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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 22 — Oct. 27, 2008
  • pp: 17826–17841

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Plasmon mediated confocal dark-field microscopy

Marcus Schmelzeisen, Jacqueline Austermann, and Maximilian Kreiter  »View Author Affiliations


Optics Express, Vol. 16, Issue 22, pp. 17826-17841 (2008)
http://dx.doi.org/10.1364/OE.16.017826


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Abstract

An efficient mode for scanning confocal dark-field microscopy through a thin gold film is established that takes advantage of the intermediate excitation of surface plasmons both in the excitation and in the emission process. This concept is verified by experimental investigation of the effective point-spread function, the intensity distribution of the scattered radiation and by comparison with a classical dark-field geometry. The wavelength-dependence of both the signal strength and the point-spread function are discussed.

© 2008 Optical Society of America

1. Introduction

Optical plasmonic resonances on nanometer-sized metal particles are frequently accompanied by strongly enhanced fields which are localized in volumes by far smaller than diffraction-limited foci that can be achieved by far-field illumination. Based on this field localization, several physical effects with significant potential for applications have been discussed as surface enhanced spectroscopies [1

J. R. Lakowicz, “Radiative Decay Engineering: Biophysical and Biomedical Applications,” Anal. Biochem. 298, 1–24(2001). [CrossRef] [PubMed]

, 2

A. Otto, I. Mrozek, H. Grabhorn, and W. Akemann, “Surface-enhanced raman-scattering,” J. Phys. Condens. Matter 4, 1143–1212 (1992). [CrossRef]

], sensing [3

K. A. Willets and R. P. Van Duyne, “Localized surface plasmon resonance spectroscopy and sensing,” Annu. Rev. Phys. Chem. 58, 267–297 (2007). [CrossRef]

] as well as enhanced light collection [4

D. M. Schaadt, B. Feng, and E. T. Yu, “Enhanced semiconductor optical absorption via surface plasmon excitation in metal nanoparticles,” Appl. Phys. Lett. 86(2005). [CrossRef]

] and emission [5

K. Joulain, J. P. Mulet, F. Marquier, R. Carminati, and J. J. Greffet, “Surface electromagnetic waves thermally excited: Radiative heat transfer, coherence properties and Casimir forces revisited in the near field,” Surf. Sci. Rep. 57, 59–112 (2005). [CrossRef]

]. For a detailed investigation of plasmonic resonances, single-object spectroscopy is mandatory since big variations between individual resonators are rather the rule than an exception. The power of optical single-particle spectroscopy has been demonstrated by investigations on resonances on metal rods [6

C. Sönnichsen, S. Geier, N. E. Hecker, G. von Plessen, J. Feldmann, H. Ditlbacher, B. Lamprecht, J. R. Krenn, F. R. Aussenegg, V. Z. H. Chan, J. P. Spatz, and M. Moller, “Spectroscopy of single metallic nanoparticles using total internal reflection microscopy,” Appl. Phys. Lett. 77, 2949–2951 (2000). [CrossRef]

, 7

C. Sönnichsen, T. Franzl, T. Wilk, G. von Plessen, J. Feldmann, O. Wilson, and P. Mulvaney, “Drastic reduction of plasmon damping in gold nanorods,” Phys. Rev. Lett. 88(2002). [CrossRef] [PubMed]

] and on platelets with different simple geometrical contours [8

S. Schultz, D. R. Smith, J. J. Mock, and D. A. Schultz, “Single-target molecule detection with nonbleaching multicolor optical immunolabels,” Proc. Natl. Acad. Sci. USA 97, 996–1001 (2000). [CrossRef] [PubMed]

, 9

J. J. Mock, M. Barbic, D. R. Smith, D. A. Schultz, and S. Schultz, “Shape effects in plasmon resonance of individual colloidal silver nanoparticles,” J. Chem. Phys. 116, 6755–6759 (2002). [CrossRef]

]. In these studies, a customized dark field optical microscope with evanescent illumination and a commercial dark field system were used. Metal surfaces as a support for plasmonic structures bear the advantage of direct electrical contact and the availability of a well-developped, simple means of surface chemical modifications based on thiols. In particular, a special type of resonances occurs if particles are placed close to a metal surface [10

P. K. Aravind and H. Metiu, “The effects of the interaction between resonances in the electromagnetic response of a sphere-plane structure - applications to surface enhanced spectroscopy,” Surf. Sci. 124, 506–528 (1983). [CrossRef]

–13

T. Okamoto and I. Yamaguchi, “Optical absorption study of the surface plasmon resonance in gold nanoparticles immobilized onto a gold substrate by self-assembly technique,” J. Phys. Chem. B 107, 10321–10324 (2003). [CrossRef]

]. Here, a very strong field enhancement between particle and surface may be achieved due to the ultimate small gaps that can be defined by molecular spacers. For their investigation, scattering spectroscopy on individual objects represents a central experimental technique.

The presence of the gold surface, though, due to the presence of surface plasmons, significantly modifies the coupling between the local resonator and incident or outgoing plane waves in a way that standard dark field microscopy is quite inefficient to study these objects. Similar plasmon-induced modifications have been discussed for the direction-dependent emission from dye molecules [14

Y. M. Gerbshtein, I. A. Merkulov, and D. N. Mirlin, “Transfer of Luminescence-center energy to surface plasmons,” JETP Lett. 22, 35–36 (1975).

] and for their excitation efficiency for illumination from different directions [15

H. Knobloch, H. Brunner, A. Leitner, F. Aussenegg, and W. Knoll, “Probing the evancescent field of propagating plasmon surface-polaritons by fluorescence and raman spectroscopy,” J. Chem. Phys. 98, 10093–10095 (1993). [CrossRef]

] which strongly modifies the appearance of single dyes in fluorescence microscopy [16

F. D. Stefani, K. Vasilev, N. Bocchio, N. Stoyanova, and M. Kreiter, “Surface-plasmon-mediated single-molecule fluorescence through a thin metallic film,” Phys. Rev. Lett. 94 (2005). [CrossRef] [PubMed]

]. For individual scatterers, analogous direction dependences have been shown [17

B. Hecht, H. Bielefeldt, L. Novotny, Y. Inouye, and D. W. Pohl, “Local excitation, scattering, and interference of surface plasmons,” Phys. Rev. Lett. 77, 1889–1892 (1996). [CrossRef] [PubMed]

].

