Journal of the Korean Physical Society, Vol. 47, August 2005, pp. S130∼S134

Surface Plasmon-Mediated Near-Field Optical Addressing of Single Nano-Objects: A Simple Analysis A. Drezet∗ Institut f¨ ur Experimentalphysik, Karl Franzens Universit¨ at Graz, Universit¨ atsplatz 5 A-8010 Graz, Austria

M. Brun,† J. C. Woehl and S. Huant‡ Laboratoire de Spectrom´etrie Physique, CNRS UMR5588, Universit´e Joseph Fourier Grenoble, BP 87, 38402 Saint Martin d’H`eres cedex, France (Received 6 September 2004) Recently, we have presented a scheme for remotely addressing single nano-objects by means of near-field optical microscopy that makes only use of one of the most fundamental properties of electromagnetic radiation: its polarization [M. Brun et al., Europhys. Lett. 64, 634 (2003)]. A medium containing optically active nanoobjects is covered with a thin metallic mask presenting sub-wavelength holes. When the optical tip is positioned some distance away from a hole, surface plasmons in the metal coating are generated which, by turning the polarization plane of the excitation light, transfer the excitation towards a chosen hole and induce emission from the underlying nano-objects. This paper gives a quantitative description of this addressing method. PACS numbers: 07.79.Fc, 78.67.-n, 73.20.Mf Keywords: Optical addressing, Surface plasmons, Quantum dots

I. INTRODUCTION Ultimate control of light requires the combined ability of confining photons to extremely small dimensions, i.e. much smaller than their wavelength, and manipulating them in a well-controlled state to address at will optically active single nanometer-scaled objects, such as single molecules, semiconductor QDs or nanocrystals. Routes in this direction have been opened up by Near-field Scanning Optical Microscopy (NSOM) [1] : for instance, early single molecule detection [2] with NSOM using conventional optical tips has recently evolved to NSOM imaging with a single molecule serving as a point-like light source [3]. Another example is the quantum corral [4], an elegant realization of Scanning-Tunneling Microscopy, which has been extended recently to the optics world by NSOM [5]. Recently, we have presented a simple optical method aimed at remotely addressing single nanoobjects by means of NSOM [6]. In essence, our method makes only use of one of the most fundamental properties of an electromagnetic field with respect to a scalar field: its polarization state, which is controlled at the apex of a NSOM tip [7]. This polarization control allows to launch the excitation to active objects by means of ∗ E-mail:

[email protected]; Address: LETI-DTS, CEA-Grenoble, 17 rue des Martyrs, 38054 Grenoble Cedex 9, France. ‡ E-mail: [email protected] † Present

surface plasmons propagating in a hollowed metal mask deposited on the material that includes the active elements as shown schematically in Fig. 1. The aim of this article is to give a simple quantitative analysis of this promising addressing procedure.

II. RADIATION OF A NSOM TIP IN FRONT OF A MIRROR In order to describe theoretically the experiment [6] we first need a consistent model to describe the radiation of light by a NSOM tip. Such model valid both in the optical far-field and near-field of the aperture has been obtained previously using analytical calculations of the electromagnetic field produced by a truncated conical antenna with a sub-wavelength aperture. This model has been tested experimentally and reproduces quantitatively far-field [8, 9] and near-field [10] observations. For the present purpose the essential result is that we can in a first approximation consider the emitting aperture as a dipolar source. The aperture is then equivalent to a system of two oscillating dipoles (an electric dipole and a magnetic one) located in the plane of the aperture and perpendicular to each other. The theory shows that the electric dipole Ptip is collinear to the incident polarization of the incoming laser injected into the fiber. In reality, because of mechanical stress inside the fiber core, the polarization is changed during propagation and it is

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Surface Plasmon-Mediated Near-Field Optical Addressing of· · · – A. Drezet et al.

only the polarization in the very last few micrometers before the tip apex which is important. However the polarization can be conserved to be linear within more than 90 % accuracy in such a way that during the experiment of Ref. [6] we can consider having a linear polarization at the fiber apex with a direction which must be characterized in situ as explained below. If we write Mtip the magnetic dipole then we have [8] Mtip = 2n × Ptip

(1)

where n is the normal to the aperture outwardly oriented. The factor two appearing in this equation is a general property of dipoles associated with sub-wavelength apertures as found originally by Bethe for a particular configuration. It is important to note that the dipoles used here are completely different in orientation from the dipoles used in the well known Bethe-Bouwkamp Model [11] valid for an aperture in a plane screen illuminated by a plane wave. Following the Bethe-Bouwkamp analysis the electric dipole is perpendicular to the aperture plane and colinear to the normal component of the incident electric field. This is in contradiction with our model and we show on Fig. 2A the electric near-field of a NSOM tip which has a clear asymptotic dipolar character when looked at large distance from the opening. The validity of our analysis is confirmed by far-field goniometrical observations [7] and near-field measurements on fluorescent nanospheres [10] which together refute the Bethe-Bouwkamp model in the present NSOM configuration. We attribute such discrepancy between both models to the high confinement and inhomogeneity of the electromagnetic field at the tip apex and to the particular conical geometry of the metallic cladding.

