We measure the far-field intensity of vortex beams going through nanoholes. The process is analyzed in terms of helicity and total angular momentum. It is seen that the total angular momentum is preserved in the process, and helicity is not. We compute the ratio between the two transmitted helicity components, γm,p. We observe that this ratio is highly dependent on the helicity (p) and the angular momentum (m) of the incident vortex beam in consideration. Due to the mirror symmetry of the nanoholes, we are able to relate the transmission properties of vortex beams with a certain helicity and angular momentum, with the ones with opposite helicity and angular momentum. Interestingly, vortex beams enhance the γm,p ratio as compared to those obtained by Gaussian beams.
The interaction of light with nano-apertures is a problem that has been carefully studied by many scientists. In particular, the work of Ebbesen et al.1 showing that nano-apertures could have an extraordinary transmission due to the coupling of light and surface plasmon polaritons (SPPs) opened up lots of possibilities among different fields in optics, micro-manipulation, biophysics and condensed-matter2,3,4,5. Lots of these studies have focused on how the nano-apertures couple with SPPs depending on many different parameters6,7. Some others have focused on the radiation diagram of these nano-apertures and have studied how light with linear polarization normal () or parallel () to the plane of incidence are transmitted through the apertures both theoretically and numerically8,9,10,11,12. Finally, sub-wavelength nano-apertures have also been used to study the interaction between SPPs and the angular momentum (AM) of light13,14,15,16,17,18,19,20.
Even though the first studies about the AM of light date back to the beginning of the twentieth century21, it was not until the 1990s when its use rapidly extended across different disciplines. The seminal finding that triggered much of the following developments was carried out by Allen and co-workers. In22, the authors established a connection between the topological charge of paraxial vortex beams and their AM content. The finding implied that the AM content of optical beams could be controlled using available holography techniques - first Computer Generated Holograms (CGHs) and later Spatial Light Modulators (SLMs)23,24,25,26,27,28. Since then, the AM of light has been used in many diverse fields such as quantum optics29,30, optical manipulation31,32, optical communications33,34 or astrophysics35,36. Here, we present for the first time experimental far-field intensity recordings of the transmission of vortex beams through single nanoholes. We project the transmitted field into its two helicity components. Then, we observe that the mirror symmetry of the nanoholes constrains the transmission process of different modes up to a large extent. As it will be shown later on, the transmitted intensity of vortex beams with total AM m and helicity p is equal to the transmitted intensity of vortex beams with AM −m and helicity −p. Finally, we compute the ratio between the two transmitted helicity components, which we denote as γm,p and whose definition is given by eq. (6). Not only we observe an enhanced helicity transference with respect to a Gaussian excitation37, but also when investigating how this quantity changes with the size of the hole, we observe that the curves present structural differences.
The article is organised as follows. First, the optical set-up used to carry out the measurements is described. Second, the beams of light used to excite the nanoholes are mathematically characterized. Third, the interaction between the incident light and the sample is explained from the point of view of symmetries and conserved quantities. Then, a description of the methodology used to measure the transmitted light is given. Finally, the far-field measurements of the transmission of vortex beams through the nanoholes are presented and discussed.
The experimental set-up is similar to the one used in20, and it is depicted in Fig. 1. A CW laser operating at λ = 633 nm is used to generate a light beam. The laser produces a collimated, linearly polarized Gaussian beam. The beam is expanded with a telescope (lenses L1-L2) to match the dimensions of the chip of an SLM. After the telescope, the polarization state of the Gaussian beam is modified with a linear polarizer (P1) and a half-wave plate (HW1) to maximize the efficiency of the SLM. The SLM creates a vortex beam (see Fig. 1b) by displaying an optimized pitchfork diffraction grating20,25,26,38. Proper control of the pitchfork hologram allows for the creation of a phase singularity of order l in the center of the beam, i.e. the phase of the beam twists around its center from 0 to 2πl radians in one revolution. Note that when l = 0, the SLM behaves simply as a mirror. Because the SLM produces different diffraction orders, a modified 4-f filtering system with the lenses L3-L4 and an iris (I) in the middle is used to filter the non-desired orders of diffraction. Lens L3 Fourier-transforms the beam, and the iris selects the first diffraction order and filters out the rest. Then, lens L4 is used to match the size of the back-aperture of the microscope objective that will be used to focus the beam down to the sample. Before focusing, the preparation of the input beam is finished by setting its polarization to either left circular polarization (LCP) or right circular polarization (RCP). This is done with a linear polarizer (P2) and a quarter-wave plate (QW1). Since the beam is collimated, this change of polarization does not appreciably affect the spatial shape of the input beam. At this point, a water-immersion (NA = 1.1) microscope objective (MO1) is used to focus the circularly polarized paraxial vortex beam onto the sample (S). The sample is a set of single isolated nanoholes drilled on a 200 nm gold film, deposited on top of a microscope slide. The sizes of the nanoholes range from 200–450 nm (see Table 1), and they are located 50μm apart from their closest neighbours. Only one single nanohole is probed at a time, and the beam is centered with respect to the nanohole with a nanopositing stage. The transmitted light through the nanohole, which is scattered in all directions, is collected with MO2, whose NA = 0.9. MO2 collimates the transmitted light, i.e. the transmitted beam is paraxial once it leaves MO2. Afterwards, the beam goes through another quarter-wave plate (QW2), which transforms the two orthogonal circular polarization states into two orthogonal linear ones. A linear polarizer P3 is placed after QW2, which acts as an analyser. That is, P3 selects one of the two circular polarization states existing before QW2. Finally, a CCD camera records the intensity of the selected component. Notice that the main difference with respect to the set-up used in20 is in the detection scheme, which allows for a complete characterization of the polarization of the collimated transmitted beam.