Here, we investigate the role of intermediate surface plasmon excitation for dark field microscopy on objects close to a metal surface. We point out in how far the intermediate excitation of surface plasmons can be used to obtain high quality dark field micrographs through a thin metal film.

2. Experimental details

Microscope glass slides (clean white, 1 mm thick, Menzel Gläser GmbH, Braunschweig, Germany) were cleaned by immersion in a 2 % detergent solution (Helllmanex, Hellma GmbH & Co KG, Müllheim, Germany) in ultra pure-water (milli-Q: 18.2 MΩ cm, Millipore, Billerica, USA), followed by rinsing with ultra pure water and ethanol (HPLC grade, Fisher Scientific, Leicestershire, United Kingdom). After 2 h in a tube furnace at 550 °C under a nitrogen atmosphere, 1.5 nm Chromium (99.9 %, Unicore Materials, Balzers, Liechtenstein) and 50 nm Gold (99.99% granulate 2.0–3.0 mm, Balzers) were thermally evaporated (Balzers BLS 500, Balzers, Lichtenstein) on the clean glass substrates.

In order to deposit dielectric spheres on a gold film all the following steps were performed in a clean room. Stock solutions of polystyrene (PS) colloids (Nanobead NIST, Polysciences Inc, USA) stored at 4 °C were allowed to warm to room temperature. The dispersion was sonicated for 2 minutes to distribute the colloids. In order to obtain a suitable particle density on the surface a volume of the stock solution was diluted 1:100 in ethanol (HPLC grade). A fraction of 70 µL was drop-cast on the gold substrates and the surface dried with a stream of nitrogen before the solution evaporated.

Gold nanoparticles were synthesized following a procedure by Frens [18

G. Frens, “Controlled Nucleation for the Regulation of the Particle Size in Monodisperse Gold Suspensions,” Nature Phys. Sci. 241, 20–22 (1973).

]. In brief 100 ml of an 0.25 mM aqueous solution of hydrogen tetrachloroaurate(III) trihydrate (HAuCl4·3 H2O, 99.995%, Sigma-Aldrich) was heated under reflux and stirring to the boiling point. Then 2 ml of an 0.015 M aqueous solution of trisodium citrate dihydrate (HOC(CO2Na)(CH2CO2Na)2·2 H2O, Sigma-Aldrich) was injected quickly. The solution changed color from black to dark red in the first few minutes. After keeping the mixture boiling for another 10 minutes the heating source was removed and the solution was allowed to cool down slowly to room temperature while stirring. The particles prepared that way had a mean diameter of 100 nm as determined by scanning electron microscopy (Gemini 1530, Zeiss-LEO, Oberkochen, Germany). For particle deposition the glass substrates with the evaporated gold film were immersed in a 1 mM ethanol solution of 1-aminoethanethiol hydrochloride (AET) (HSCH2CH2NH2·HCl, Sigma Aldrich) for 17 hours. In this process, the thiol group of the AET attaches to the gold surface and forms a monolayer. After the substrates were rinsed with a mixture of 1:1 ethanol/water they were immersed in the freshly prepared colloidal gold suspension for 5 minutes. Subsequently they were again rinsed with distilled water and dried with nitrogen. The gold nanoparticles were strongly immobilized by this process.

The scattering spectra of individual nanoparticles were obtained using a customized scanning confocal optical microscope (SCOM) in dark field mode and a fiber-coupled CCD spectrometer (Andor Shamrock SR303i, Andor Technology, Belfast, Northern Ireland). In the confocal setup light from a He-Ne laser (633 nm) or a Xenon lamp (Osram XBO 150) was coupled into an optical fiber, acting as a point-like light source and providing a divergent beam with an approximately Gaussian profile (TEM00-mode). The illumination beam was collimated by an achromatic lens (f=150 mm) and directed to an oil immersion (nglass=noil=1.503) microscope objective (60x, NA=1.4, Plan-Apo, Nikon GmbH) by a 50/50 beam splitter. In the focus of the objective the sample was mounted on top of a piezoelectric stage (Tritor 101 CAP, Piezosystems GmbH, Jena, Germany). The scattered light from the sample was collected by the same objective and separated from the illumination beam by the splitter. After its way through a confocal arrangement (2x lenses, f=100 mm, 100 µm pinhole) the remaining light was focused (lens, f=30 mm) for scanning onto an avalanche photo diode (APD) (Perkin Elmer, Optoelectronics Inc., USA) or into a fiber connected to the CCD spectrometer for spectral analysis. For light scattering measurements two blocks are inserted into the beam before the beam splitter (B1) and the confocal arrangement (B2) to realize dark field conditions (Compare Fig. 1(a)). In order to image the radiation patterns of the scattering objects the confocal pinhole was removed and the back focal plane of the microscope objective was imaged in the block plane B2 and again on the chip of a CCD device (uEye UI-2250-M-GL, IDS Imaging Development Systems GmbH, Obersulm, Germany) by a suited combination of lenses.

For taking particle spectra Xenon illumination was used. The piezoelectric stage of the confocal microscope was moved to a position of an identified particle and the light of this position IObj analyzed for 100 s by the spectrometer. As a reference for the gold film I(Ref) the procedure was repeated for a clean position in the neighbourhood of the particle. After measuring all selected particles the background count IDark was determined with a closed lamp shutter and the lamp spectrum IXe with a mirror as sample. From this, spectra for an object σ 0=(IObj-IDark)/(IXe-IDark) and the associated reference σ R=(IRef-IDark)/(IXe-IDark) were obtained, yielding a scattering strength of one particle σ=σ 0-σR .

3. Concept

In general, dark field microscopy relies on the selection of a limited angular spectrum illuminating the specimen. The specular reflection from this light is then blocked such that only scattered light is reaching the detection path. An experimental realization of this concept is schematically depicted in Fig. 1(a). Two blocks B1 and B2 are placed in the excitation and detection path of the scanning confocal optical microscope, respectively.

Fig. 1. (a) Schematic representation of a dark field microscope. (b) Reflectivity and field enhancement for plane wave illumination with a wavelength of 633 nm in p polarization (Dielectric constants are εglass=2.25901, εgold=-11.8321+i·1.36442, εair=1, the thickness of the gold film is dAu=50 nm. (c) Same as (b) for s polarization. (d) Magnified view on (b).

