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Virtual Journal for Biomedical Optics

| EXPLORING THE INTERFACE OF LIGHT AND BIOMEDICINE

  • Editor: Gregory W. Faris
  • Vol. 2, Iss. 10 — Oct. 31, 2007
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Effect of the incident angle on the electric near field of a conical probe for plane wave and Gaussian beam illumination

A. V. Goncharenko and Yia-Chung Chang  »View Author Affiliations


Optics Express, Vol. 15, Issue 18, pp. 11517-11529 (2007)
http://dx.doi.org/10.1364/OE.15.011517


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Abstract

We consider a simple analytical model for the electric near field of a semi-infinite conical probe and apply it to study the incident angle dependence of the field for the case of side illumination by both the plane wave and the Gaussian beam. The electric near field is shown to peak when approaching the grazing incidence. In some cases, a peak can also occur at an incident angle somewhat below 90°. The results obtained are in qualitative agreement with those for a thin semi-infinite wire and previously published results for the finite-size conical probes.

© 2007 Optical Society of America

1. Introduction

In spite of a great body of publications devoted to near-field optics and its numerous applications, especially to scanning near-field optical microscopy (SNOM), so far many aspects of the near-field theory remain uncovered. Among them, optimizing illumination conditions can be considered as an important issue affecting the performance of the experimental setup. Apart from the optical illumination scheme [1

1. A. Bouhelier, “Field-enhanced scanning near-field optical microscopy,” Microsc. Res. Tech. 69, 563–579 (2006). [CrossRef] [PubMed]

], the illumination conditions include the state of polarization of the incident beam, the angle of incidence, and parameters of the laser beam, such as waist width and divergence angle, as well as the focal spot position (usually one deals with Gaussian beam illumination).

It should be noted that all the above results are based on numerical computations. Naturally, the computations have been performed for finite-size tips and for specific materials only. Of course, this limits generality of the results obtained. To our knowledge, no analytical analysis for the incident angle dependence of the electric near field has been given in the literature. Although the analytical treatment is possible only for the semi-infinite tips, due to their relatively large lengths the real probes may frequently be considered as semi-infinite.

Furthermore, due to the complicated nature of various near-field phenomena, many people use different simplifying assumptions. At the same time, their validity and the range of applicability are on frequent occasions unclear. One of the frequently used approximations is dealing with plane-wave illumination. Meantime, as the light sources in the near-field optics are usually lasers, it would look more natural to deal with Gaussian beams to correctly interpret or predict experimental results. However, the literature on near-field optics of the Gaussian beams is highly limited. So, the range of applicability of the widely-used plane wave approximation has been something of a mystery.

In this work, we are going to address the above questions, namely, (i) how the electric near field of the semi-infinite material cone depends on the incident angle for the case of plane wave illumination and (ii) what happens when we are dealing with Gaussian beam illumination. The main goal of the present work is to find general regularities in the behavior of the near field as a function of illumination conditions using a simple mathematical model. There is no question that the proper solution of the above problem could be of practical significance.

2. Electric near field of a semi-infinite cone for plane wave illumination

It is well known that both the electric and magnetic fields may be expressed in terms of a pair of scalar Debye potentials U and V. So, in spherical coordinates the components of the electric field are (Er=(2r+k 2)(rU) E θ=r -1rθ(rU)+ikη(sinθ)-1φV, E φ=(r sinθ)-1r∂φ(rU)-ikη∂θV, where η is the intrinsic impedance of the surrounding medium and k=ω/c is the wave number. The Debye potentials for a sharp semi-infinite cone are known; if the cone is illuminated by a linearly polarized plane wave incident from the direction θ 0, φ 0 [see Fig. 1(a)], they are [12

12. J. J. Bowman, J. J. Bowman, T. B. A. Senior, and P. L. E. Uslenghi, eds., (Hemisphere, New York, 1987).

]

Fig. 1. Two schemes of illumination of a conical probe: side illumination (a) and total internal reflection illumination (b).
U=2ikm=0Λmνsexp(iνsπ2)jνs(kr)Pνsm(cosθ){νs(νs+1)0πα[Pνsm(cosα)]2sinαdα}1
×[msinm(φφ0)cosβ(sinθ0)1Pνsm(cosθ0)+cosm(φφ0)sinβθ0Pνsm(cosθ0)]
(1a)

and

V=2iηkm=0Λmμsexp(iμsπ2)jμs(kr)Pμsm(cosθ){μs(μs+1)0πα[Pμsm(cosα)]2sinαdα}1
×[mcosm(φφ0)cosβθ0Pμsm(cosθ0)msinm(φφ0)sinβ(sinθ0)1Pμsm(cosθ0)],
(1b)

where P ν m(…) is the associated Legendre function, j(…) is the spherical Bessel function, β is the polarization angle (the angle between the incident electric field and the plane of incidence), Λ m is the Neumann symbol (Λ m=1 at m=0 and Λ m=2 at m≥1), and νs and µs are the separation parameters which can be determined from subsidiary conditions.

In the near zone (kr≪1), keeping only the leading term in Eqs. (1) leads to the Bowman’s expression for the electric near field of the cone [12

12. J. J. Bowman, J. J. Bowman, T. B. A. Senior, and P. L. E. Uslenghi, eds., (Hemisphere, New York, 1987).

], namely

E(r,θ)=r̂Er+θ̂EθE0C(kr)ν1Pν1(cosθ0)cosβ[r̂+θ̂νθ]Pν(cosθ)
(2)

with E 0 the amplitude of the incident field, and the constant C=exp(12iπ(ν+1))π[sinα·2ν1Γ(ν+12)Pν1(cosα)νPν(cosα)]1. In what follows, we consider only the case of p-polarization which is of most practical interest; thus, we take β=0. It should be noted that both Felsen and Bowman have considered only the case of the perfectly conducting cone. In this case the boundary conditions for the separation parameter ν give a simple equation Pν(-cosα)=0. For the material cone with a finite permittivity this equation should be replaced by [13

13. J. Van Bladel, “Field singularities at the tip of a Dielectric Cone,” IEEE Trans. Antennas Propag. 33, 893–895 (1985). [CrossRef]

,14

14. M. Idemen, “Confluent Tip Singularity of the Electromagnetic Field at the Apex of a Material Cone,” Wave Motion 38, 251–277 (2003). [CrossRef]

]

ε1Pν+1(cosα)Pν(cosα)+ε2Pν+1(cosα)Pν(cosα)+(ε1ε2)cosα=0.
(3)

Pν1(cosθ)=12ν(ν+1)sinθ·2F1(2+ν,1ν;2;1cosθ2)
(4)

where 2 F 1(…) is the hypergeometrical function, it immediately follows that E θ→0 as θ→0. At the same time, both functions, Pν(cosθ) and P 1 ν(cosθ) as well, rise with the angle θ if the parameter ν<0 [17