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In the experimental set-up [12, 13] the optical signal emitted at the first focus FA of an elliptic mirror is collected by a multimode fiber located at the second focus FB . If we consider a point-like source in the region of FA the fraction of energy which can in principle be collected is equal to the energy emitted in the half space z ≥ 0 beyond the sample. In the present case the source consists of two oscillating dipoles and we write Ptot ( Mtot ) the electric (magnetic) dipole moment of the punctual source. In the radiation zone (i.e. at distance r of the source very large compared to the wavelength λ) the electromagnetic fields take on the limiting forms [14] EF F ' k 2 (ˆ r × Ptot ) × ˆ r BF F ' ˆ r × EF F ,

eikr eikr − k 2 (ˆ , r × Mtot ) r r (2)

where ˆ r is the unit position vector directed from the source to the direction of the observation point. The time-averaged power radiated per unit solid angle by the oscillating source is dS c = Re[r2ˆ r · EF F × B∗F F ], dΩ 8π

(3)

i.e. for r → +∞ ck 4 2 dS 2 2 = [P + Mtot − (ˆ r · Ptot ) dΩ 8π tot 2 − (ˆ r · Ptot ) + 2ˆ r · (Ptot × Mtot )].

(4)

A careful integration done over the half solid angle for z ≥ 0 gives [15] S=

ck 4 2 3 2 [Ptot + Mtot + (Px My − Py Mx )]. 3 2

(5)

Now, due to the presence of the sample and in particular of the aluminum mask in front of the tip aperture, the electromagnetic field emerging from the NSOM tip

Fig. 1. Principle of the addressing method demonstrated experimentally in Ref. [6]. A medium containing optically active nanoobjects (here semiconductor quantum dots: QDs) is covered with a thin metallic mask presenting sub-wavelength holes. When the optical tip is positioned some distance away from a hole, surface plasmons in the metal coating are generated which, by turning the polarization plane of the excitation light, transfer the excitation towards a chosen hole and induce emission (at lower energy) from the underlying nano-objects.

Fig. 2. Calculated electric field lines at the tip apex. In A, the tip is free standing while it is facing (in the near-field) a metal mask in B. Here the model of Ref. [8] is applied to both the optical tip and to its image in the aluminium mirror.

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Journal of the Korean Physical Society, Vol. 47, August 2005

is reflected back and we can associate two image dipoles to the tip dipoles. In the z ≥ 0 half space the total electromagnetic far-field is subsequently radiated by the two dipoles Ptot = Ptip + Pimage Mtot = Mtip + Mimage .

(6)

It is important to remark that the dipolar approximation neglects higher-order contribution such as the quadrupolar and octopolar terms in the radiation field. The contribution of each multipoles to the total intensity is a function IE,M (l, m) of the integer l, m and depends on the electric (E) or magnetic (M) nature of the multipole.
We have IE,M (l + 1, m) /IE,M (l, m) ∼ O k 2 d2 where d is the feedback distance and k the wave vector of the incoming light. For d = 10 nm and λ = 600 nm, we have k 2 d2 ' 10−2 and consequently such contributions can be neglected in a first approximation. The image theory is only an approximation (for a more complete electromagnetic theory of dipole radiation in the presence of a partially reflecting halfspace, see Sommerfeld [16, 17] and Chance [18]). Rigourously written the electromagnetic field E, B radiated by the 2 tip dipoles in presence of the mirror is i ∇ × (Gm · Mtip ) k i B = Gm · Mtip − ∇ × (Ge · Ptip ) k E = Ge · Ptip +

(7)

where Ge,m is the dyadic Green function [19] representing the electric (magnetic) field radiated by three unit orthogonal electric (magnetic) dipoles located in front of a mirror. Such Green tensors can be written in an integral form and take simple forms only in the special case of a perfect metal for which we have