In principle, the blocks may have any form in two dimensions as long as they are complementary in the sense that their superposition blocks the entire beam. Note that reflection at the surface corresponds to a point reflection of the beam profile about the optical axis such that blocking is achieved if the sum of B1 and the point reflected B2, seen along the beam propagation, covers the entire plane. Due to diffraction, some overlap is generally required for good blocking. Usually, rotationally symmetric blocks, i.e. a disc and an annulus are employed to keep the symmetry of the imaging system. As soon as objects are to be investigated through a thin metal film, this geometry becomes unsuitable as will be explained in the following.

The type of sample that is to be investigated is depicted in Fig. 1(a). An oil immersion objective focuses the incident light on the surface of a microscope coverslip which is covered by a gold film with a thickness of approximately 50 nm. Behind the film, a polarizable object is placed whose scattering cross-section is to be studied. We assume a polarizability only perpendicular to the sample surface. Excitation is due to a plane wave with amplitude EB1,x in the plane where B1 is placed. Only fields within the area AB1 which corresponds to rays collected by the objective are contributing. We will consider two different linear polarizations of the exciting light with the electrical field vector being parallel and perpendicular to the y coordinate, denoted in the following by the subscripts ⊥ and ‖, respectively. The light field not blocked by B1 is focused by the objective. The polarizable unit emits radiation which is transformed to plane waves in the area AB2 which is defined as the analogue of AB1 in the detection path. These waves are partially removed by the block B2, the transmitted fraction is detected. The detectable power P which is proportional to the expected scattering signal can be calculated as

P= cn ε0 [ α0 λ2 k2 d4π λ02]2 A B2 [ EzFP ( rFP)]2 × ∑ Δ A B2 Δ A B2 A B2 { [ Tzx ( r B2)]2 + [ Tzy ( r B2)]2}

Tzx= [ R −1AR]zx· 1 cos (θ) Tzy= [ R −1AR]zy· 1 cos (θ)
(1)

where the local field EFP,z that drives the polarizable unit is given by

E FP,z= −ik A B1 2πd ∑ Δ A B1 Δ A B1 A B1 eikr Tzx ( r B1) E B1,x
(2)

for ⊥ polarization and a corresponding expression with x replaced by y for ‖ polarization. The Matrix A is defined as

A= ( E p,x 0 0 0 E s,y 0 E p,z 0 0)

The matrix components represent local electrical fields at the position of the polarizable unit upon illumination with a plane wave with unit electrical field which may be termed ‘enhancement factors’. Ep,x and Ep,z correspond to transverse-magnetic (p) polarization and denote Cartesian field components in a coordinate system where the plane of incidence equals the x-z-plane. Es,y is the corresponding field component for transverse-electric (s) polarization. R is a rotation matrix that connects this Cartesian system with the fixed one which is defined relative to the block edge and linear polarization of the incident light (compare Fig. 1(a)).

Further physical quantities in eqn. 1 are the speed of light in vacuum c, the index of refraction of the immersion medium n, the wave vector k of the incident light, and the focal length d of the microscope objective. rFP , rB1 and rB2 are vectors in the focal plane, AB1 and AB2, respectively. The angle θ is measured between the objects surface normal and the angle of incidence of a ray which is emanating from rB1 or rB2 and focused on the sample (compare Fig. 1(a)). For numerical integration, AB1 and AB2 are decomposed in small pieces ΔAB1 ΔAB2 and the corresponding contributions are summed. α0 and λ0 are arbitrarily chosen constants. A detailed derivation of eqns 1 and 2 is provided in the appendix.

The field EFP,z that drives the polarizable unit in z-direction is essentially obtained by coherently summing over AB1 contributions with a magnitude which is proportional to Tzy and Tzx for ⊥ and ‖ polarization, respectively. The expression in curly brackets in equation (1) can be interpreted as a detection efficiency and is obtained by an incoherent summation of a similar expression over AB2 .

Finally, the important information lies in the field enhancement factors Es,y , Ep,x and Ep,z . They are plotted for illumination with a red He-Ne laser line (633 nm) in Fig. 1(b–d) as a function of the angle of incidence θ. In addition, the corresponding reflectivities are displayed. One notes that for s polarization the field at the back side of the metal film is quite weak. For p polarization, a strong field is seen in a narrow angular range, accompanied by a minimum in reflectivity, at maximum a field enhancement factor of almost 8 is reached. This effect is well known as ‘surface plasmon excitation in Kretschmann-configuration’ [19

E. Kretschmann and H. Raether, “Radiative Decay of Non Radiative Surface Plasmons Excited by Light,” Zeitschrift Für Naturforschung Part A 23, 2135–2136 (1968).

] and has important consequences for the signal obtained in dark field microscopy through a thin metal film. Obviously, the magnitude Ep,z dominates the experimental signal. Different from standard microscopy where contributions everywhere in AB1 and AB2 are of similar weight, both excitation and emission are dominated by contributions in the very narrow angular range corresponding to intermediate surface plasmon excitation. As a consequence, the standard way of utilizing circular symmetric complementary blocks with some overlap will either prevent efficient excitation or efficient emission and in turn produce only weak signals.

Fig. 2. (a) |Tzx(rB1)|2, b) |Tzy(rB1)|2 (c) Intensity in B2 (Term in curly brackets in equation (1)). (d) Same as (c) if a linear block is introduced. The scaling does not represent the full range of values which goes up to 55.

A solution for this problem is the use of complementary linear blocks in the excitation and emission pathway that both allow for a significant coupling to propagating surface plasmons. In Fig. 2(a) and (b) the moduli squared of the quantities Tzx (rB1 ) and Tzy (rB1 ) are displayed, representing effective intensities which contribute to the excitation for ⊥ and ‖ polarization, respectively. Only the radiation passing a block which is displaced by 0.5 mm from the optical axis as used in the experiment was plotted. In both cases, an efficient excitation is expected. Fig. 2(c) shows the quantity |Tzx (rB1 )|2+|Tzy (rB1 )|2 corresponding to the unpolarized intensity in AB2 due to a z dipole in the focus. Fig. 2(d) finally shows the intensity that remains detectable if a second linear block is introduced in AB2. In the reminder of this manuscript we will first show that this experimental approach of ‘surface plasmon-mediated dark field microscopy’ allows for a sensitive measurement of resonators through a thin gold film. Then we will compare the experimentally obtained patterns with the theoretical prediction based on the simplified model of a point-like polarizable unit. Finally, we will investigate the wavelengthdependence of the plasmon-mediated dark field signal and discuss implications for spectroscopy on this kind of system.