17. R. N. Hall, “The application of non-integral Legendre functions to potential problems,” J. Appl. Phys. 20, 925–931 (1949). [CrossRef]

]. As a result, the total electric near field E(r,θ)=Er2+Eθ2 has a prominent minimum in the direction along the probe axis, i.e., at θ=0 [18

18. Our numerical simulations show that this statement is also valid for ν>0.

]. This contradicts to results of numerous computations (see, e.g., [5

5. R. Bachelot, F. H’Dhili, D. Barchiesi, G. Lerondel, R. Fikri, P. Royer, N. Landraud, J. Peretti, F. Chaput, G. Lampel, J. P. Boilot, and K. Lahlil, “Apertureless near-field optical microscopy: a study of the local tip field enhancement using photosensitive Azobenzene-containing films,” J. Appl. Phys. 94, 2060–2072 (2003). [CrossRef]

,15

15. R. M. Roth, N. C. Panoiu, M. M. Adams, R. M. Osgood, C. C. Neacsu, and M. B. Raschke, “Resonant-Plasmon Field enhancement from asymmetrically illuminated Conical Metallic-Probe Tips,” Opt. Express 14, 2921–2931 (2006). [CrossRef] [PubMed]

]). Thus, to become a useful tool for various near-field studies, the Bowman’s solution needs some modification, taking proper account of finite curvature of the probe’s apex.

One of approaches allowing one to take into account the nonzero radius of curvature of the conical probe is based on the introduction of a “core” geometry, namely, the sphere on the orthogonal cone [19

19. A. V. Goncharenko, J. K. Wang, and Y. C. Chang, “Electric Near-Field Enhancement of a Sharp Semi-Infinite Conical Probe: Material and Cone Angle Dependence,” Phys. Rev. B 74, 235442 (2006). [CrossRef]

]. One can show that the electric field near the sphere could be approximately estimated as

E(r,θ)E0C(kr)ν1[1+ν+1νB(r,r0)]Pν1(cosθ0)[r̂+θ̂νθ1B(r,r0)1+ν+1νB(r,r0)]Pν(cosθ)
(5)

with r0 the sphere radius, B(r,r0)=S(r0,r)2ν+1, and S=(ε2ε1)(ε2+ν+1νε1). As was noted earlier [20

20. W. P. Dyke, J. K. Trolan, W. W. Dolan, and G. Barnes, “The Field Emitter: Fabrication, Electron Microscopy, and Electric Field Calculations,” J. Appl. Phys. 24, 570–576 (1953). [CrossRef]

,21

21. J. C. Wiesner and T. E. Everhart, “Point-Cathode Electron Sources — Electron Optics of the Initial Diode Region,” J. Appl. Phys. 44, 2140–2148 (1973). [CrossRef]

], azimuthally invariant equipotential surfaces of the above “core” system (“bowling” pin) constitute a class of the shapes [22

22. D. Hu, M. Micic, N. Klymyshyn, Y. D. Suh, and H. P. Lu, “Correlated Topographic and Spectroscopic Imaging beyond Diffraction limit by Atomic Force Microscopy Metallic Tip-Enhanced Near-Field Fluorescence Lifetime Microscopy,” Rev. Sci. Instrum.74, 3347–3355 (2003). [CrossRef]

]. In the near-field approximation, the equipotential surfaces satisfy the equation [19

19. A. V. Goncharenko, J. K. Wang, and Y. C. Chang, “Electric Near-Field Enhancement of a Sharp Semi-Infinite Conical Probe: Material and Cone Angle Dependence,” Phys. Rev. B 74, 235442 (2006). [CrossRef]

]

ψ1(r,θ)=ψ0(kr)ν[1S(r0r)2ν+1]Pν(cosθ).
(6)

Fig. 2. Equipotential surfaces for the “bowling” pin geometry.

Equations (2) and (5) give the same dependence of the electric near field on the angle θ 0, namely, E(r,θ)∝P 1 ν(cosθ 0). It is therefore of interest to take a look at the behavior of the Legendre function of the first order, P 1 ν(cosθ). The dependence of the function P 1 ν(cosθ 0) on the incident angle θ i=π-θ 0 for different ν is given in Fig. 3. As evident from the figure, the dependence is not trivial. Indeed, for small (both positive and negative) values of ν the function drops monotonically to zero. As 2F1(2+ν,1ν;2;1cosθ2)1asθ0, P 1 ν(cosθ 0)∝ν(ν+1)sinθ 0 at small θ 0=π-θ i, see Eq. (4). At the same time, as ν becomes close to unity, a peak of the function ()0 P 1 ν(cosθ 0) can occur. This peak can occur at incident angles in excess of about 53°, and only for the values of the parameter ν in excess of about 0.85 (see Fig. 4). The peak position is near π/2 as ν tends to unity.

Fig. 3. Absolute value of the associated Legendre function P 1ν(cosθ 0) vs the incident angle θ i.
Fig. 4. Dependence of the incident angle θi at which P 1 ν(cosθ 0) peaks vs the parameter ν.

As the incident angle approaches zero, the function P 1 ν(cosθ0) grows without bound. It is, however, necessary to keep in mind that the incident angle is limited by the cone semiangle, and the above calculations lose their physical meaning for θ i<α. Thus, the electric near field is maximal near grazing incidence. In Sec. 4 we show that such a behavior of the incident angle dependence of the electric near field can be considered qualitatively in terms of the antenna theory.

3. Electric near field of a semi-infinite cone for an incident Gaussian beam

So far, we have dealt with the plane wave illumination only. Let us now see what happens if the incident wave is a Gaussian laser beam. Here we consider only a Gaussian intensity profile which is characteristic of the commonly used fundamental laser mode or the light radiated by a single-mode fiber. In the following, the problem is treated in terms of the angular spectrum of plane waves (see, e.g., [23

23. C. G. Chen, P. T. Konkola, J. Ferrera, R. K. Heilmann, and M. L. Schattenburg, “Analysis of Vector Gaussian Beam Propagation and the Validity of Paraxial and Spherical Approximations,” J. Opt. Soc. Am. A 19, 404–412 (2002). [CrossRef]

,24

24. P. C. Chaumet, “Fully vectorial highly nonparaxial beam close to the waist,” J. Opt. Soc. Am. A 23, 3197–3202 (2006). [CrossRef]

]).

In as much as now we are able to calculate the electric near field for the plane wave illumination for all angles of incidence, in this section we are going first to calculate the plane wave spectrum of the Gaussian beam at the desired position (i.e., to find the expansion of the Gaussian beam in terms of the elementary plane waves) and then combine the near fields from each spectral component to determine the total near field.

