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], 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).

While the role of the incident beam polarization is more or less clear, especially for scattering-type SNOM (s-SNOM) [2–6], less attention was paid to the effect of the incident angle on the electric near field. Surprisingly, we found no experimental study of this effect. It is standard practice to keep the angle of incidence to be constant. In addition, the cantilever body limits the minimal value of the incident angle. As to numerical studies, Hagmann [7] has considered the electric field at the apex of perfectly conducting tips having the shape of a conical frustrum with two hemispherical ends and a paraboloid. His results show that the electric near field grows rapidly as the incident angle decreases. The incident angle θ_max at which the field is maximal is close to the null grazing angle. Sun and Shen [8] have computed the incident angle dependence of the electric field beneath the apex of a silver conical tip for the tip lengths up to 3.5 λ (about 1.7 µm). Their results show that the electric near field can be a nonmonotonic function of the incident angle. As the tip length increases, the incident angle at which the field is maximal approaches to the null grazing angle. In addition, a few weaker peaks can occur at higher angles. Finally, Esteban et al. [9] have dealt with silicon conical tips of up to 3 µm length capped by two hemispherical ends. In particular, the computations evidence that a single peak of the electric near field for the 3 µm tip occurs at θ _max≅40° when the cone semiangle α=10°.

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

A rigorous treatment of the electromagnetic fields near an illuminated cone is, generally speaking, a challenge, even for the relatively simple 2D case of a wedge [10]. Fifty years ago serious efforts were mounted by Felsen to solve the problem of electromagnetic scattering by a semi-infinite perfectly conducting cone in terms of the scalar Green’s function [11]. Later, Bowman obtained expressions for the electric and magnetic fields scattered by a semi-infinite cone for arbitrary polarization and direction of incident wave using the dyadic Green’s function technique [12]. The eigenfunction expansion to determine the dyadic Green’s function includes the Bessel and Legendre functions. Thus, a formal analytical solution describing electromagnetic scattering by a cone is known; it can be written in the form of an infinite modal series. It should be, however, noted that in attempting to get numerical results from the analytical solution, a difficulty arises related to the slow convergence of the series. Generally speaking, this problem for arbitrary material cone is poorly studied. Fortunately, when dealing with the near-field problems, as the parameter kr is small (k is the wave number), it is frequently possible to keep only the leading (azimuthally independent) term in the expansion. This simplifies matters, allowing one to present results in an easy-to-use form convenient for various near-field studies.

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 (E_r=(∂²_r+k ²)(rU) E _θ=r ^-1∂_r∂_θ(rU)+ikη(sinθ)^-1φV, E _φ=(r sinθ)^-1∂r∂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]

 

Fig. 1. Two schemes of illumination of a conical probe: side illumination (a) and total internal reflection illumination (b).

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and

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 ν_s and µ_s 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], namely

with E ₀ the amplitude of the incident field, and the constant $C=\exp \left(-\frac{1}{2}i\pi \left(\nu +1\right)\right)\sqrt{\pi }{\left[\sin \alpha ·2^{\nu -1}\Gamma \left(\nu +\frac{1}{2}\right)P_{\nu }^1\left(-\cos \alpha \right)\frac{\partial }{\partial \nu }{P}_{\nu }\left(-\cos \alpha \right)\right]}^{-1}$. 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 P_ν(-cosα)=0. For the material cone with a finite permittivity this equation should be replaced by [13,14]

Let us take a closer look at Eq. (2). Although the geometry of the problem allows the probe to be asymmetrically illuminated, the near field E does not depend on the azimuthal angle φ in this approximation. The validity of the approximation is based on the fact that just the longitudinal (directed along the probe axis and, therefore, independent of the angle φ) component of the incident field is enhanced and hence makes a major contribution to the total electric near field [2]. The simulation results (see, e.g. [15]) also confirm that the electric near field is nearly azimuthally invariant. However, it is necessary to keep in mind that Eq. (2) may become inaccurate when the leading term in the eigenfunction expansion is so small that the next terms (which, generally, have other dependence on the incident angle) may not be neglected. This can take place when the angle θ ₀ is close to zero [Eq. (2) erroneously predicts zero electric near field at θ ₀=0]. Furthermore, Eq. (2) has been obtained for the sharp (rounded-off) cone that limits its applicability for the real probes. Indeed, the electric field in Eq. (2) behaves singularly, E(r,θ)→∞ as r→0, that is impossible in reality. Besides, it is easy to see that because E_θ(r, θ)∝∂P_ν(cosθ)/∂θ=P ¹ _ν(cosθ) and the associated Legendre function P ¹ _ν(cosθ) may be represented as [16]

where ₂ F ₁(…) is the hypergeometrical function, it immediately follows that E _θ→0 as θ→0. At the same time, both functions, P_ν(cosθ) and P ¹ _ν(cosθ) as well, rise with the angle θ if the parameter ν<0 [17]. As a result, the total electric near field $E(r,\theta )=\sqrt{E_r^2+E_{\theta }^2}$ has a prominent minimum in the direction along the probe axis, i.e., at θ=0 [18]. This contradicts to results of numerous computations (see, e.g., [5,15]). 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]. One can show that the electric field near the sphere could be approximately estimated as

with r₀ the sphere radius, B(r,r₀)=S(r₀,r)^2ν+1, and $S=\left({\epsilon }_2-{\epsilon }_1\right)⁄\left({\epsilon }_2+\frac{\nu +1}{\nu }{\epsilon }_1\right)$. As was noted earlier [20,21], azimuthally invariant equipotential surfaces of the above “core” system (“bowling” pin) constitute a class of the shapes [22]. In the near-field approximation, the equipotential surfaces satisfy the equation [19]