4. Experimental validation and comparison to theory

Fig. 3. (a–b) Measured scattering signals from a polystyrene sphere (d=300 nm) for incident light of λ=633 nm polarized parallel (‖) and perpendicular (⊥) to the block edge. (c–d) Calculated signal assuming a pointlike dipole in z-direction.

First, we present an experiment designed to validate the expectation that efficient imaging is possible with the proposed plasmon-mediated dark field microscopy. A test sample was prepared by coating a microscope coverslip with a 50 nm gold film. The gold surface was decorated with polystyrene spheres with a diameter of 300 nm. In Fig. 3 the resulting patterns for the two linear polarizations under investigation are displayed. In both cases, a clear signal is observed. For ⊥ polarization, a slightly elongated spot is seen with two ‘streaks’. In ‖-polarization, we see a nodal plane crossing the Gaussian focal point, indicating a dominant contribution of a z-polarized dipole [20

B. Sick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85, 4482–4485 (2000). [CrossRef] [PubMed]

] with similar streaks. Note the similarity to the patterns that are obtained without B2 which have been discussed in detail and measured by near field microscopy [21

A. Bouhelier, F. Ignatovich, A. Bruyant, C. Huang, G. Colas des Francs, J. C. Weeber, A. Dereux, G. P. Wiederrecht, and L. Novotny, “Surface plasmon interference excited by tightly focused laser beams,” Opt. Lett. 32, 2535–2537 (2007). [CrossRef] [PubMed]

]. The calculations for a point-like z-dipole are shown in Fig. 3(c) and (d). Taking into account the simplicity of the model, the shape of the patterns is reproduced quite well. The lateral extension of these patterns is around 1 µm for the main maximum, including the streaks a total size of the pattern of the order of 2 µm is obtained. Only objects which are separated by at least this distance can be individually addressed in a straightforward manner.

Fig. 4. Light intensity distribution in AB2, recorded with a CCD camera. (a) Specular reflection from a mirror, no blocks. (b) Scattered light from a polystyrene sphere in dark field mode. (c) Light scattered by a ‘clean’ part of the gold film.
Fig. 5. Dark-field micrographs of 60 nm Au colloids on Au at 633 nm for different block shapes as shown in the insets. (a) Circular symmetric blocking. (b) Double-linear blocking. (c) Double-linear blocking with additional central block preventing light al low θ to pass B1.

The role of intermediate excitation of surface plasmons in the emission process can be nicely demonstrated by imaging the light intensity distribution in a plane which is an image of the objectives back focal plane and is placed behind B2. Directions of light rays emitted from the sample correspond to positions in this image plane. The recorded images are shown in Fig. 4. Figure 4(a) shows the specular reflected light from a mirror in the focal plane if no blocks are present. The approximately homogenous beam profile from AB1 is reproduced in AB2. In the image, the angles of incidence θ corresponding to the positions in AB2 are indicated: 0° corresponds to rays on the optical axis of the system. The angle of incidence corresponding to the outermost rays is determined by the numerical aperture of the microscope objective, yielding 68° for the NA=1.4 objective used. Figure 4(b) shows the light intensity in AB2 if a polystyrene sphere is measured in plasmon-mediated dark field mode. A bright ring is observed at a position which corresponds to plasmon-mediated emission. Since the detection is unpolarized, this pattern corresponds to Fig. 2 (d). We note that in addition to the plasmon-mediated light, some experimental background intensity is seen which does not vanish if a clean part of the gold surface is investigated (Fig. 4(c)).

Fig. 6. Calculated patterns for a pointlike dipole in z-direction, field enhancement factors and reflectivities for four selected wavelengths and two polarization directions. The white cross indicates the Gaussian focal point.

Figure 5 shows dark-field micrographs of the same area of a gold film decorated with gold nanoparticles that were obtained with differently shaped blocks B1 and B2, sketched in the inset. With two circularly symmetric blocks (disc of 5 mm, pinhole of 1 mm), no objects can be seen (a). Choosing two linear blocks, several objects with different strength are readily identified (b). Figure 5(c) finally demonstrates that the experimental background may be reduced by adding an annulus to the linear block of 5 mm diameter in B1, thus efficiently reducing the background signal which dominates for these values of θ. This represents the direct experimental proof that the proper choice of block shapes allows for a very efficient detection of scattering objects through intermediate surface plasmon excitation.

Since the optical properties of gold vary strongly through the visible spectral region, the signal obtained in plasmon-mediated dark field microscopy will be wavelength dependent. Fig. 6 displays calculated patterns as well as the enhancement factors Ep,x (θ) and Ep,z (θ) for four representative wavelengths. At 450 nm (a), gold is a good absorber and only a weak local electrical field with correspondingly weak patterns is seen. Note that the intensity scales are adjusted for optimum dynamic range for each wavelength. The patterns are very localized due to the short wavelength. At 525 nm some feature is seen in R(θ) that can be interpreted as a surface plasmon with accompanying field enhancement, reflected in maxima in Ep,x and Ep,z . The enhancement is moderate but the resonance covers a broad angular range. At this wavelength, the strongest scattering signals are obtained. A further increase in wavelength to 633 nm leads to a sharp surface plasmon resonance. At resonance, the field increases significantly, nonetheless the scattering signal is getting weaker. Apparently, the increase in maximum field strength is overcompensated by the decrease in angular range where the enhancement occurs. At 850 nm, the angular range where field enhancement occurs is reduced further while the amplification factor does not increase. As a consequence, the signal has further weakened significantly. We note that even when focusing in free space, the lateral extension of the focal spot scales with the wavelength λ, as a consequence, the maximum exciting field scales with λ -1 and the maximum scattering signal with λ -2. This effect gives an additional contribution to the observed intensity reduction at increasing λ.

Fig. 7. (a) Maximum scattering intensity as a function of wavelength for a dipole in z-direction. (b) Displacement of the object position from the Gaussian focal point for maximum intensity and ⊥ polarization.