As shown in Fig. 1a, we prepare monochromatic paraxial circularly polarized vortex beams, which are tightly focused onto the nano-apertures. The mathematical expression of their electric field (before focusing) is given by:
where is the circular polarization unitary vector, with being the horizontal and vertical polarization vectors, and p = ±1 the handedness of the beam; Am,p is a normalisation constant; w0 is the beam waist; k is the wavenumber, k = 2π/λ with λ the single wavelength in consideration; and (ρ, ϕ, z) are the cylindrical coordinates. An implicit harmonic exp(−iωt) dependence is assumed, where ω = 2πc/λ is the angular frequency of light, and c is the speed of light in vacuum. Notice that m − p is the topological charge of the beam created by the SLM, i.e. m − p = l. The origin of this formula can be seen in the collimated limit of Bessel beams with well defined helicity37,39. Also, note that given the definition of , p = 1 refers to LCP and p = −1 to RCP. In fact, it can be proven that p is the value of the helicity of the beam in the paraxial approximation37,39. In this approximation, are eigenstates of the helicity operator. The helicity operator can be expressed as Λ = (∇ ×)/k for monochromatic beams39,40. Thus, 37,39. This fact is very useful for our purposes, as it enables us to identify collimated circularly polarized vortex beams as states of well-defined helicity. States of well-defined helicity have eigenvalue +1 (−1) when their plane wave decomposition yields only left (right) handed polarized waves39,41 [p170]. Since the helicity operator is the generator of duality transformations42,43,44, will be invariant under duality transformations within the paraxial approximation. In a similar manner, are eigenstates of the z component of the AM operator, Jz. That is, . Then, because Jz is the generator of rotations along the z axis41,45, will be symmetric under these transformations, too.
In order to understand the forthcoming results, here we describe the light-matter interaction from the point of view of symmetries and conserved quantities39. The main advantage of this description is that it easily allows for the prediction of different light-matter interaction in a qualitative way. In contrast, if quantitative predictions need to be made, full Maxwell equations (or dyadic Green tensor) solvers are needed. In particular, the description of the laser beams given by eq. (1) would not be used to compute the light matter-interaction. Instead, the incident field given by eq. (1) would have to be transformed by the aplanatic lens model given in46 [3.5] to yield the focused field .
The nanoholes used in the experiment are almost perfectly round. Therefore, they are symmetric under rotations along the optical axis, which we will denote as z without loss of generality. In addition, the sample is also symmetric under mirror transformations with respect to any plane containing the z axis. Now, we will consider that all the properties of the sample are inherited by a linear integro-differential operator , which can be found using the Green Dyadic formalism. Then, fulfils the following commutation rules:
where Rz and are the operators that generate rotations along the z axis and mirror transformations with respect to a plane that contains the z axis, respectively. The first equality is a consequence of the fact that Jz is the generator of rotations along the z axis, i.e. Rz(θ) = exp(−iθJz). Because of this, if a nanohole interacts with a beam which is an eigenstate of Rz, Jz, or with eigenvalue ν, the result of the interaction will still be an eigenstate of the same operator with the same eigenvalue ν. Here, it is important to note that the microscope objectives, which we model as aplanatic lenses, do not change the helicity or the AM momentum content of the beam39,47,48. That is, using eq. (2) notation, . This is schematically depicted in Fig. 1c, where it can be seen that the incident red helix at the back of the MO1 keeps its color when it is focused by it. Hence, given an incident beam of the kind , the focused beam, denoted as , will keep the eigenvalues of Jz and Λ equal to m and p. Then, the transmitted field through a nanohole due to the focused vortex beam can be computed as . Here, notice that the subindices m, p refer to the eigenvalues of Jz and Λ for the incident beam. Notwithstanding, as mentioned earlier, due to the cylindrical symmetry of the problem, will also be an eigenstate of Jz with value m. However, the helicity of the incident beam is not preserved in the interaction. This is a consequence of the fact that duality symmetry is broken by the nanohole and the multilayer system37,39,44,49,50. After the light-matter interaction has taken place, MO2 collects most of (its NA = 0.9) and collimates it, thus retrieving a paraxial beam. Because MO2 also preserves helicity and AM, the collimated transmitted field keeps the same eigenvalue of Jz as , i.e. . In contrast, , as the sample scatters light in both helicity components because it is not dual-symmetric37,39,44,49. That is, the transmitted collimated field can be decomposed as:
where and within the paraxial approximation. This is schematically displayed in Fig. 1c, where it is seen that after the sample S an additional blue helix appears. Now, both and can be modelled using an expression similar to that given by eq. (1):
where Am,p(ρ) and Bm,p(ρ) depend on the NA of MO1 and MO2. We will denote as the direct component, since it maintains the polarization state . Consequently, the topological charge of is still m − p. The other orthogonal component is , and we will denote it as crossed component. The crossed component has a polarization state when the incident state is . Due to the cylindrical symmetry, the value of the AM along the z axis must be preserved. Therefore, as it can be seen in eq. (4), when p changes to −p, the topological charge of the beam goes to m + p. That is, the crossed component is a vortex beam whose optical charge differs in 2p units with respect to the incident beam , or with respect to the direct transmitted component . This effect was observed by Chimento and co-workers using an incident Gaussian beam and a 20 μm circular aperture51. Similarly, in37, we measured the same phenomenon using LCP Gaussian beams and nanoholes ranging from 100–550 nm. Here, for the first time, we measure the same phenomenon using different incident vortex beams. That is, we record the far-field intensity patterns of vortex beams propagating through nanoholes. In fact, the far-field patterns are measured for the two transmitted helicity components. Notice that these measurements differ from the work previously done in20 where only the total transmitted intensity power was measured; and they also differ from17, where the measurements and simulations were done in the near-field. Now, in order to measure the two transmitted helicity components, we use the CCD camera, QW2, and P3 (see Fig. 1a). As mentioned earlier, QW2 does the following polarization transformation:
Because both and are paraxial fields, their intensity patterns are not perturbed by this transformation. Thus, their intensity profiles can be singled out by P3. Finally, the CCD camera records the intensity profile of and separately (see Fig. 1a). The experiment has been carried out for six different incident fields: three different vortex beams with topological charge l = m − p = −1, 0, 1 are created by the SLM and each of them is right and left circularly polarized (p = −1, 1). Following the notation in eq. (1), the six beams used to carry out the experiments are: . In Fig. 2, we show the CCD images of the transmitted direct component , when the six incident beams go through a5, i.e. a nanohole with diameter Φ = 333 nm (see Table 1). A choice of a different nanohole does not change the images qualitatively. Looking at the intensity patterns displayed in Fig. 2, it can be seen that the direct component of the light transmitted through the nanohole has the same features as the incident beam. That is, both are roughly cylindrically symmetric, they have the same helicity, and the same AM. The AM content of the mode can be inferred from the order of the optical singularity in the center of the beam and the helicity of the mode: following our notation for paraxial vortex beams put forward in eq. (1), m = l + p. In principle, the order of the optical singularity cannot be inferred from an intensity plot. However, in our experiment we are able to confidently assess the absolute value of the order of the singularity. The reason becomes especially clear looking at the images of the transmitted crossed component. Figure 3 depicts the recorded CCD images for when the values of p and l of the incident beam are p = −1, 1 and l = m − p = −1, 0, 1. Hence, the image taken on the row p = 1 and column l = 1 corresponds to . That is, is the crossed component of an incident vortex beam going through the nanohole. Looking at eqs. (4), it is seen that should be a vortex beam with a topological charge of order l′ = m + p = 3. However, instead of observing a vortex of charge l′ = 3, three singularities of charge l = 1 are observed. This occurs because higher order phase singularities are very unstable and prone to split into first order singularities52,53,54,55. Thus, in the current scenario, a phase singularity of order l′ will split into |l′| singularities of order sign(l′). The instabilities mainly arise from the imperfections in centering the beam with respect to the sample, as well as the tolerances of the linear polarizer P3. Then, measuring the number of phase singularities and the helicity of the beam enables us to experimentally verify that the output patterns are consistent with the AM along the z axis being conserved within the experimental errors. Finally, one could think that the patterns shown in Fig. 3 depend on the transmissivity of the sample. However, as it will be seen hereafter, samples a3−6 have significantly different transmissivities and the patterns observed for the crossed component were very similar to the one given by a5 and shown in Fig. 3.