G0e = k 2 ek ek + ∇∇k GD + k 2 e⊥ e⊥ + ∇∇⊥ GN (8) and

G0m = k 2 ek ek + ∇∇k GN + k 2 e⊥ e⊥ + ∇∇⊥ GD . (9) Here the first (second) term in the right-hand side of the equation acts only on the parallel (normal) components of the dipoles (∇k,⊥ [...] = ek,⊥ · ∂[...]/∂xk,⊥ ). The two scalar Green functions GD,N satisfy respectively the Dirichlet and Neumann conditions for the infinite plane 0 boundary problem. We have GD,N = eikR /R ∓ eikR /R0 where R denotes the distance between the observation point M and the point-like source and R0 the distance between M and the geometrical image point located in the z ≤ 0 half space. For a real metal such as aluminum we cannot rigourously identify the Green tensors with the ones given by Eqs. (8) and (9). However it was observed by Sommerfeld [16] and Van der Pol [20] in the context of antenna radiation theory that in the approximation

Fig. 3. Electric density energy around the tip (logarithmic scale) calculated by neglecting the contribution of the magnetic dipole (left) and including it (right). This figure has been computed using the point-like dipole(s) describing the optical tip and its image in the aluminium mirror.

where the sources are in the immediate neighborhood of the surface, the metal can be considered as perfect in the analysis of the propagating field at large distance z from the surface as it is the case in the experimental configuration. Accepting the dipole model of the image theory as a good approximation, we obtain the electric field lines shown in Fig. 2B. The importance of the effect of the magnetic dipole on the radiated field is shown in Fig. 3: obviously, such contribution cannot be neglected.

III. RADIATION OF A BETHE APERTURE EXCITED BY A NSOM TIP So far we have not considered the presence of the nanohole in the mask. The electromagnetic field produced by the tip and its image excites the hole which reacts in a first approximation as a Bethe-Bouwkamp two-dipole source. Such source is characterized by electric and magnetic dipoles given by [11,14] PBB = −

a3 E⊥ , 3π

MBB = 2

a3 Bk 3π

(10)

where E⊥ and Bk are respectively the normal component of the electric field and the tangential component of the magnetic field calculated at the center of the nanohole (radius a) and produced by the tip and its image in absence of the nanohole. Because the two electric dipoles of the tip and of its image compensate each other the electric field is essentially quadrupolar in the vicinity of the tip as we can see on Fig. 2B [21]. However the magnetic field is essentially dipolar and dominates the electric field in the region close to the tip aperture. This means that the nanohole is principally excited by a magnetic field

Surface Plasmon-Mediated Near-Field Optical Addressing of· · · – A. Drezet et al.

Fig. 4. Experimental (left) and simulated (right) reflection images of sub-wavelength holes in a metal film. The simulated image is calculated using the dipolar-interaction model between the tip and a nanohole described in the text. The contrast ratio between dark and bright is approximately 1 to 5 both in the experiment and in the simulation.

and that the Bethe-Bouwkamp source is of magnetic nature because |MBB |  |PBB |. Using this fact we can 4 a3 Bk |2 where simplify Eq. (5) in S (x, y) ' ck3 |2Mtip +2 3π Bk is the field in the point x, y in the plane z = 0 produced by the tip. This point x, y corresponds to the position of the center of nanohole relative to the intersection of the tip symmetry axis with the z = 0 plane at x = y = 0. Using such an approach we have previously explained [6] the experimental reflectivity images of the hollowed mask (taken at the same wavelength as the excitation). However it appears that we obtain a better agreement if we consider the complete field produced by the two dipoles without neglecting any contribution. The result is represented on Fig. 4 and the main finding is that the image is characterized by two bright spots separated by a dark one (in the simplified interpretation given in Ref. [6], the bright spots appear brighter than here). The electric polarization is represented by an white arrow perpendicular to the axis joining the two bright spots. This effect is a direct consequence of the importance of the magnetic field in this experiment and will be not present without the magnetic dipole of the tip. In the above discussion, we have neglected the possible contribution of the addressed nano-objects. This is based on the assumption that they do not contribute to the reflection images because they emit light at longer wavelengths. IV. ADDRESSING QUANTUM DOTS USING A SURFACE PLASMON WAVE Surface plasmons (SPs) are electromagnetic waves confined at the interface between a metal and a dielectric medium such as air or glass [22]. Such waves are strongly exponentially damped in the direction perpendicular to the interface and we cannot expect to see a direct manifestation of such waves at distances much higher that