In Fig. 7(a) the maximum intensity in the calculated scattering patterns is displayed for an object with wavelength-independent scattering cross-section across the visible spectral range. This representation reproduces the maximum signal strength around 540 nm and the significant reduction for shorter and longer wavelengths. Fig. 7(a) can be interpreted as a transmission function of the entire optical system including the gold film. To recover wavelength-dependent scattering cross-sections, it is necessary to normalize any measured signal by this transmission function.

A further important effect is seen in the patterns in Fig. 6 which is a direct consequence of the block shapes that are not circular symmetric: maximum intensity is achieved if the scatterer is displaced from the Gaussian focal point. Fig. 7(b) shows this displacement for ⊥ polarization across the visible range. The variations are not completely negligible but small enough to allow for spectroscopy in plasmon-mediated dark field mode across the visible range by adjusting the microscope to maximum intensity before recording a spectrum.

Figure 8 shows as an example scattering spectra from a gold particle, taken at different positions of a pattern. Around the intensity maximum, the spectra do not vary much and show all identical features: Peaks at 760 nm, 610 nm and a shoulder around 520 nm. We assign them tentatively to resonances of the coupled system of the metal particle with the metal surface. At larger distance from the maximum, the relative strength of short-wavelength features decreases rapidly, as is directly intelligible due to the smaller focus for smaller wavelength.

Fig. 8. (a–b) Spectra from a gold particle of 115 nm diameter on top of a 50 nm gold film and a cysteamin spacer layer (1 nm) in white light illumination, taken for different positions as absolute (a) and normalised values (b). In (c) positions where spectra are taken are indicated on top of a lateral intensity distribution obtained in scanning mode with white light illumination.

6. Conclusion

In summary, we have shown that dark field microscopy through a thin metal film can be performed very efficiently if a non-symmetric blocking scheme is used that allows for the intermediate excitation of surface plasmons both for excitation and for detection. This concept was demonstrated by comparing lateral patters obtained experimentally with the theoretical prediction and by direct monitoring of the emission pattern. The resulting signal is strongly wavelength-dependent with a maximum efficiency around 530 nm. This effect, together with a wavelength-dependent lateral displacement between the Gaussian focal point and the maximum in the scattering pattern, should be considered for spectroscopy.

Appendices

Appendix

In this appendix, the details for the calculation of the plasmon-mediated dark field signal are explained. Essentially, three independent steps are taken first: The calculation of the 3D–field distribution in the focus (A1), the modelling of a dipole with wavelength-independent scattering cross-section (A2) and the collected radiation from a dipole in the focus (A3). Finally, these results are combined to yield an expression that allows modelling the plasmon-mediated dark field microscopy (A4).

A1: Field distribution in the focus

For the following discussion the microscope objective is modelled as an aplanatic system [22

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light , 6 ed. (Cambridge University Press, 1997). [PubMed]

]. The geometry under consideration is sketched in Figure A1. The optical system is illuminated by a homogenous linearly polarized plane wave along the optical axis. A coordinate system is chosen such that x is parallel to the electrical field vector of the incident wave and z antiparallel to the lights propagation direction.

Fig.A1. 1. Geometry under consideration, (a) 2D-sketch, (b) 3D-sketch illustrating the definition ψ. (c) Energy conservation of connected areas on the focal plane

The microscope objective is modelled [22

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light , 6 ed. (Cambridge University Press, 1997). [PubMed]

] as a device transforming each ‘ray’ of the parallel incident beam with the electrical field E⃗ B1 in the block plane 1 (B1) to a source situated at the focal sphere of the system (immaterial sphere centered at the Gaussian focal point with Radius R equal to the focal length f). This source has an electrical field E⃗′ FS tangential to the focal sphere which may be decomposed in a p and s polarized component where the polarization direction refers to the plane of incidence of this particular ‘ray’. For illustration, such a ray is sketched in Figure S1b.

( EFS p′ EFS s' 0)= ( cos (ψ) sin (ψ) 0 −sin (ψ) cos (ψ) 0 0 0 1)· ( E B1x E B1y 0) cos (θ)

E→FS′= cos (θ)R E→ B1

To determine E→FS′, one needs to rotate the incident field E⃗ B1 around the z-axis to a coordinate system where the plane of incidence coincides with the x-z plane, this rotation is expressed by the matrix R. Identical energy flux through connected areas on the focal plane and in B2 is guaranteed by multiplication with cos(θ)1/2 : P∝E2 FSAFS =E2 B1AB1 (compare Figure A1c). The dash indicates that fields are expressed in a rotated coordinate system that varies with the position in B1 that is considered. The angles ψ and θ are defined according to Figure A1b. These fields are propagated towards the focal region as spherical waves according to the Huygens-Fresnel principle [22

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light , 6 ed. (Cambridge University Press, 1997). [PubMed]

]. In the absence of a multilayer system, the total field strength in the focus is obtained by summing contributions ΔAFS from the entire focal sphere:

E→FP=∑Δ AFS [ − ik 2π E→FS′] eikdd

with the distance d. Close to the Gaussian focal point kd≫1 and therefore one may approximate

e ik (d)dd= e ik (d) ( R+r)d≈ e i k (R) ( R+r)d= e i k (R) (R)d e i k (R)r

where the first (constant) overall phase factor may be neglected. Thus:

E→FP= ∑Δ AFS [ − ik 2π E→FS′] eikrd = ∑ Δ AFS E→ FP,C′

The C indicates that those are contributions to the total field which remain to be summed up. When focussing through a multilayer system, each incident plane wave contribution E⃗ inc ′ FP,C will give rise to a local field E⃗′ FP,C at the specified position which is calculated by the transfer matrix algorithm [23

P. Yeh, Optical Waves in Layered Media (John Wiley & Sons Inc, Hoboten, 1988).

] for the locally p and s polarized components. The connection between incident field amplitude and local field amplitude may be expressed as three ‘enhancement factors’ which are combined in the matrix A.

( E FP,C x′ E FP,C y′ E FP,C z′)= ( A p,x 0 0 0 A s,y 0 A p,z 0 0) ( E FP,C inc, p′ E FP,C inc, s′ 0) E FP,C′=A E FP,c inc′

For the summation, all these contributions have to be transformed to the common coordinate system.