To obtain the corresponding expansion for the 3D case, we assume that the electric field in the local coordinates near the focal plane is [see Fig. 1(a)]

Ei(x',y',z')E0exp[(x'+y')2w02]exp(ikz')
(7)

where w 0 is the beam waist width (here and below the time dependent factor exp(iωt) is omitted). The field can be expanded in terms of the elementary plane waves as (see, e.g., [24

24. P. C. Chaumet, “Fully vectorial highly nonparaxial beam close to the waist,” J. Opt. Soc. Am. A 23, 3197–3202 (2006). [CrossRef]

]) Ex'(r)=1(2π)2dpdqAx'(p,q)exp[ik(px'+qy'+tz')], Ey'(r)=1(2π)2dpdqAy'(p,q)exp[ik(px'+qy'+tz')], Ez'(r)=1(2π)2dpdq[ptAx'(p,q)+qtAy'(p,q)]exp[ik(px'+qy'+tz')], with t=1p2q2 and

Ai(p,q)=k2dx'dy'Ei(x',y',0)exp[ik(px'+qy')]=2πf2exp(p2+q24f2),
(8)

It is convenient to express p and q via the local azimuthal angle φ′ in the plane (x′, y′): p=sinθ′cosφ′; q=sinθ′sinφ′. Then, the total electric near field may be evaluated as

Enf(x',y',z')=E02πf2π2π2exp(14w02k2sin2θ')exp(ikz'cosθ')sinθ'cosθ'dθ'
02πE(α,θ0,θ',φ')exp[ik(x'sinθ'cosφ'+y'sinθ'sinφ')]dφ',
(9)

where the limits of integration are chosen in such a way that the propagating (nonevanescent) waves are taken into account only. This issue will be discussed later, in Sec. 4.

As was noted earlier, we consider the azimuthally invariant electric field. This simplifies the problem, reducing it to two-dimensional. So, for the 2D incident Gaussian beam the electric field at any point is (see, e.g., [25

25. Em. E. Kriezis, P. K. Pandelakis, and A. G. Papagiannakis, “Diffraction of a Gaussian Beam from a Periodic Planar Screen,” J. Opt. Soc. Am. A 11, 630–636 (1994). [CrossRef]

] for more detail)

Ei(x',z')=E0kw02ππ2π2exp(14w02k2sin2θ')exp[ik(x'sinθ'+z'cosθ')]cosθ'dθ'.
(10)

Then the total electric near field may be written as a superposition of the fields generated by the elementary plane waves, i.e.,

Enf(x',z')=E0kw02ππ2π2E(α,θ0θ')exp(14w02k2sin2θ')exp[ik(x'sinθ'+z'cosθ')]cosθ'dθ'.
(11)

x'=(xsx)cosθi+(zsz)sinθi,z'=(xsx)sinθi(zsz)cosθi.
(12)

where sx and sz are the shifts of the center of the focal spot relative to the apex [see Fig. 1(a)].

To evaluate the effect of the Gaussian beam illumination on the electric near field, in Fig. 5 we present our computations of the electric field enhancement at the point r=5 nm, θ=0° as a function of the beam waist width for different values of the incident angle. The cone semi-angle is chosen to be as for Fig. 2, i.e., α=25° and the separation parameter ν=-0.141. Both sx and sz are assumed to be zero. Similar computations for ν=0.505 (this approximately corresponds to silicon at the same wavelength) are also given for comparison. The results clearly show that (i) the near field grows as the incident angle decreases and (ii) the near field is a nonmonotonic function of the beam waist width, i.e., the electric near field has a peak at a value of w0. At the same time, at w0>1 µm (typical size of the focal spot) the electric near field enhancement depends only slightly on the beam waist width.

Fig. 5. Electric near field enhancement for a silver (solid lines) and silicon (dashed lines) conical probe at the point r=5 nm, θ=0° vs the beam waist radius.

4. Discussion

While a rigorous interpretation of the results obtained for the plane-wave illuminated cone is not straightforward, it is of interest to consider a simpler model, namely, a thin perfectly conducting wire. In this connection, it should be noted that an expression for the scattering cross section of a thin semi-infinite cone is analogous to that of a thin semi-infinite wire [12

12. J. J. Bowman, J. J. Bowman, T. B. A. Senior, and P. L. E. Uslenghi, eds., (Hemisphere, New York, 1987).

]. Both the electric and magnetic fields of the long wire can be considered quantitatively in terms of the traveling-wave currents [26

26. R. Cicchetti and A. Faraone, “On the optical behavior of the Electromagnetic Field excited by a Semi-Infinite Traveling-Wave Current,” IEEE Trans. Antennas Propag. 53, 4015–4025 (2005). [CrossRef]

]. Evaluation of the electric field near the end of the long wire is of interest on its own. Such a problem, however, is beyond the scope of the present work. Instead, we give a brief description of radiation properties of a long wire antenna and then demonstrate their connection with the current and electric near field.

Consider a transmitting long wire antenna. The traveling current of the antenna produces radiated fields which can be calculated in a standard way (see, e.g., [27

27. C.A. Balanis, Antenna Theory. Analysis and Design (Wiley, New York, 1997).

]). In the general case, the field distribution is a multilobe pattern where the number of lobes depends on the antenna length l. The peaks of the lobes occur at the angles

θn=cos1(1λ2lmn),
(13)

where mn=0.742, 2.93, 4.96, … for the first, second, and so third maxima [27

27. C.A. Balanis, Antenna Theory. Analysis and Design (Wiley, New York, 1997).

]. As the antenna becomes longer, the position of the largest (main) lobe (n=1) shifts to the antenna axis. At the same time, the subsidiary lobes shrink in the opposite direction. Furthermore, it results from the reciprocity theorem that the radiating and receiving patterns of any antenna are identical. It means that in the case of a receiving antenna, an obliquely incident electromagnetic wave induces the maximal current near the grazing incidence [notice that accurate evaluation of the induced current yields I (x′, y′, z′)=0 at θ i=0, i.e, the zero current at exact grazing incidence (see, e.g., [28

28. D.C. Chang, S.W. Lee, and L. Rispin, “Simple formula for current on a cylindrical receiving antenna,” IEEE Trans. Antennas Propag. 26, 683–690 (1978). [CrossRef]

])]. Correspondingly, the maximal current causes the maximal scattered electric field which may be found from Pocklington’s equation [27

27. C.A. Balanis, Antenna Theory. Analysis and Design (Wiley, New York, 1997).

]. So, if the origin coincides with the wire end, two components of the electric field for the thin wire in the cylindrical coordinates are

Ez(r)=i4πωε[0II(z')(z'2+k2)G(rr')dz'],
(14a)
Eρ(r)=i4πωε[0II(z')ρ'z'G(rr')dz'],
(14b)

where lis the wire length and G(r⃗-r⃗′) is the free space Green’s function. The above results for the wire, obviously, agree with our results for the incident angle dependence of the electric field near the cone apex. Because the incident angle θi cannot be less than the cone semiangle, the electric near field shows an increase as θi approaches its grazing value. Besides, the results are in line with the above-mentioned simulations by Sun and Shen [8

8. W. X. Sun and Z. X. Shen, “Optimizing the Near Field around Silver Tips,” J. Opt. Soc. Am. A 20, 2254–2259 (2003). [CrossRef]

] for the electric field near the apex of the finite-size rounded silver tip.