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the considered optical wavelength λ. It is possible to see indirectly a SP by considering for example the exchange of energy between propagating 3D waves and creeping 2D SP as in the ATR method [22]. However we can detect a SP by recombining a fraction of its energy in a 3D propagating mode using a scattering defect such as a small hole [23, 24]. In the present case, it is such a method which is used because SP modes are excited by the NSOM tip located in front of the aluminum surface and are subsequently scattered by the nanohole. Our previous treatment of the interaction of the tip radiation with the metal plane was obtained under the assumption that the metal is perfect. If this is justified in studies considering the far-field radiation of a NSOM tip only, the finite permittivity  (ω) of the metal must be taken into account in order to have a surface plasmon. Such an effect was predicted originally by Sommerfeld [16,17] who observed that a point-like dipole located on a metal surface radiates 2D waves that are exponentially damped not only perpendicular to the surface but also along the direction of propagation in the interface plane. Such attenuated waves take the asymptotical √ form ψ (R) ∝ eikSP R e−κSP R / R at long p distance R from the source, were kSP + iκSP = (ω/c) / ( + 1) defines the plasmon wavevector dispersion rule. In addition it is well known that an electric dipole parallel to the metal plane generates a plasmon propagating essentially along the direction of the polarization of the dipole [17,23–25]. The angular dispersion of the intensity in the plane follows a simple cos2 (φ) law characteristic of a 2D dipole. Because the tip is not only an electric dipole in our configuration the contribution to the plasmon field carried by the field generated by the magnetic dipole can be questioned. In order to define the plasmon emission and its propagation along the surface we need to express the Green tensors Ge,m for a metal with finite permittivity. Using the method of the Hertz vector we obtain the formula

˜ D + k 2 e⊥ e⊥ + ∇∇⊥ G ˜N Ge = k 2 ek ek + ∇∇k G
2 + k e⊥ ek + ∇ek ∇⊥ H (11) and

˜ N + k 2 e⊥ e⊥ + ∇∇⊥ G ˜D Gm = k 2 ek ek + ∇∇k G
+ k 2 e⊥ ek + ∇ek ∇⊥ H. (12) ˜ D,N are generalization of GD,N defined in the Here G upper medium by Z +∞ 2iλ ˜ D = GD + G dλ J0 (λρ) eik1z (z+h) k1z + k2z 0 Z +∞ 2i1 λk2z ˜ N = GN + G dλ J0 (λρ)  k 2 1z + 1 k2z 0 ·eik1z (z+h)

(13) p

where k1z,2z = (1,2 k 2 − λ2 ) and Im (k1z,2z ) ≥ 0 for the two medium with index 1 = 1 (air) and 2 =  (ω)

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Journal of the Korean Physical Society, Vol. 47, August 2005

REFERENCES

(metal). Similarly we have Z +∞ 2 (2 − 1 ) λ2 H = cos (φ) dλ 2 k1z + 1 k2z k1z + k2z 0 ·J1 (λρ) eik1z (z+h) .

(14)

The integral terms contain the exponentially damped plasmon contribution. However, the dominant contribution to the plasmon comes from the electric dipole with GD because the optical index of the metal appears in the denominator of the integrand in H and in GN . It is possible (see [17, 26]) to calculate this electric plasmon contribution and we √ obtain asymptotically ψ (R) ∝ cos (φ) eikSP R e−κSP R / R. We have observed this behavior in our experiment [6]. It was clearly visible that by rotating the laser polarization at the tip apex we can change the intensity of the luminescence emitted by the quantum dots [6] located in the active medium below the nanohole. The angular dispersion follows a cos2 (φ) law as expected for SP mode. It can be noted that the polarization reference can be controlled using the reflectivity images described above. Then the plasmon propagates essentially perpendicularly to the axis joining the two bright spots. This cos2 (φ) law can be expected from the property of electromagnetic field at the apex of the NSOM tip. It is indeed well known that the transverse electric mode TE11 which propagates in the fiber with metal cladding exhibits a surface charge distribution varying as cos2 (φ) where φ is the angular coordinate of the point on the metal if the polarization axis is identified with the x axis and the fiber symmetry axis with the z axis [10]. In addition, by changing the relative positions of the hole and of the tip we could observe experimentally [6] a damping in the excitation that follows approximately the e−2κSP R /R rule. The damping length is approximately 4.2 micrometers in agreement with values for aluminum in this optical domain (wavelength 515 nm) where  = −38.79 + 10.38i. V. CONCLUSION Different possibilities of this new method of addressing by surface plasmons could be exploited in the future. For example it will be possible to address different optical elements in vivo a fact which has clear technological implications. The magnetic dipole playing an important role in the description process, it could be possible to excite a magnetic element such as a quantum dot doped with a magnetic impurity. The possibility of selective addressing could also have important implications for quantum optics at the nanoscale using plasmons.

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