( E FP,Cx E FP,Cy E FP,Cz)= ( cos (ψ) −sin (ψ) 0 sin (ψ) cos (ψ) 0 0 0 1)( E FP,C x′ E FP,C y′ E FP,C z′) E→ FP,C= R −1 E→ FP,C′

And the field is finally obtained by integration of all those contributions

E→FP=∑Δ AFS [ − ik 2π E→FS] eikrd = −ik 2πd∑ Δ AFS eikr R −1AR E→ B1 cos (θ)

When summing over contributions from B1 with ΔAB1 =ΔAFS cos(θ)

E→FP= −ik 2πd∑Δ A B1 eikr R −1AR E→ B1= −ik A B1 2πd∑ ΔA B1 A B1eikr R −1AR E→ B1· 1 cos (θ)
(A1)

Here, we consider illumination with a linearly polarized plane wave in B1, thus Ex B1 is the only non vanishing component in the B1 plane. A block partly covering B1 is identified with a certain area in B1. It is modelled by setting the corresponding plane wave contributions to 0.

A2: Source of the scattering signal

This field gives rise to a oscillating charge distribution, represented as a dipole p⃗:

p→= ( αxx αxy αxz αyx αyy αyz αzx αzy αzz) ( EFPx EFPy EFPz)

In order to compare dipoles with identical scattering cross-sections, the polarizability α will depend on wavelength. Since the radiated power P from a point dipole in free space is given as

P= ck43 ∣ p→∣2 4π ε0

We may model a z-polarized scatterer with wavelength-independent scattering cross-section by introducing the wavelength-dependent polarizability

αzz= k02 k2 α0= λ2 λ02 α0
(A2)

where a constant factor α0 /λ2 0 may be chosen arbitrarily. Experimentally, this dipole represents the source of the scattered field which is finally detected.

A3: Emitted radiation

A3.1 Reciprocity

We assume that all intensity passing the block B2 in the detection path finally reaches the detector. Then, calculation of the detectable power (photon rate) requires knowledge of the field components Ex B2 and Ey B2 E in B2 due to this dipole p: The reciprocity theorem [24

R. E. Collin, Field Theory of Guided Waves (IEEE Press, New York, 1991).

] states for current distributions J⃗B2 and J⃗FP in B2 in the focal plane which radiate the electrical fields E⃗B2 and E⃗FP:

∫ dV E→ FP⋅ J→ B2= ∫ dV E→ B2⋅ J→ FP

If we ascribe the two currents to point dipoles p⃗ FP and p⃗B2, we obtain

E→ p B2 ( r→ FP)⋅ p→ FP= E→ p FP ( r→ B2)⋅ p→ B2

Thus (assuming dipoles with unit strength), the problem of calculating the x-component of the field in Ex B2 due to the radiating dipole p⃗ FP in the focal plane is equivalent to the calculation of projection of the field on pFP ⃗ due to a dipole p⃗B2 with identical strength situated at B2, if p⃗ B2 is chosen parallel to the x coordinate. This situation is sketched for a z-oriented dipole p⃗ FP in Figure S2. For the y component, an analogous argument holds. (Note: the electrical fields right behind the focal sphere and in B1 are assumed to be identical)

A3.2 Evaluation of the fields
Fig.A2. 2. Illustration of reciprocity

Let us now consider Ex B2 (RB2 ) the x-component of the electrical field in B2 which is calculated as the z-component of the field due to the dipole p⃗ B2 (rB2 ) which is oriented parallel to the x axis. Its contribution is decomposed in a p and an s component, giving rise to plane wave contributions travelling towards the focal plane

E→ sp= p→ sp k2 e i k→ d→ ∣ d→∣ 1 4π ε0

p and s denote the polarizations relative to the plane of incidence. By interaction with the multilayer system, this radiation gives rise to a local field at the dipole position. Therefore, the strategy to calculate this focal field Ez FP,C , due to an incident local plane wave from the block plane as outlined in Section A1 can be applied again when this field is assumed to have the amplitude in x-direction:

E˜ B2x =p k2 1 4π ε0 1 d cos (θ)

Here, the correction factor 1 cos (θ) must be applied when transforming ΔAFS→ΔAB2. We obtain with the same steps:

• transformation to a local coordinate system where the (x’, z’) plane equals the plane of incidence for this ray.

• calculation of the individual local fields for the p/s component by the transfer matrix algorithm,

• rotation back to the fixed coordinate system

E˜ FP= R −1⋅A⋅R⋅ e^x E˜ B2x= [ R −1AR]( r B2)⋅ e^x E˜ B2x= [ R −1AR]( r B2) e^xp k2 1 4π ε0 1 d cos⁡(θ)

Now, this quantity equals the field component Ex B2 due to reciprocity (see section A3.1) and we obtain for an arbitrarily oriented dipole p⃗

E B2x= ( px py pz)· [ ( [ R −1AR]xx [ R −1AR]xy [ R −1AR]xz [ R −1AR]yx [ R −1AR]yy [ R −1AR]yz [ R −1AR]zx [ R −1AR]zy [ R −1AR]zz) ( 1 0 0)] k2 4π ε0 1 d cos (θ)
(A3a)

And in an analogous manner

E B2y= ( px py pz)· [ ( [ R −1AR]xx [ R −1AR]xy [ R −1AR]xz [ R −1AR]yx [ R −1AR]yy [ R −1AR]yz [ R −1AR]zx [ R −1AR]zy [ R −1AR]zz) ( 1 0 0)] k2 4π ε0 1 d cos (θ)
(A3b)

If we assume now a pure z dipole we obtain (A2), (A3)

( px py pz)= ( 0 0 EFPz ( λ λ0)2 α0)

E B2x= EFPz ( λ λ0)2 α0 [ R −1AR]zx ( r B2) k2 1 4π ε0 1 d cos (θ)

E B2y= EFPz ( λ λ0)2 α0 [ R −1AR]zy ( r B2) k2 1 4π ε0 1 d cos (θ)

The energy flux of this field is

dP d A B2= cn E→. D→ =cn ε0 ∣ E→∣2=

cn ε0 [ α0 λ2 k2 d4π λ02]2 [ EzFP ( rFP)]2 { [ [ R −1AR]zx ( r B2)]2 + [ [ R −1AR]zy ( r B2)]2}· 1 cos (θ)
(A4)

with the dielectric displacement D⃗, the speed of light c and the refractive index n. The expression (A4) factorizes into

i) a constant pre-factor,

ii) The modulus squared of the excitation field projected on the polarizable unit

iii) a contribution that depends exclusively on rB2 . From this, the intensity is obtained by integrating over the non covered part of B2

P= cn ε0 [ α0 λ2 k2 d4π λ02]2 A B2[ EzFP ( rFP)]2×

∑ Δ A B2 A B2 { [ 1 cos (θ) [ R −1AR]zx ( r B2)]2 + [ 1 cos (θ) [ R −1AR]zy ( r B2)]2}
(A5)

Together with (A1), this expression forms the basis of the discussion in the main manuscript.

