sensors make use of surface plasmon waves propagating along a metal - dielectric interface to measure ultra-small variations in the dielectric refractive index in the vicinity of the interface with resolutions down to 10−7 - 10−8 RIU (refractive index units) [46]. As a rule SPR-sensors are built around the prism-based Kretschmann configuration, diffraction gratings, integrated or fiber waveguides [46]. Among various types of SPR sensors fiber optic based configurations appear particularly promising due to their inherent advantages such as variable gauge length, no need for mechanical adjustment of moving parts, miniaturization and remote sensing capabilities, as well as the potential to drive down the costs of biosensing systems [1, 5, 6]. The best examples of fiber optic SPR-sensors use single-mode (SM) fibers [1, 6]. However to fabricate such sensors a part of the fiber cladding needs to be removed either chemically or mechanically, which brings down the reliability and longevity of the sensor, as well as leads to certain technological difficulties. Another major drawback of fiber optic SPR-sensors is typically a lower resolution (10−5-10−6) as compared to bulk systems, e.g. based on Kretschmann configuration [1]. Thus a search for novel configuration of single-mode optical fiber-based SPR-sensors free from these limitations appears to be timely and of considerable importance.

In Ref [7]. it was shown that surface plasmon resonance can be excited in a standard single-mode fiber without breaking its structural integrity by applying a metal film straight upon its optical cladding. The interaction of the fundamental mode guided by the fiber core with the metal film is realized in this case indirectly, through the intermediary of cladding modes of the bent fiber. This phenomenon can be used as the basis for precision single-mode SPR-sensors of a new type. However, for an adequate description of the operation of such sensors a detailed analysis of light propagation in bent SM optical fibers with metalized cladding is required, which is the subject of this paper.

2. Methodology and results

The object under investigation is a standard SMF28-type single-mode optical fiber made up of a silica glass core and cladding, and a polymer coating with the following refractive indexes: n1 = 1.4504, n2 = 1.4447, n3~1.5, and outer radii: ρ1 = 4.15 um, ρ2 = 62.5 um, ρ3 = 125 um, respectively. A section of the fiber is stripped off the polymer jacket and coated with a thin silver film, after which the fiber is bent with a constant bend radius. Then the bent section is immersed in a medium with a refractive index to be measured n0, which is supposed to be between 1.3 and 1.44. Light is launched into and output from the bent segment 2 through the straight sections of the same fiber – 1 and 3 (Fig. 1). The analysis of light propagation in the presented structure is performed numerically using the eigenmode expansion method (EME) [8]. For simplicity the cylindrical step-index SM fiber is approximated with an equivalent 2D slab waveguide with a graded index profile calculated using the effective-index method (Fig. 1, inset 1) [9]. Such an approximation inevitably leaves out some modes of section 2, but greatly simplifies calculation, while keeping intact the physical essence of the processes under study [9].

 figure: Fig. 1

Fig. 1 Schematic of the waveguiding structure under study and its modes: 1 – input straight waveguide section, 2 – bent waveguide section, 3 – output straight waveguide section, 4 – waveguide core, 5 – waveguide cladding, 6 – polymer jacket, 7 – silver film, 8 – electric field amplitude profile of one of the two modes of section 2 responsible for coupling light guided by the core to surface plasmons (the profile of the second mode is similar to this one and not shown), 9 – electric field profile of the fundamental mode of sections 1 and 3. Inset 1 – refractive index profile of a standard SM fiber (n1) and effective graded index profile of the equivalent slab waveguide n’1. Inset 2 – electric field profile of the fundamental mode of the bent SM waveguide with an infinite cladding. Inset 3 - electric field profile of a cladding whispering gallery mode of the bent waveguide. Inset 4 - electric field profile of the symmetric surface plasmon mode guided by the metal - surrounding medium interface.

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Commercial mode-solving software by Lumerical Solutions, Inc. based on a finite-difference algorithm was used to find the modes supported by each of the waveguide sections. The numerical results show that, as expected, sections 1 and 3 have only one guided fundamental mode (FM) (Fig. 1, curve 9), while the metalized section 2 due to the total internal reflection at the outer surface of the cladding (although somewhat attenuated by the metal film) supports many modes. However only two among them are excited with considerable efficiency – the ones whose profiles are similar in the core region to the straight waveguide fundamental mode. At the resonant SPR wavelength the profiles of these modes acquire a sharp peak at the metal - surrounding medium interface and a large (15-20 dB) increase in attenuation. The characteristic electric field profile of one of these modes is shown in Fig. 1, curve 8.