Another effect, a possible appearance of the electric near field peak for the cones with ν>0.85, seemingly, does not admit a simple interpretation. One can show, however, that similar effect may be explained in terms of the first Born approximation. Indeed, it is easy to see that in the weak dielectric contrast limit, i.e., when ε 2ε 1, the parameter ν→1. In this case, the applied (incident) electric field could be considered as a good approximation for the internal field of the scatterer. So, for a thin cone or a thin wire the internal field is

E(r')E0exp(ikzz')=E0[cos(kz'cosθi)+isin(kz'cosθi)]=E0Φ(z')
(15)

with Φ(z′) the phase factor. Then, the Lippmann-Schwinger equation for z-component of the electric field, which is of interest in our case, takes the form

Ez(r)=E0z(r)+KVdr'[Gxz(r,r')E0x(r')+Gyz(r,r')E0y(r')+Gzz(r,r')E0z(r')]=E0(r)sinθ0
×[1+KVdr'Φ(z')Gzz(r,r')]+E0(r)cosθ0KVdr'Φ(z')[sinφ0Gzx(r,r')+cosφ0Gzy(r,r)],
(16)

where Glk(r⃗,r⃗′) is the free-space Green’s tensor [29

29. O. J. F. Martin, C. Girard, and A. Dereux, “Generalized field propagator for Electromagnetic Scattering and Light Confinement,” Phys. Rev. Lett. 74, 526–529 (1995). [CrossRef] [PubMed]

] and the constant K=k 2(ε 2-ε 1). It should be emphasized that the tensor Glk(r⃗,r⃗′), generally, contains nondiagonal components which can be of considerable importance in the near zone. Thus, the z-component of the electric near field results from two terms. As the K factor is small, the first term gives roughly the sinusoidal dependence on the incident angle, while the second one gives roughly the cosinusoidal dependence. When the parameter ν is very nearly unity, only the first term contributes to the near field. In contrast, as the parameter ν decreases, the second term becomes significant. Its contribution to the total electric near field increases when θi=π-θ0→0. However, it should be remembered that the angle θi is limited by its grazing value. Thus, a peak in the incident angle dependence of the electric near field occurs; its position shifts to θi=π/2 as the dielectric contrast ε21→0. Of course, as the parameter ν is not too close to unity, the first Born approximation becomes invalid.

Although the cones with ν>0.85 are not of great practical interest from the point of view of a strong near-field enhancement [19

19. A. V. Goncharenko, J. K. Wang, and Y. C. Chang, “Electric Near-Field Enhancement of a Sharp Semi-Infinite Conical Probe: Material and Cone Angle Dependence,” Phys. Rev. B 74, 235442 (2006). [CrossRef]

], it must be kept in mind that the above estimation for the parameter ν is obtained for the semi-infinite cone having weak dielectric contrast. So one can expect that the nonmonotonic behavior of the near field against the incident angle for the finite-size real cones could also occur at ν<0.85. This assumption has been supported by the numerical results for the metal tips [8

8. W. X. Sun and Z. X. Shen, “Optimizing the Near Field around Silver Tips,” J. Opt. Soc. Am. A 20, 2254–2259 (2003). [CrossRef]

], as well as for the dielectric ones [9

9. R. Esteban, R. Vogelgesang, and K. Kern, “Simulation of Optical Near and Far Fields of Dielectric Apertureless Scanning Probe,” Nanotechnology 17, 475–482 (2006). [CrossRef]

]. In the latter case, the maximal near field was found to be at θi≅40° for a 3µm silicon tip with ε=17.76+0.51i (λ=488 nm) and α10° (corresponding ν≅0.74).

Let us now consider our results for the electric near field in the case of Gaussian beam illumination. Our first result (a growth of the near field with decreasing incident angle) becomes clear from the preceding discussion. As to the growth of the near field with decreasing beam waist width, a similar result was obtained earlier [9

9. R. Esteban, R. Vogelgesang, and K. Kern, “Simulation of Optical Near and Far Fields of Dielectric Apertureless Scanning Probe,” Nanotechnology 17, 475–482 (2006). [CrossRef]

]. At the same time, our calculations predict the existence of the peak of the near field at w0 λ/2, i.e., close to the diffraction limit. Mathematically, its origin is clear. Indeed, taking a look at the right side of Eq. (10) we see that the electric near field results from the competition of two factors. The first factor, w 0, tends to increase the field while the second one, exp(-w20k2sin2 θ′/4), in contrast, tends to decrease it. As a result, a peak in the dependence of Enf on w 0 occurs at a value of the beam waist width. It should be, however, noted that although, theoretically, the minimal value of the beam width can reach (in vacuum) about λ/4, the production of such tightly focused beams involves considerable technical difficulties [30

30. N. I. Petrov, “Focusing of beams into subwavelength area in an inhomogeneous medium,” Opt. Express 9, 658–673 (2001). [CrossRef] [PubMed]

].

The simulations show [31

31. N. I. Petrov, “Evanescent and propagating fields of a strongly focused beam,” J. Opt. Soc. Am. 20, 2385–2389 (2003). [CrossRef]

] that as the spot size becomes close to λ/2, evanescent waves, not considered in the present work, must be taken into account for an accurate description of the electric field. Our computations (not shown here) evidence that extending the integration limits in Eq. (9) to take into account the evanescent fields does not critically affect the electric near field. Indeed, as shown in [24

24. P. C. Chaumet, “Fully vectorial highly nonparaxial beam close to the waist,” J. Opt. Soc. Am. A 23, 3197–3202 (2006). [CrossRef]

], for w 0=λ/2, the contribution of the evanescent part to the total electric field near the beam axis is small even close to the waist. In the off-axis case, the contribution can reach 10% for the z′ component and 1.2% for the x′ component as x′/λ<1. It is natural to assume that the evanescent waves could further enhance the electric near field. At the same time, for the incident angles 70–90° (the case under consideration) contribution of Ez′ to the total near field is expected to be small. Recent results by Esteban et al. [9

9. R. Esteban, R. Vogelgesang, and K. Kern, “Simulation of Optical Near and Far Fields of Dielectric Apertureless Scanning Probe,” Nanotechnology 17, 475–482 (2006). [CrossRef]

], where terms of fifth order in the diffraction angle are included in the description of the associated fields, show that the electric field enhancement is not dramatic for tightly focused beams. In our case, for w 0>2λ≈µm (a typical experimental situation) Gaussian beam illumination has only a slight effect on the electric near field. So one can state that the plane wave approximation is justified in this region.