Acknowledgment

We acknowledge financial support from the ‘Bundesministerium für Bildung und Forschung’ (03N8702).

References and links

1.

J. R. Lakowicz, “Radiative Decay Engineering: Biophysical and Biomedical Applications,” Anal. Biochem. 298, 1–24(2001). [CrossRef] [PubMed]

2.

A. Otto, I. Mrozek, H. Grabhorn, and W. Akemann, “Surface-enhanced raman-scattering,” J. Phys. Condens. Matter 4, 1143–1212 (1992). [CrossRef]

3.

K. A. Willets and R. P. Van Duyne, “Localized surface plasmon resonance spectroscopy and sensing,” Annu. Rev. Phys. Chem. 58, 267–297 (2007). [CrossRef]

4.

D. M. Schaadt, B. Feng, and E. T. Yu, “Enhanced semiconductor optical absorption via surface plasmon excitation in metal nanoparticles,” Appl. Phys. Lett. 86(2005). [CrossRef]

5.

K. Joulain, J. P. Mulet, F. Marquier, R. Carminati, and J. J. Greffet, “Surface electromagnetic waves thermally excited: Radiative heat transfer, coherence properties and Casimir forces revisited in the near field,” Surf. Sci. Rep. 57, 59–112 (2005). [CrossRef]

6.

C. Sönnichsen, S. Geier, N. E. Hecker, G. von Plessen, J. Feldmann, H. Ditlbacher, B. Lamprecht, J. R. Krenn, F. R. Aussenegg, V. Z. H. Chan, J. P. Spatz, and M. Moller, “Spectroscopy of single metallic nanoparticles using total internal reflection microscopy,” Appl. Phys. Lett. 77, 2949–2951 (2000). [CrossRef]

7.

C. Sönnichsen, T. Franzl, T. Wilk, G. von Plessen, J. Feldmann, O. Wilson, and P. Mulvaney, “Drastic reduction of plasmon damping in gold nanorods,” Phys. Rev. Lett. 88(2002). [CrossRef] [PubMed]

8.

S. Schultz, D. R. Smith, J. J. Mock, and D. A. Schultz, “Single-target molecule detection with nonbleaching multicolor optical immunolabels,” Proc. Natl. Acad. Sci. USA 97, 996–1001 (2000). [CrossRef] [PubMed]

9.

J. J. Mock, M. Barbic, D. R. Smith, D. A. Schultz, and S. Schultz, “Shape effects in plasmon resonance of individual colloidal silver nanoparticles,” J. Chem. Phys. 116, 6755–6759 (2002). [CrossRef]

10.

P. K. Aravind and H. Metiu, “The effects of the interaction between resonances in the electromagnetic response of a sphere-plane structure - applications to surface enhanced spectroscopy,” Surf. Sci. 124, 506–528 (1983). [CrossRef]

11.

H. Xu and M. Käll, “Surface-Plasmon-Enhanced Optical Forces in Silver Nanoaggregates,” Phys. Rev. Lett. 89, 246802 (2002). [CrossRef] [PubMed]

12.

E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, “A hybridization model for the plasmon response of complex nanostructures,” Science 302, 419–422 (2003). [CrossRef] [PubMed]

13.

T. Okamoto and I. Yamaguchi, “Optical absorption study of the surface plasmon resonance in gold nanoparticles immobilized onto a gold substrate by self-assembly technique,” J. Phys. Chem. B 107, 10321–10324 (2003). [CrossRef]

14.

Y. M. Gerbshtein, I. A. Merkulov, and D. N. Mirlin, “Transfer of Luminescence-center energy to surface plasmons,” JETP Lett. 22, 35–36 (1975).

15.

H. Knobloch, H. Brunner, A. Leitner, F. Aussenegg, and W. Knoll, “Probing the evancescent field of propagating plasmon surface-polaritons by fluorescence and raman spectroscopy,” J. Chem. Phys. 98, 10093–10095 (1993). [CrossRef]

16.

F. D. Stefani, K. Vasilev, N. Bocchio, N. Stoyanova, and M. Kreiter, “Surface-plasmon-mediated single-molecule fluorescence through a thin metallic film,” Phys. Rev. Lett. 94 (2005). [CrossRef] [PubMed]

17.

B. Hecht, H. Bielefeldt, L. Novotny, Y. Inouye, and D. W. Pohl, “Local excitation, scattering, and interference of surface plasmons,” Phys. Rev. Lett. 77, 1889–1892 (1996). [CrossRef] [PubMed]

18.

G. Frens, “Controlled Nucleation for the Regulation of the Particle Size in Monodisperse Gold Suspensions,” Nature Phys. Sci. 241, 20–22 (1973).

19.

E. Kretschmann and H. Raether, “Radiative Decay of Non Radiative Surface Plasmons Excited by Light,” Zeitschrift Für Naturforschung Part A 23, 2135–2136 (1968).

20.

B. Sick, B. Hecht, and L. Novotny, “Orientational imaging of single molecules by annular illumination,” Phys. Rev. Lett. 85, 4482–4485 (2000). [CrossRef] [PubMed]

21.

A. Bouhelier, F. Ignatovich, A. Bruyant, C. Huang, G. Colas des Francs, J. C. Weeber, A. Dereux, G. P. Wiederrecht, and L. Novotny, “Surface plasmon interference excited by tightly focused laser beams,” Opt. Lett. 32, 2535–2537 (2007). [CrossRef] [PubMed]

22.

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light , 6 ed. (Cambridge University Press, 1997). [PubMed]

23.

P. Yeh, Optical Waves in Layered Media (John Wiley & Sons Inc, Hoboten, 1988).

24.