As an example, in Fig. 2, we show the equipotential surfaces computed for the “core” geometry with the cone parameters α=25° and ν=-0.141, that corresponds to silver at λ=633 nm, and r ₀=1 nm. Practical applicability of such an approach is related to the fact that, in principle, one of the equipotential surfaces could be more or less well fitted to the geometry of a real cone. If R is the radius of inscribed sphere (i.e., the radius of curvature of the rounded cone) then the maximal electric field occurs at its surface, i.e., at r=R. Thus, within the framework of the above model, the electric near field at r > R is of the form (5) where r ₀ can be considerer as a phenomenological fitting parameter of the model.

 

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

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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 ${}_2{F}_1(2+\nu ,1-\nu ;2;\frac{1-\cos \theta }{2})\to 1\mathrm{as}\theta \to 0$, 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.

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Fig. 4. Dependence of the incident angle θ_i at which P ¹ _ν(cosθ ₀) peaks vs the parameter ν.

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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., [23,24]).

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)]

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]) $E_{x\text{'}}\left(\overrightarrow{r}\right)=\frac{1}{{\left(2\pi \right)}^2}{\int }_{-\infty }^{-\infty }\mathrm{dp}{\int }_{-\infty }^{\infty }\mathrm{dq}{A}_{x\text{'}}(p,q)\exp \left[\mathrm{ik}\left(px\text{'}+qy\text{'}+tz\text{'}\right)\right],$ $E_{y\text{'}}\left(\overrightarrow{r}\right)=\frac{1}{{\left(2\pi \right)}^2}{\int }_{-\infty }^{-\infty }\mathrm{dp}{\int }_{-\infty }^{\infty }\mathrm{dq}{A}_{y\text{'}}(p,q)\exp \left[\mathrm{ik}\left(px\text{'}+qy\text{'}+tz\text{'}\right)\right],$ $E_{z\text{'}}\left(\overrightarrow{r}\right)=-\frac{1}{{\left(2\pi \right)}^2}{\int }_{-\infty }^{-\infty }\mathrm{dp}{\int }_{-\infty }^{\infty }\mathrm{dq}\left[\frac{p}{t}{A}_{x\text{'}}(p,q)+\frac{q}{t}{A}_{y\text{'}}(p,q)\right]\exp \left[ik\left(px\text{'}+qy\text{'}+tz\text{'}\right)\right],$ with $t=\sqrt{1-p^2-q^2}$ and

with the beam parameter f=(kw₀)^-1. We note that t is real for the propagating waves and imaginary for the evanescent waves [24]. The local zenith angle θ′ may be related to the space frequencies p and q as $\cos (\theta \text{'})=t=\sqrt{1-p^2-q^2}$.

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

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] for more detail)

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

We note that the (x, z) coordinates corresponding to the origin which coincides with the cone apex are related to the local (x′, z′) coordinates through

where s_x and s_z 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 s_x and s_z 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.

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

where m_n=0.742, 2.93, 4.96, … for the first, second, and so third maxima [27]. 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])]. Correspondingly, the maximal current causes the maximal scattered electric field which may be found from Pocklington’s equation [27]. So, if the origin coincides with the wire end, two components of the electric field for the thin wire in the cylindrical coordinates are

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 θ_i cannot be less than the cone semiangle, the electric near field shows an increase as θ_i approaches its grazing value. Besides, the results are in line with the above-mentioned simulations by Sun and Shen [8] 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 ₂→ε 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

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

where G_lk(r⃗,r⃗′) is the free-space Green’s tensor [29] and the constant K=k ²(ε ₂-ε ₁). It should be emphasized that the tensor G_lk(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 θ_i=π-θ₀→0. However, it should be remembered that the angle θ_i is limited by its grazing value. Thus, a peak in the incident angle dependence of the electric near field occurs; its position shifts to θ_i=π/2 as the dielectric contrast ε₂-ε₁→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], 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], as well as for the dielectric ones [9]. In the latter case, the maximal near field was found to be at θ_i≅40° for a 3µm silicon tip with ε=17.76+0.51i (λ=488 nm) and α≅10° (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]. 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 E_nf 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].

The simulations show [31] 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], 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 E_z′ to the total near field is expected to be small. Recent results by Esteban et al. [9], 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]. According to [15], 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]. 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 [34,35], for designing ‘single-molecule’ fluorescent probes [36], or for handling single molecules [37].

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], 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(k_xx+k_yy+k_zz)]. 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, $k_z=ik\sqrt{\sin {\theta }_1-{\sin }^2{\theta }_c}$, is purely imaginary (here θ_c is the critical angle). On the other hand, this component can be formally written as k_z=k cosθ_i. 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 cosθ_i. 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 [8,15], 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]). 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., [39,40]).

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., [15,41–43]). 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.