5. Final comments and conclusions

A few comments on practical importance and applicability of our results are now in order. First of all, it is vital to note the possible role of a substrate which is usually present in the experiment. A rigorous treatment of the substrate effect requires solution of the self-consistent problem akin to Eq. (4). In practice, interaction between the probe and substrate can result in depolarization of the scattered field that complicates the issue [32

32. C. Durkan and I. V. Shvets, “Polarization effects in Reflection-Mode Scanning Near-Field Optical Microscopy,” J. Appl. Phys. 83, 1837–1843 (1998). [CrossRef]

]. According to [15

15. R. M. Roth, N. C. Panoiu, M. M. Adams, R. M. Osgood, C. C. Neacsu, and M. B. Raschke, “Resonant-Plasmon Field enhancement from asymmetrically illuminated Conical Metallic-Probe Tips,” Opt. Express 14, 2921–2931 (2006). [CrossRef] [PubMed]

], the effect of the incident angle on the enhancement of the z-component of the electric near field is weak in the case of a gold rounded tip near the gold substrate. For the distance between the tip and the substrate d=4 nm, the maximal enhancement is achieved at about 45°. Furthermore, in the case of the dielectric substrate, even simple consideration of the field intensity in the tip-substrate gap within the framework of the Fresnel theory may be a good approximation as the phase of the reflected field is properly taken into account [33

33. O. J. F. Martin and C. Girard, “Controlling and Tuning Strong Optical Field Gradients at a Local Probe Microscope Tip Apex,” Appl. Phys. Lett. 70, 705 (1997). [CrossRef]

]. At the same time, a knowledge of the electric near field of the free-standing tip can be of interest on its own, for example, for Raman spectroscopy, when a scattering particle is attached to the tip [34

34. M. S. Anderson, “Locally enhanced Raman Spectroscopy with an Atomic Force Microscope,” Appl. Phys. Lett. 76, 3130–3132 (2000). [CrossRef]

,35

35. M. S. Anderson and W. T. Pike, “A Raman-Atomic Force Microscope for Apertureless-Near-Field Spectroscopy and Optical Trapping,” Rev. Sci. Instrum. 73, 1198–1203 (2002). [CrossRef]

], for designing ‘single-molecule’ fluorescent probes [36

36. D. Richards, “Near-Field Microscopy: throwing light on the Nanoworld,” Phil. Trans. R. Soc. Lond. A 361, 2843–2857 (2003). [CrossRef]

], or for handling single molecules [37

37. K. O. Greulich, “Single molecule studies of DNA and RNA,” ChemPhysChem 6, 2458–2471 (2005). [CrossRef] [PubMed]

].

Although we have considered the case of illumination with propagating plane waves only, the case of evanescent field illumination [say, when the evanescent field is generated by total internal reflection (TIR) at a dielectric-air interface [1

1. A. Bouhelier, “Field-enhanced scanning near-field optical microscopy,” Microsc. Res. Tech. 69, 563–579 (2006). [CrossRef] [PubMed]

], see Fig. 1(b)] could be considered in a similar way. In the general case, the incident field is of the form E0(r⃗)=E0 exp(ik⃗r⃗)=E0exp[i(kxx+kyy+kzz)]. So, for the propagating waves (side illumination) all components of the wave vector are real. At the same time, for the evanescent waves (TIR illumination) the z-component of the wave vector, kz=iksinθ1sin2θc, is purely imaginary (here θc is the critical angle). On the other hand, this component can be formally written as kz=k cosθi. It means that geometry with evanescent wave illumination may be formally considered assuming the incident angle to be complex that ensures a purely imaginary value of cosθi. Such an approach may be approximately valid if the conical probe is located sufficiently far from the interface, i.e., when the boundary effects on the dielectric surface are neglected.

If it is possible to neglect the azimuthal dependence of the electric near field, the Bowman’s representation (2) for the near field, as well as its generalization (5) can be considered as good ones, especially when dealing with the incident angle dependence. In some cases, however, it is impossible. So, according to some data [8

8. W. X. Sun and Z. X. Shen, “Optimizing the Near Field around Silver Tips,” J. Opt. Soc. Am. A 20, 2254–2259 (2003). [CrossRef]

,15

15. R. M. Roth, N. C. Panoiu, M. M. Adams, R. M. Osgood, C. C. Neacsu, and M. B. Raschke, “Resonant-Plasmon Field enhancement from asymmetrically illuminated Conical Metallic-Probe Tips,” Opt. Express 14, 2921–2931 (2006). [CrossRef] [PubMed]

], for distances very close to the apex, a double-lobed optical pattern can occur in the incidence plane that cannot be described within the framework of the above approximation. One more example is the above Born approximation which, although correct for the low dielectric contrast only, takes into account coupling between transverse and longitudinal oscillations that can occur in the near zone.

When dealing with Gaussian beam illumination, the electric near field is calculated in a crude way due to our choice of the problem parameters, namely, near-normal incidence and observation point close to the beam axis. In the general case, however, a more sophisticated technique must be used to accurately evaluate the near field for tightly focused beam illumination, taking into account both the higher-mode corrections and evanescent fields. This challenging problem is of interest, in particular, in connection with the rapid development of such applications as optical trapping and optical manipulation (see, e.g., the review [38

38. K. C. Neuman and S. M. Block, “Optical Trapping,” Rev. Sci. Instrum. 75, 2787–2809 (2004). [CrossRef]

]). Finally, we did not consider interesting polarization effects on the electric near field of a conical probe which could occur, in particular, for radially or azimuthally polarized light beams (see, e.g., [39

39. L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal Field Modes probed by Single Molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001). [CrossRef] [PubMed]

,40

40. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized beam,” Phys. Rev. Lett. 91, 233901 (2003). [CrossRef] [PubMed]

]).

It comes as a surprise but, to our knowledge, so far nobody measured the incident angle dependence of the electric field near the apex of a probe. Conventionally, the incident angle is kept fixed in the corresponding equipment and it ranges around 50–70° (see, e.g., [15

15. R. M. Roth, N. C. Panoiu, M. M. Adams, R. M. Osgood, C. C. Neacsu, and M. B. Raschke, “Resonant-Plasmon Field enhancement from asymmetrically illuminated Conical Metallic-Probe Tips,” Opt. Express 14, 2921–2931 (2006). [CrossRef] [PubMed]

,41

41. D. Mehtani, N. Lee, R. D. Hartschuh, A. Kisliuk, M. D. Foster, A. P. Sokolov, and J. F. Maguire, “Nano-Raman Spectroscopy with side-illumination optics,” J. Raman Spectrosc. 36, 1068–1075 (2005). [CrossRef]

43

43. Q. Nguyen, R. Ossikovski, and J. Schreiber, “Contrast enhancement on Crystalline Silicon in polarized reflection mode tip-enhanced Raman Spectroscopy,” Opt. Commun. 274, 231–235 (2007). [CrossRef]

]). Meantime, our results evidence that a considerable gain in the near-field enhancement can be obtained simply by decreasing the incident angle.