R. E. Collin, Field Theory of Guided Waves (IEEE Press, New York, 1991).

OCIS Codes
(180.1790) Microscopy : Confocal microscopy
(240.6680) Optics at surfaces : Surface plasmons
(290.5820) Scattering : Scattering measurements

ToC Category:
Microscopy

History
Original Manuscript: July 3, 2008
Revised Manuscript: August 14, 2008
Manuscript Accepted: August 14, 2008
Published: October 20, 2008

Virtual Issues
Vol. 3, Iss. 12 Virtual Journal for Biomedical Optics

Citation
Marcus Schmelzeisen, Jacqueline Austermann, and Maximilian Kreiter, "Plasmon mediated confocal dark-field microscopy," Opt. Express 16, 17826-17841 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-22-17826


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References

  1. J. R. Lakowicz, "Radiative Decay Engineering: Biophysical and Biomedical Applications," Anal. Biochem. 298, 1-24 (2001). [CrossRef] [PubMed]
  2. A. Otto, I. Mrozek, H. Grabhorn, and W. Akemann, "Surface-enhanced raman-scattering," J. Phys. Condens. Matter 4, 1143-1212 (1992). [CrossRef]
  3. K. A. Willets and R. P. Van Duyne, "Localized surface plasmon resonance spectroscopy and sensing," Annu. Rev. Phys. Chem. 58, 267-297 (2007). [CrossRef]
  4. D. M. Schaadt, B. Feng, and E. T. Yu, "Enhanced semiconductor optical absorption via surface plasmon excitation in metal nanoparticles," Appl. Phys. Lett. 86(2005). [CrossRef]
  5. K. Joulain, J. P. Mulet, F. Marquier, R. Carminati, and J. J. Greffet, "Surface electromagnetic waves thermally excited: Radiative heat transfer, coherence properties and Casimir forces revisited in the near field," Surf. Sci. Rep. 57, 59-112 (2005). [CrossRef]
  6. C. Sönnichsen, S. Geier, N. E. Hecker, G. von Plessen, J. Feldmann, H. Ditlbacher, B. Lamprecht, J. R. Krenn, F. R. Aussenegg, V. Z. H. Chan, J. P. Spatz, and M. Moller, "Spectroscopy of single metallic nanoparticles using total internal reflection microscopy," Appl. Phys. Lett. 77, 2949-2951 (2000). [CrossRef]
  7. C. Sönnichsen, T. Franzl, T. Wilk, G. von Plessen, J. Feldmann, O. Wilson, and P. Mulvaney, "Drastic reduction of plasmon damping in gold nanorods," Phys. Rev. Lett. 88, 077402 (2002). [CrossRef] [PubMed]
  8. S. Schultz, D. R. Smith, J. J. Mock, and D. A. Schultz, "Single-target molecule detection with nonbleaching multicolor optical immunolabels," Proc. Natl. Acad. Sci. USA 97, 996-1001 (2000). [CrossRef] [PubMed]
  9. J. J. Mock, M. Barbic, D. R. Smith, D. A. Schultz, and S. Schultz, "Shape effects in plasmon resonance of individual colloidal silver nanoparticles," J. Chem. Phys. 116, 6755-6759 (2002). [CrossRef]
  10. P. K. Aravind and H. Metiu, "The effects of the interaction between resonances in the electromagnetic response of a sphere-plane structure - applications to surface enhanced spectroscopy," Surf. Sci. 124, 506-528 (1983). [CrossRef]
  11. H. Xu and M. Käll, "Surface-Plasmon-Enhanced Optical Forces in Silver Nanoaggregates," Phys. Rev. Lett. 89, 246802 (2002). [CrossRef] [PubMed]
  12. E. Prodan, C. Radloff, N. J. Halas, and P. Nordlander, "A hybridization model for the plasmon response of complex nanostructures," Science 302, 419-422 (2003). [CrossRef] [PubMed]
  13. T. Okamoto and I. Yamaguchi, "Optical absorption study of the surface plasmon resonance in gold nanoparticles immobilized onto a gold substrate by self-assembly technique," J. Phys. Chem. B 107, 10321-10324 (2003). [CrossRef]
  14. Y. M. Gerbshtein, I. A. Merkulov, and D. N. Mirlin, "Transfer of Luminescence-center energy to surface plasmons," JETP Lett. 22, 35-36 (1975).
  15. H. Knobloch, H. Brunner, A. Leitner, F. Aussenegg, and W. Knoll, "Probing the evancescent field of propagating plasmon surface-polaritons by fluorescence and raman spectroscopy," J. Chem. Phys. 98, 10093-10095 (1993). [CrossRef]
  16. F. D. Stefani, K. Vasilev, N. Bocchio, N. Stoyanova, and M. Kreiter, "Surface-plasmon-mediated single-molecule fluorescence through a thin metallic film," Phys. Rev. Lett. 94 (2005). [CrossRef] [PubMed]
  17. B. Hecht, H. Bielefeldt, L. Novotny, Y. Inouye, and D. W. Pohl, "Local excitation, scattering, and interference of surface plasmons," Phys. Rev. Lett. 77, 1889-1892 (1996). [CrossRef] [PubMed]
  18. G. Frens, "Controlled Nucleation for the Regulation of the Particle Size in Monodisperse Gold Suspensions," Nature Phys. Sci. 241, 20-22 (1973).
  19. E. Kretschmann and H. Raether, "Radiative Decay of Non Radiative Surface Plasmons Excited by Light," Zeitschrift Für Naturforschung Part A 23, 2135-2136 (1968).
  20. B. Sick, B. Hecht, and L. Novotny, "Orientational imaging of single molecules by annular illumination," Phys. Rev. Lett. 85, 4482-4485 (2000). [CrossRef] [PubMed]
  21. A. Bouhelier, F. Ignatovich, A. Bruyant, C. Huang, G. Colas des Francs, J. C. Weeber, A. Dereux, G. P. Wiederrecht, and L. Novotny, "Surface plasmon interference excited by tightly focused laser beams," Opt. Lett. 32, 2535-2537 (2007). [CrossRef] [PubMed]
  22. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 6 ed., (Cambridge University Press, 1997). [PubMed]
  23. P. Yeh, Optical Waves in Layered Media (John Wiley & Sons Inc, Hoboten, 1988).
  24. R. E. Collin, Field Theory of Guided Waves (IEEE Press, New York, 1991).

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