Due to ohmic losses the modes of section 2 do not obey the conjugated form of the orthogonality relation [10, 11]. Therefore when analyzing the coupling of guided light from section 1 to section 2, excitation coefficients for the modes of section 2 are calculated from the following equation:

ap=1NpE1×Hp·zdx
where a more general unconjugated form of the orthogonality relation is used: Ep×Hpzdx=δpqNp [11]. Here E1 is the electric field of the fundamental mode of section 1, Np, Ep, Hp are, respectively, normalization, electric and magnetic fields of section 2 modes, δpq – Kronecker delta, z – unit vector along the fiber axis. An equation similar to Eq. (1) is used to compute the excitation coefficient for FM in section 3.

As a result of numerical analysis a power ratio between fundamental modes in sections 1 and 3 was computed, which gives the transmission coefficient of the whole structure under study. The spectral dependence of this coefficient is shown in Fig. 2(a) for the case when the waveguide is bent in a full circle i.e. bend length is L = 2πR, where R is the bend radius. As one might expect the transmission spectrum of the waveguiding structure features a resonant dip indicating the excitation of surface plasmon resonance. According to the simulation results, the resonant wavelength of the dip λSPR can be controlled by adjusting the bend radius. For example, it falls within the standard telecommunication C-band (1530–1565 nm) for a bend radius of 7.5 mm. However for the demonstration of the proposed technique we chose U-band (1625–1675 нм), R = 7.1 mm since the sensitivity of the resonant wavelength to the surrounding medium refractive index (SPR/dn0) here is slightly (10-15%) higher than that at R = 7.5 mm.

 figure: Fig. 2

Fig. 2 Calculated transmission coefficient of the studied structure: a – transmission spectra computed at two values of n0 (1.4216 and 1.4226, respectively) for d = 20 nm (1 and 2), 30 nm (3 and 4) and 40 nm (5 and 6). In the inset: SPR wavelength dependence on the refractive index n0 for d = 30 nm; b – transmission coefficient vs. n0 at a fixed wavelength λ = 1.66 um; dn1 and dn2 dependences on n0.

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The depth and spectral selectivity of the dip, as well as the sensitivity SPR/dn0 depend strongly on the metal film thickness d. By running multiple optimization routines it was numerically worked out that the maximum values of these parameters are attained at d = 30 nm. This is illustrated in Fig. 2(a), which shows the calculated transmission spectra for two values of refractive index n0 (1.4216 and 1.4226) when d = 20, 30 and 40 nm, R = 7.1 mm. The inset of Fig. 2(a) demonstrates the calculation results for the resonant wavelength dependence on the refractive index of the surrounding medium for the optimum value of metal film thickness d = 30nm. As one can see, this dependence is nearly linear, with the SPR wavelength shift amounting to ~80 nm per 4•10−3 change in n0, which corresponds to a sensitivity of ~20 um/RIU. It is higher than that of many known fiber optic SPR sensors and only slightly lower than that of the best of them [1, 2, 5]. Furthermore, the SPR spectral dip turns out to be much narrower than that in many other fiber-based SPR-refractometers, which potentially may bring about additional increase in refractometric measurement resolution in the wavelength modulated detection regime.

Further analysis reveals that for the waveguiding structure under study even more promising appears to be intensity-modulated detection regime. Figure 2(b) shows the dependence of the transmission coefficient on the surrounding medium refractive index at a fixed wavelength λ = 1.66 um for d = 30 nm.

As one can see, an extreme variation in transmission is observed reaching 60 dB for ~1.5•10−3 change in the refractive index. The intensity sensitivity can be increased even further by extending the length of the metalized waveguide section. However the simulation results indicate that for 1.5 and more waveguide loops the dependence of transmission on the refractive index may become nonmonotonous near the maximum attenuation due to the noises arising from other modes of section 2, which impedes refractometric measurement by the proposed method. So the optimal length of the bent section is chosen to be one full loop with the bend radius of 7.1 mm.