To conclude, we have considered the incident angle dependence of the electric near field of a semi-infinite conical probe for the plane wave as well for the Gaussian beam incidence. In the case of the plane wave illumination, when taking into account only the leading (azimuthally independent) term in the near field expansion, the field is shown to peak near grazing incidence. It is shown that this effect can be explained in terms of antenna theory. In addition, for the weak dielectric contrast, a peak can also occur at near-normal incidence that agrees with the first Born approximation. In the case of the Gaussian beam illumination, the near field is shown to increase with decreasing both the incident angle and the beam waist width. At the same time, as the Gaussian beam is not too tightly focused, the electric near field enhancement is close to that as for the plane wave illumination.

Acknowledgment

The authors would like to thank M. Dvoynenko for helpful discussions.

References and links

1.

A. Bouhelier, “Field-enhanced scanning near-field optical microscopy,” Microsc. Res. Tech. 69, 563–579 (2006). [CrossRef] [PubMed]

2.

L. Novotny, R. X. Bean, and X. S. Xie, “Theory of Nanometric Optical Tweezers,” Phys. Rev. Lett. 79, 645–648 (1997). [CrossRef]

3.

O. J. F. Martin and C. Girard, “Controlling and tuning strong optical field gradients at a local probe microscope tip apex,” Appl. Phys. Lett. 70, 705–707 (1997). [CrossRef]

4.

S. Takahashi and A. V. Zayats, “Near-Field Second-Harmonic Generation at a Metal Tip Apex,” Appl. Phys. Lett. 80, 3479–3481 (2002). [CrossRef]

5.

R. Bachelot, F. H’Dhili, D. Barchiesi, G. Lerondel, R. Fikri, P. Royer, N. Landraud, J. Peretti, F. Chaput, G. Lampel, J. P. Boilot, and K. Lahlil, “Apertureless near-field optical microscopy: a study of the local tip field enhancement using photosensitive Azobenzene-containing films,” J. Appl. Phys. 94, 2060–2072 (2003). [CrossRef]

6.

R. Ossikovski, Q. Nguen, and G. Picardi, “Simple model for the polarization effects in tip-enhanced Raman Spectroscopy,” Phys. Rev. B 75, 045412 (2007). [CrossRef]

7.

M. J. Hagmann, “Intensification of Optical Electric Fields caused by the Interaction with a Metal Tip in Photofield Emission and Laser-Assisted Scanning Tunneling Microscopy,” J. Vac. Sci. Technol. B 15, 597–601 (1997). [CrossRef]

8.

W. X. Sun and Z. X. Shen, “Optimizing the Near Field around Silver Tips,” J. Opt. Soc. Am. A 20, 2254–2259 (2003). [CrossRef]

9.

R. Esteban, R. Vogelgesang, and K. Kern, “Simulation of Optical Near and Far Fields of Dielectric Apertureless Scanning Probe,” Nanotechnology 17, 475–482 (2006). [CrossRef]

10.

M. A. Salem, A. H. Kamel, and A. V. Osipov, “Electromagnetic fields in the presence of an infinite dielectric wedge,” Proc. R. Soc. A462, 2503–2522 (2006). [CrossRef]

11.

L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Wiley, 1994) and references therein. [CrossRef]

12.

J. J. Bowman, J. J. Bowman, T. B. A. Senior, and P. L. E. Uslenghi, eds., (Hemisphere, New York, 1987).

13.

J. Van Bladel, “Field singularities at the tip of a Dielectric Cone,” IEEE Trans. Antennas Propag. 33, 893–895 (1985). [CrossRef]

14.

M. Idemen, “Confluent Tip Singularity of the Electromagnetic Field at the Apex of a Material Cone,” Wave Motion 38, 251–277 (2003). [CrossRef]

15.

R. M. Roth, N. C. Panoiu, M. M. Adams, R. M. Osgood, C. C. Neacsu, and M. B. Raschke, “Resonant-Plasmon Field enhancement from asymmetrically illuminated Conical Metallic-Probe Tips,” Opt. Express 14, 2921–2931 (2006). [CrossRef] [PubMed]

16.

H. Bateman and A. Erdelyi, Higher Transcendental Functions, (McGraw-Hill, 1985) Vol. 1.

17.

R. N. Hall, “The application of non-integral Legendre functions to potential problems,” J. Appl. Phys. 20, 925–931 (1949). [CrossRef]

18.

Our numerical simulations show that this statement is also valid for ν>0.

19.

A. V. Goncharenko, J. K. Wang, and Y. C. Chang, “Electric Near-Field Enhancement of a Sharp Semi-Infinite Conical Probe: Material and Cone Angle Dependence,” Phys. Rev. B 74, 235442 (2006). [CrossRef]

20.

W. P. Dyke, J. K. Trolan, W. W. Dolan, and G. Barnes, “The Field Emitter: Fabrication, Electron Microscopy, and Electric Field Calculations,” J. Appl. Phys. 24, 570–576 (1953). [CrossRef]

21.

J. C. Wiesner and T. E. Everhart, “Point-Cathode Electron Sources — Electron Optics of the Initial Diode Region,” J. Appl. Phys. 44, 2140–2148 (1973). [CrossRef]

22.

D. Hu, M. Micic, N. Klymyshyn, Y. D. Suh, and H. P. Lu, “Correlated Topographic and Spectroscopic Imaging beyond Diffraction limit by Atomic Force Microscopy Metallic Tip-Enhanced Near-Field Fluorescence Lifetime Microscopy,” Rev. Sci. Instrum.74, 3347–3355 (2003). [CrossRef]

23.

C. G. Chen, P. T. Konkola, J. Ferrera, R. K. Heilmann, and M. L. Schattenburg, “Analysis of Vector Gaussian Beam Propagation and the Validity of Paraxial and Spherical Approximations,” J. Opt. Soc. Am. A 19, 404–412 (2002). [CrossRef]

24.

P. C. Chaumet, “Fully vectorial highly nonparaxial beam close to the waist,” J. Opt. Soc. Am. A 23, 3197–3202 (2006). [CrossRef]

25.

Em. E. Kriezis, P. K. Pandelakis, and A. G. Papagiannakis, “Diffraction of a Gaussian Beam from a Periodic Planar Screen,” J. Opt. Soc. Am. A 11, 630–636 (1994). [CrossRef]

26.

R. Cicchetti and A. Faraone, “On the optical behavior of the Electromagnetic Field excited by a Semi-Infinite Traveling-Wave Current,” IEEE Trans. Antennas Propag. 53, 4015–4025 (2005). [CrossRef]

27.

C.A. Balanis, Antenna Theory. Analysis and Design (Wiley, New York, 1997).

28.

D.C. Chang, S.W. Lee, and L. Rispin, “Simple formula for current on a cylindrical receiving antenna,” IEEE Trans. Antennas Propag. 26, 683–690 (1978). [CrossRef]

29.

O. J. F. Martin, C. Girard, and A. Dereux, “Generalized field propagator for Electromagnetic Scattering and Light Confinement,” Phys. Rev. Lett. 74, 526–529 (1995). [CrossRef] [PubMed]

30.

N. I. Petrov, “Focusing of beams into subwavelength area in an inhomogeneous medium,” Opt. Express 9, 658–673 (2001). [CrossRef] [PubMed]

31.

N. I. Petrov, “Evanescent and propagating fields of a strongly focused beam,” J. Opt. Soc. Am. 20, 2385–2389 (2003). [CrossRef]

32.

C. Durkan and I. V. Shvets, “Polarization effects in Reflection-Mode Scanning Near-Field Optical Microscopy,” J. Appl. Phys. 83, 1837–1843 (1998). [CrossRef]

33.

O. J. F. Martin and C. Girard, “Controlling and Tuning Strong Optical Field Gradients at a Local Probe Microscope Tip Apex,” Appl. Phys. Lett. 70, 705 (1997). [CrossRef]

34.

M. S. Anderson, “Locally enhanced Raman Spectroscopy with an Atomic Force Microscope,” Appl. Phys. Lett. 76, 3130–3132 (2000). [CrossRef]

35.

M. S. Anderson and W. T. Pike, “A Raman-Atomic Force Microscope for Apertureless-Near-Field Spectroscopy and Optical Trapping,” Rev. Sci. Instrum. 73, 1198–1203 (2002). [CrossRef]

36.

D. Richards, “Near-Field Microscopy: throwing light on the Nanoworld,” Phil. Trans. R. Soc. Lond. A 361, 2843–2857 (2003). [CrossRef]

37.

K. O. Greulich, “Single molecule studies of DNA and RNA,” ChemPhysChem 6, 2458–2471 (2005). [CrossRef] [PubMed]

38.

K. C. Neuman and S. M. Block, “Optical Trapping,” Rev. Sci. Instrum. 75, 2787–2809 (2004). [CrossRef]

39.

L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, “Longitudinal Field Modes probed by Single Molecules,” Phys. Rev. Lett. 86, 5251–5254 (2001). [CrossRef] [PubMed]

40.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized beam,” Phys. Rev. Lett. 91, 233901 (2003). [CrossRef] [PubMed]

41.

D. Mehtani, N. Lee, R. D. Hartschuh, A. Kisliuk, M. D. Foster, A. P. Sokolov, and J. F. Maguire, “Nano-Raman Spectroscopy with side-illumination optics,” J. Raman Spectrosc. 36, 1068–1075 (2005). [CrossRef]

42.

C. C. Neacsu, J. Dreyer, N. Behr, and M. B. Raschke, “Scanning-probe Raman Spectroscopy with single-molecule sensitivity,” Phys. Rev. B 73, 193406 (2006). [CrossRef]

43.

Q. Nguyen, R. Ossikovski, and J. Schreiber, “Contrast enhancement on Crystalline Silicon in polarized reflection mode tip-enhanced Raman Spectroscopy,” Opt. Commun. 274, 231–235 (2007). [CrossRef]

OCIS Codes
(180.5810) Microscopy : Scanning microscopy

ToC Category:
Microscopy

History
Original Manuscript: April 30, 2007
Revised Manuscript: June 13, 2007
Manuscript Accepted: June 27, 2007
Published: August 27, 2007

Virtual Issues
Vol. 2, Iss. 10 Virtual Journal for Biomedical Optics

Citation
A. V. Goncharenko and Yia-Chung Chang, "Effect of the incident angle on the electric near field of a conical probe for plane wave and Gaussian beam illumination," Opt. Express 15, 11517-11529 (2007)
http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-15-18-11517