3. Discussion

Insight into the processes behind the formation of a narrow dip in the transmission spectrum of the studied waveguiding structure is provided by the coupled-mode theory. In its context the modes of section 2 can be viewed as result of coupling between three modes: the fundamental mode of a bent SM waveguide with an infinite cladding (Fig. 1, inset 2); whispering gallery mode (WGM) supported by the bent waveguide’s cladding (Fig. 1, inset 3); and the symmetric surface plasmon mode (SPM) guided along the boundary between the metal and surrounding medium (Fig. 1, inset 4). Figure 3 shows the calculated dispersion curves for these three modes at a fixed value of n0 around 1.42.

 figure: Fig. 3

Fig. 3 Dispersion curves for the fundamental mode of the bent SM waveguide with an infinite cladding, WGM, and SPM: 1 – schematic representation of the spectral region where WGM is effectively coupled to SPM; 2 - schematic representation of the spectral region where FM is effectively coupled to WGM. Inset 1 – electric field profile of section 2 mode at λ1 ≅ 1.43 um, which can be interpreted as a result of coupling between WGM and SPM. Inset 2 - electric field profile of section 2 mode at λ2 ≅ 1.61 um, which can be interpreted as FM – WGM – SPM coupling.

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As seen from Fig. 3, at λ1 ≅ 1.43 um the propagation constants of WGM and SPM become equal which implies resonant coupling between these modes. The numerical results show that, the spectral range around λ1 where effective energy transfer occurs from WGM to SPM is defined by the metal film thickness. It is known that in the Kretschmann configuration using metal films thinner than 45-50 nm leads to broadening of the spectral (or angular) range in which the incident light energy is coupled to surface plasmons [5]. In a similar way, for the 30-nm-thick metal film, which is found to be optimal for our configuration, the coupling between WGM and SPM takes place in a wide spectral range around λ1 (region 1 in Fig. 3) including the wavelength λ2 ≅ 1.61 um, where a resonant coupling occurs between the fundamental mode of the bent waveguide and WGM. It is around λ2 where the two modes with a characteristic profile depicted in Fig. 1, curve 8 arise in the modal spectrum of section 2, which can be interpreted as a result of FM – WGM – SPM coupling. Calculations show that the strong coupling between FM and WGM takes place in a much narrower spectral range (region 2 in Fig. 3) than that of SPM – WGM coupling, and it is this fact that is responsible for such a narrow dip in the transmission spectrum around λ2, where a strong energy transfer occurs from the light guided by the waveguide core to surface plasmons through the intermediary of a whispering gallery mode. When the bent waveguide section is long enough e.g. of a full loop length, the attenuation in it and, correspondingly, the depth of the dip in the transmission spectrum become very large. A change in the surrounding medium refractive index results in the change of SPM propagation constant, which disrupts coupling for, at least, two of the three coupled modes and brings about an abrupt decrease of guided light attenuation at the wavelength λ2. This explains the high sensitivity of the waveguiding structure under study to the refractive index of the surrounding medium in the intensity measurement mode.

If the length of section 2 is significantly larger than one loop, power carried by the two modes responsible for the transfer of energy from the guided light to surface plasmons drops, by the end of the bent section, to a level comparable with the power of other modes of section 2. The latter modes have very low excitation efficiency by the fundamental mode of section 1 but are much less lossy since they are not effectively involved in SPR excitation. This can be interpreted as weak coupling between the fundamental mode and other WGMs and is the reason for the nonmonotonous dependence of the transmission coefficient on the refractive index n0 near the maximum attenuation when L>>2πR (R = 7.1 mm).

In the intensity measurement mode the minimum detectable change in the measured refractive index is defined by the intensity noise level. As a first approximation let us assume that the noise in an SPR refractometric system is due to two major factors: laser intensity noise and photoreceiver noise. Then the detection limit of the system dn can be expressed as a sum of two terms:

dn=dn1+dn2=PN1P0Tdn0dT+PN2P0dn0dT,
where PN1 – equivalent noise power of the laser source, P0 – laser source power, T – transmission coefficient, PN2 – equivalent noise power of the photoreceiver. The estimate is valid under the assumption that a frequency-stabilized laser is used with a linewidth much narrower than the width of SPR dip in the transmission spectrum of the studied structure. Figure 2(b) shows the calculation results for dn1 and dn2 as functions of n0 obtained for the values of P0, PN1, PN2 that are typical of modern laboratory equipment. As can be seen, the detection limit comes to about 10−8 and is mainly dominated by the term dn1 which is responsible for the laser noise. By using standard noise reduction techniques for intensity-modulated optical signals (e.g. power referencing, laser source modulation and frequency-selective detection) the value of dn1 can be drastically reduced, which promises further increase in accuracy.

4. Conclusions

Thus the processes of excitation of surface plasmon waves in a metal film applied to the cladding of a standard bent single-mode optical fiber have been studied numerically. It was shown that by adjusting the waveguide bend radius and metal film thickness one can achieve effective coupling between the fundamental mode guided by the waveguide core and the symmetric surface plasmon mode propagating along the metal – surrounding medium interface through the intermediary of whispering gallery mode supported by the bent waveguide’s cladding. This effect was demonstrated to allow for precision refractometry both in the wavelength and intensity measurement modes with a resolution down to 10−8 RIU. Usage of standard noise reduction techniques for intensity-modulated optical signals promises further increase in accuracy.

Acknowledgments

The research was partially supported by the Russian Academy of Sciences programs (grant No. 12-I-P24-02; “Far East” program’s grants No. 4.6, 3.2.18) and Russian Foundation for Basic Research (grant No. 14.02.31558).

References and links

1. X. Guo, “Surface plasmon resonance based biosensor technique: a review,” J Biophotonics 5(7), 483–501 (2012). [CrossRef]   [PubMed]  

2. Y. Chen and H. Ming, “Review of surface plasmon resonance and localized surface plasmon resonance sensor,” Photonic Sensors 2(1), 37–49 (2012). [CrossRef]  

3. P. Zijlstra, P. M. R. Paulo, and M. Orrit, “Optical detection of single non-absorbing molecules using the surface plasmon resonance of a gold nanorod,” Nat. Nanotechnol. 7(6), 379–382 (2012). [CrossRef]   [PubMed]  

4. G. Xiao and W. J. Bock, eds., Photonic Sensing: Principles and Applications for Safety and Security Monitoring (Wiley, 2012).

5. J. Homola, Surface Plasmon Resonance Based Sensors (Springer, 2006).

6. B. D. Gupta and R. K. Verma, “Surface plasmon resonance-based fiber optic sensors: principle, probe designs, and some applications,” J. Sens. 2009(1), 979761 (2009).

7. Yu. N. Kulchin, O. B. Vitrik, A. V. Dyshlyuk, and Zh. Zhou, “Conditions for surface plasmon resonance excitation by whispering gallery modes in a bent single mode optical fiber for the development of novel refractometric sensors,” Laser Phys. 23(8), 085105 (2013). [CrossRef]  

8. D. Gallagher “Photonics CAD Matures,” LEOS newsletter, February 2008, 8 – 14 (2008).

9. K. S. Chiang, “Analysis of optical fibers by the effective-index method,” Appl. Opt. 25(3), 348–354 (1986). [CrossRef]   [PubMed]  

10. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).

11. R. Sammut and A. W. Snyder, “Leaky modes on a dielectric waveguide: orthogonality and excitation,” Appl. Opt. 15(4), 1040–1044 (1976). [CrossRef]   [PubMed]  

References

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  1. X. Guo, “Surface plasmon resonance based biosensor technique: a review,” J Biophotonics 5(7), 483–501 (2012).
    [Crossref] [PubMed]
  2. Y. Chen and H. Ming, “Review of surface plasmon resonance and localized surface plasmon resonance sensor,” Photonic Sensors 2(1), 37–49 (2012).
    [Crossref]
  3. P. Zijlstra, P. M. R. Paulo, and M. Orrit, “Optical detection of single non-absorbing molecules using the surface plasmon resonance of a gold nanorod,” Nat. Nanotechnol. 7(6), 379–382 (2012).
    [Crossref] [PubMed]
  4. G. Xiao and W. J. Bock, eds., Photonic Sensing: Principles and Applications for Safety and Security Monitoring (Wiley, 2012).
  5. J. Homola, Surface Plasmon Resonance Based Sensors (Springer, 2006).
  6. B. D. Gupta and R. K. Verma, “Surface plasmon resonance-based fiber optic sensors: principle, probe designs, and some applications,” J. Sens. 2009(1), 979761 (2009).
  7. Yu. N. Kulchin, O. B. Vitrik, A. V. Dyshlyuk, and Zh. Zhou, “Conditions for surface plasmon resonance excitation by whispering gallery modes in a bent single mode optical fiber for the development of novel refractometric sensors,” Laser Phys. 23(8), 085105 (2013).
    [Crossref]
  8. D. Gallagher “Photonics CAD Matures,” LEOS newsletter, February 2008, 8 – 14 (2008).
  9. K. S. Chiang, “Analysis of optical fibers by the effective-index method,” Appl. Opt. 25(3), 348–354 (1986).
    [Crossref] [PubMed]
  10. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).
  11. R. Sammut and A. W. Snyder, “Leaky modes on a dielectric waveguide: orthogonality and excitation,” Appl. Opt. 15(4), 1040–1044 (1976).
    [Crossref] [PubMed]