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References

  1. A. Bouhelier, "Field-enhanced scanning near-field optical microscopy," Microsc. Res. Tech. 69, 563-579 (2006). [CrossRef] [PubMed]
  2. L. Novotny, R. X. Bean, and X. S. Xie, "Theory of Nanometric Optical Tweezers," Phys. Rev. Lett. 79, 645-648 (1997). [CrossRef]
  3. O. J. F. Martin and C. Girard, "Controlling and tuning strong optical field gradients at a local probe microscope tip apex," Appl. Phys. Lett. 70, 705-707 (1997). [CrossRef]
  4. S. Takahashi and A. V. Zayats, "Near-Field Second-Harmonic Generation at a Metal Tip Apex," Appl. Phys. Lett. 80, 3479-3481 (2002). [CrossRef]
  5. R. Bachelot, F. H’Dhili, D. Barchiesi, G. Lerondel, R. Fikri, P. Royer, N. Landraud, J. Peretti, F. Chaput, G. Lampel, J. P. Boilot, and K. Lahlil, "Apertureless near-field optical microscopy: a study of the local tip field enhancement using photosensitive Azobenzene-containing films," J. Appl. Phys. 94, 2060-2072 (2003). [CrossRef]
  6. R. Ossikovski, Q. Nguen, and G. Picardi, "Simple model for the polarization effects in tip-enhanced Raman Spectroscopy," Phys. Rev. B 75, 045412 (2007). [CrossRef]
  7. M. J. Hagmann, "Intensification of Optical Electric Fields caused by the Interaction with a Metal Tip in Photofield Emission and Laser-Assisted Scanning Tunneling Microscopy," J. Vac. Sci. Technol. B 15, 597-601 (1997). [CrossRef]
  8. W. X. Sun and Z. X. Shen, "Optimizing the Near Field around Silver Tips," J. Opt. Soc. Am. A 20, 2254-2259 (2003). [CrossRef]
  9. R. Esteban, R. Vogelgesang, and K. Kern, "Simulation of Optical Near and Far Fields of Dielectric Apertureless Scanning Probe," Nanotechnology 17, 475-482 (2006). [CrossRef]
  10. See, e.g., M. A. Salem, A. H. Kamel, and A. V. Osipov, "Electromagnetic fields in the presence of an infinite dielectric wedge," Proc. R. Soc. A 462, 2503-2522 (2006). [CrossRef]
  11. See L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Wiley, 1994) and references therein. [CrossRef]
  12. J. J. Bowman, The Cone, in Electromagnetic and Acoustic Scattering by Simple Shapes, J. J. Bowman, T. B. A. Senior, P. L. E. Uslenghi, eds., (Hemisphere, New York, 1987).
  13. J. Van Bladel, "Field singularities at the tip of a Dielectric Cone," IEEE Trans. Antennas Propag. 33, 893-895 (1985). [CrossRef]
  14. M. Idemen, "Confluent Tip Singularity of the Electromagnetic Field at the Apex of a Material Cone," Wave Motion 38, 251-277 (2003). [CrossRef]
  15. R. M. Roth, N. C. Panoiu, M. M. Adams, R. M. Osgood, Jr., C. C. Neacsu, and M. B. Raschke, "Resonant-Plasmon Field enhancement from asymmetrically illuminated Conical Metallic-Probe Tips," Opt. Express 14, 2921-2931 (2006). [CrossRef] [PubMed]
  16. See, e.g., H. Bateman and A. Erdelyi, Higher Transcendental Functions, (McGraw-Hill, 1985) Vol. 1.
  17. See, e.g., R. N. Hall, "The application of non-integral Legendre functions to potential problems," J. Appl. Phys. 20, 925-931 (1949). [CrossRef]
  18. Our numerical simulations show that this statement is also valid for ν>0.
  19. A. V. Goncharenko, J. K. Wang, and Y. C. Chang, "Electric Near-Field Enhancement of a Sharp Semi-Infinite Conical Probe: Material and Cone Angle Dependence," Phys. Rev. B 74, 235442 (2006). [CrossRef]
  20. W. P. Dyke, J. K. Trolan, W. W. Dolan, and G. Barnes, "The Field Emitter: Fabrication, Electron Microscopy, and Electric Field Calculations," J. Appl. Phys. 24, 570-576 (1953). [CrossRef]
  21. J. C. Wiesner and T. E. Everhart, "Point-Cathode Electron Sources - Electron Optics of the Initial Diode Region," J. Appl. Phys. 44, 2140-2148 (1973). [CrossRef]
  22. The "bowling" pin shape can be also of interest in its own right because such a geometry can be formed by sputter coating a standard dielectric probe with a metal, see, e.g., D. Hu, M. Micic, N. Klymyshyn, Y. D. Suh, and H. P. Lu, "Correlated Topographic and Spectroscopic Imaging beyond Diffraction limit by Atomic Force Microscopy Metallic Tip-Enhanced Near-Field Fluorescence Lifetime Microscopy," Rev. Sci. Instrum. 74, 3347-3355 (2003). [CrossRef]
  23. C. G. Chen, P. T. Konkola, J. Ferrera, R. K. Heilmann, and M. L. Schattenburg, "Analysis of Vector Gaussian Beam Propagation and the Validity of Paraxial and Spherical Approximations," J. Opt. Soc. Am. A 19, 404-412 (2002). [CrossRef]
  24. P. C. Chaumet, "Fully vectorial highly nonparaxial beam close to the waist," J. Opt. Soc. Am. A 23, 3197-3202 (2006). [CrossRef]
  25. Em. E. Kriezis, P. K. Pandelakis, and A. G. Papagiannakis, "Diffraction of a Gaussian Beam from a Periodic Planar Screen," J. Opt. Soc. Am. A 11, 630-636 (1994). [CrossRef]
  26. See, e.g., R. Cicchetti and A. Faraone, "On the optical behavior of the Electromagnetic Field excited by a Semi-Infinite Traveling-Wave Current," IEEE Trans. Antennas Propag. 53, 4015-4025 (2005). [CrossRef]
  27. C.A. Balanis, Antenna Theory. Analysis and Design (Wiley, New York, 1997).
  28. D.C. Chang, S.W. Lee, and L. Rispin, "Simple formula for current on a cylindrical receiving antenna," IEEE Trans. Antennas Propag. 26, 683-690 (1978). [CrossRef]
  29. O. J. F. Martin, C. Girard, and A. Dereux, "Generalized field propagator for Electromagnetic Scattering and Light Confinement," Phys. Rev. Lett. 74, 526-529 (1995). [CrossRef] [PubMed]
  30. N. I. Petrov, "Focusing of beams into subwavelength area in an inhomogeneous medium," Opt. Express 9, 658-673 (2001). [CrossRef] [PubMed]
  31. N. I. Petrov, "Evanescent and propagating fields of a strongly focused beam," J. Opt. Soc. Am. 20, 2385-2389 (2003). [CrossRef]
  32. C. Durkan and I. V. Shvets, "Polarization effects in Reflection-Mode Scanning Near-Field Optical Microscopy," J. Appl. Phys. 83, 1837-1843 (1998). [CrossRef]
  33. O. J. F. Martin and C. Girard, "Controlling and Tuning Strong Optical Field Gradients at a Local Probe Microscope Tip Apex," Appl. Phys. Lett. 70, 705 (1997). [CrossRef]
  34. M. S. Anderson, "Locally enhanced Raman Spectroscopy with an Atomic Force Microscope," Appl. Phys. Lett. 76, 3130-3132 (2000). [CrossRef]
  35. M. S. Anderson and W. T. Pike, "A Raman-Atomic Force Microscope for Apertureless-Near-Field Spectroscopy and Optical Trapping," Rev. Sci. Instrum. 73, 1198-1203 (2002). [CrossRef]
  36. D. Richards, "Near-Field Microscopy: throwing light on the Nanoworld," Phil. Trans. R. Soc. Lond. A 361, 2843-2857 (2003). [CrossRef]
  37. K. O. Greulich, "Single molecule studies of DNA and RNA," ChemPhysChem 6, 2458-2471 (2005). [CrossRef] [PubMed]
  38. K. C. Neuman and S. M. Block, "Optical Trapping," Rev. Sci. Instrum. 75, 2787-2809 (2004). [CrossRef]
  39. L. Novotny, M. R. Beversluis, K. S. Youngworth, and T. G. Brown, "Longitudinal Field Modes probed by Single Molecules," Phys. Rev. Lett. 86, 5251-5254 (2001). [CrossRef] [PubMed]
  40. R. Dorn, S. Quabis, and G. Leuchs, "Sharper focus for a radially polarized beam," Phys. Rev. Lett. 91, 233901 (2003). [CrossRef] [PubMed]
  41. D. Mehtani, N. Lee, R. D. Hartschuh, A. Kisliuk, M. D. Foster, A. P. Sokolov, and J. F. Maguire, "Nano-Raman Spectroscopy with side-illumination optics," J. Raman Spectrosc. 36, 1068-1075 (2005). [CrossRef]
  42. C. C. Neacsu, J. Dreyer, N. Behr, and M. B. Raschke, "Scanning-probe Raman Spectroscopy with single-molecule sensitivity," Phys. Rev. B 73, 193406 (2006). [CrossRef]
  43. Q. Nguyen, R. Ossikovski, and J. Schreiber, "Contrast enhancement on Crystalline Silicon in polarized reflection mode tip-enhanced Raman Spectroscopy," Opt. Commun. 274, 231-235 (2007). [CrossRef]

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