2013 (1)

Yu. N. Kulchin, O. B. Vitrik, A. V. Dyshlyuk, and Zh. Zhou, “Conditions for surface plasmon resonance excitation by whispering gallery modes in a bent single mode optical fiber for the development of novel refractometric sensors,” Laser Phys. 23(8), 085105 (2013).
[Crossref]

2012 (3)

X. Guo, “Surface plasmon resonance based biosensor technique: a review,” J Biophotonics 5(7), 483–501 (2012).
[Crossref] [PubMed]

Y. Chen and H. Ming, “Review of surface plasmon resonance and localized surface plasmon resonance sensor,” Photonic Sensors 2(1), 37–49 (2012).
[Crossref]

P. Zijlstra, P. M. R. Paulo, and M. Orrit, “Optical detection of single non-absorbing molecules using the surface plasmon resonance of a gold nanorod,” Nat. Nanotechnol. 7(6), 379–382 (2012).
[Crossref] [PubMed]

2009 (1)

B. D. Gupta and R. K. Verma, “Surface plasmon resonance-based fiber optic sensors: principle, probe designs, and some applications,” J. Sens. 2009(1), 979761 (2009).

1986 (1)

1976 (1)

Chen, Y.

Y. Chen and H. Ming, “Review of surface plasmon resonance and localized surface plasmon resonance sensor,” Photonic Sensors 2(1), 37–49 (2012).
[Crossref]

Chiang, K. S.

Dyshlyuk, A. V.

Yu. N. Kulchin, O. B. Vitrik, A. V. Dyshlyuk, and Zh. Zhou, “Conditions for surface plasmon resonance excitation by whispering gallery modes in a bent single mode optical fiber for the development of novel refractometric sensors,” Laser Phys. 23(8), 085105 (2013).
[Crossref]

Guo, X.

X. Guo, “Surface plasmon resonance based biosensor technique: a review,” J Biophotonics 5(7), 483–501 (2012).
[Crossref] [PubMed]

Gupta, B. D.

B. D. Gupta and R. K. Verma, “Surface plasmon resonance-based fiber optic sensors: principle, probe designs, and some applications,” J. Sens. 2009(1), 979761 (2009).

Kulchin, Yu. N.

Yu. N. Kulchin, O. B. Vitrik, A. V. Dyshlyuk, and Zh. Zhou, “Conditions for surface plasmon resonance excitation by whispering gallery modes in a bent single mode optical fiber for the development of novel refractometric sensors,” Laser Phys. 23(8), 085105 (2013).
[Crossref]

Ming, H.

Y. Chen and H. Ming, “Review of surface plasmon resonance and localized surface plasmon resonance sensor,” Photonic Sensors 2(1), 37–49 (2012).
[Crossref]

Orrit, M.

P. Zijlstra, P. M. R. Paulo, and M. Orrit, “Optical detection of single non-absorbing molecules using the surface plasmon resonance of a gold nanorod,” Nat. Nanotechnol. 7(6), 379–382 (2012).
[Crossref] [PubMed]

Paulo, P. M. R.

P. Zijlstra, P. M. R. Paulo, and M. Orrit, “Optical detection of single non-absorbing molecules using the surface plasmon resonance of a gold nanorod,” Nat. Nanotechnol. 7(6), 379–382 (2012).
[Crossref] [PubMed]

Sammut, R.

Snyder, A. W.

Verma, R. K.

B. D. Gupta and R. K. Verma, “Surface plasmon resonance-based fiber optic sensors: principle, probe designs, and some applications,” J. Sens. 2009(1), 979761 (2009).

Vitrik, O. B.

Yu. N. Kulchin, O. B. Vitrik, A. V. Dyshlyuk, and Zh. Zhou, “Conditions for surface plasmon resonance excitation by whispering gallery modes in a bent single mode optical fiber for the development of novel refractometric sensors,” Laser Phys. 23(8), 085105 (2013).
[Crossref]

Zhou, Zh.

Yu. N. Kulchin, O. B. Vitrik, A. V. Dyshlyuk, and Zh. Zhou, “Conditions for surface plasmon resonance excitation by whispering gallery modes in a bent single mode optical fiber for the development of novel refractometric sensors,” Laser Phys. 23(8), 085105 (2013).
[Crossref]

Zijlstra, P.

P. Zijlstra, P. M. R. Paulo, and M. Orrit, “Optical detection of single non-absorbing molecules using the surface plasmon resonance of a gold nanorod,” Nat. Nanotechnol. 7(6), 379–382 (2012).
[Crossref] [PubMed]

Appl. Opt. (2)

J Biophotonics (1)

X. Guo, “Surface plasmon resonance based biosensor technique: a review,” J Biophotonics 5(7), 483–501 (2012).
[Crossref] [PubMed]

J. Sens. (1)

B. D. Gupta and R. K. Verma, “Surface plasmon resonance-based fiber optic sensors: principle, probe designs, and some applications,” J. Sens. 2009(1), 979761 (2009).

Laser Phys. (1)

Yu. N. Kulchin, O. B. Vitrik, A. V. Dyshlyuk, and Zh. Zhou, “Conditions for surface plasmon resonance excitation by whispering gallery modes in a bent single mode optical fiber for the development of novel refractometric sensors,” Laser Phys. 23(8), 085105 (2013).
[Crossref]

Nat. Nanotechnol. (1)

P. Zijlstra, P. M. R. Paulo, and M. Orrit, “Optical detection of single non-absorbing molecules using the surface plasmon resonance of a gold nanorod,” Nat. Nanotechnol. 7(6), 379–382 (2012).
[Crossref] [PubMed]

Photonic Sensors (1)

Y. Chen and H. Ming, “Review of surface plasmon resonance and localized surface plasmon resonance sensor,” Photonic Sensors 2(1), 37–49 (2012).
[Crossref]

Other (4)

G. Xiao and W. J. Bock, eds., Photonic Sensing: Principles and Applications for Safety and Security Monitoring (Wiley, 2012).

J. Homola, Surface Plasmon Resonance Based Sensors (Springer, 2006).

D. Gallagher “Photonics CAD Matures,” LEOS newsletter, February 2008, 8 – 14 (2008).

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).

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Figures (3)

Fig. 1
Fig. 1 Schematic of the waveguiding structure under study and its modes: 1 – input straight waveguide section, 2 – bent waveguide section, 3 – output straight waveguide section, 4 – waveguide core, 5 – waveguide cladding, 6 – polymer jacket, 7 – silver film, 8 – electric field amplitude profile of one of the two modes of section 2 responsible for coupling light guided by the core to surface plasmons (the profile of the second mode is similar to this one and not shown), 9 – electric field profile of the fundamental mode of sections 1 and 3. Inset 1 – refractive index profile of a standard SM fiber (n1) and effective graded index profile of the equivalent slab waveguide n’1. Inset 2 – electric field profile of the fundamental mode of the bent SM waveguide with an infinite cladding. Inset 3 - electric field profile of a cladding whispering gallery mode of the bent waveguide. Inset 4 - electric field profile of the symmetric surface plasmon mode guided by the metal - surrounding medium interface.
Fig. 2
Fig. 2 Calculated transmission coefficient of the studied structure: a – transmission spectra computed at two values of n0 (1.4216 and 1.4226, respectively) for d = 20 nm (1 and 2), 30 nm (3 and 4) and 40 nm (5 and 6). In the inset: SPR wavelength dependence on the refractive index n0 for d = 30 nm; b – transmission coefficient vs. n0 at a fixed wavelength λ = 1.66 um; dn1 and dn2 dependences on n0.
Fig. 3
Fig. 3 Dispersion curves for the fundamental mode of the bent SM waveguide with an infinite cladding, WGM, and SPM: 1 – schematic representation of the spectral region where WGM is effectively coupled to SPM; 2 - schematic representation of the spectral region where FM is effectively coupled to WGM. Inset 1 – electric field profile of section 2 mode at λ1 ≅ 1.43 um, which can be interpreted as a result of coupling between WGM and SPM. Inset 2 - electric field profile of section 2 mode at λ2 ≅ 1.61 um, which can be interpreted as FM – WGM – SPM coupling.

Equations (2)

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a p = 1 N p E 1 × H p ·zdx
dn=d n 1 +d n 2 = P N1 P 0 T d n 0 dT + P N2 P 0 d n 0 dT ,

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