Now, using the method described above, we have consistently measured the intensity ratio between the crossed and direct helicity components. Recently, it has been shown that this ratio is monotonously dependent on the size of the nanohole37. Here, we extend the study presented in37, where the excitation was a LCP Gaussian beam, to different incident vortex beams and we show that the size-dependence is not monotonous. We define this ratio as:
where and are the intensities of the crossed and direct component measured at the chip of the CCD camera (A). Therefore, they can be obtained as:
The results are presented in Table 1. The measured γm,p are listed as a function of parameters that we control in the experiment: the topological charge given by the SLM (l = m − p) of the paraxial incident beam and its circular polarization vector .
In Fig. 4 we plot the data from Table 1. First of all, it can be observed that the behavior described in37 is retrieved for both helicity components when l = 0. That is, when a circular nano-aperture is excited by a Gaussian beam, its γm,p ratio monotonously decreases from γp,p = 100% for small holes (with respect to the wavelength) to γp,p ≈ 0% for large holes. In fact, it is observed that the result is helicity-independent, as both helicity components yield a very similar result. Nevertheless, the behavior of γm,p for other modes is more complex. The first striking feature that can be readily observed is that γm,p are no longer monotonous functions of the diameter of the nanohole (Φ). Even though the particular case of a Gaussian beam can be explained looking at the two limits (dipolar behavior for small sizes, and diffraction theory for the large ones)37, it is clear that a more thorough study is needed to explain the behavior of vortex beams through single nanoholes. In particular, these results suggest that vortex beams with |m| ≠ 1 can resonantly couple to a nanohole. Another interesting feature is that, while in the case of an incident Gaussian beam (l = m − p = 0), it is seen that the γm,p are independent of the polarization of the incident beam, when the incident field is a vortex beam, this is no longer the case. Indeed, the first and third rows of Fig. 4 clearly show that given a paraxial vortex beam with charge l = m − p, its transmission through a single nanohole strongly depends on the value of the helicity p of the beam.
One could expect that in order to study these phenomena (and some others, too), the complete structure of the fields at the focus should be studied. In that case, the three polarization components of the focused incident field would have to be computed. Certainly, if quantitative predictions were needed, this would be the way to find them. However, the qualitative behavior of the interaction can be understood through symmetry considerations. It is important to note that while the expression of is different from eq. (1), they have the same relevant symmetry properties: well-defined AM along the z axis, well-defined helicity and broken translational symmetry in the XY plane. Then, due to their ease, we will explain the qualitative behavior of the interactions using symmetries only.
Indeed, the reason behind the different behavior of the propagation of vortex beams through nanoholes is an underlying symmetry relating the transmission of incident beams of the kind and others of the kind . More specifically, both the intensity plots shown in Figs 2 and 3, as well as the γm,p plots displayed in Fig. 4 show that the six incident beams can be grouped in three pairs: , , and , where each pair has a very distinct behavior, whereas both members of each pair share the same features. Note that each member of the pairs shares the same |m|. The formal proof of why this happens can be found in20, but the idea is the following one. If we apply a mirror symmetry operator to , we obtain except for a phase. That is, the two beams of each of the three pairs above are connected with a mirror symmetry. Then, because the sample is symmetric under , mirror symmetric beams will have mirror symmetric scattering patterns, and they will yield equal intensities. This is clearly featured in Figs 5 and 6. Both figures feature the transmitted intensity through the nanoholes for the three incident vortex beams when p = 1 (Fig. 5), and p = −1 (Fig. 6). The transmitted intensity is denoted as . The colors chosen for each plot are consistent with those used in Fig. 4. Because both figures display one of the two members of each of the three pairs of mirror symmetric beams, both figures yield a great resemblance. It is clear now why the transmittance of a Gaussian beam with a well-defined helicity through a nanohole is not dependent on its helicity value p: it is the only case where given a vortex beam with a topological charge l = m − p = 0, the two modes with helicity −p, p are mirror symmetric.
Figures 5 and 6 also show that the total transmissivity as a function of the diameter of the four beams whose |m| ≠ 1 have a non-trivial behavior. That is, their is not monotonic, unlike the Gaussian beam, whose was found to be linear in a log-log plot37. However, even though vortex beams can lead to strong helicity-transfer enhancements (Table 1 and Fig. 4), Gaussian beams still yield a much larger transmission through the nanoholes. Finally, it is seen that the γm,p ratio can yield results much larger than 100%. This is especially clear for the two modes with m = 0, which yield a γ0,p ratio much larger than 100%. That is, even though all the light incident on the nanohole has helicity p, most of the transmitted light flips its helicity value, yielding the opposite value −p. This is a big difference with respect to the results reported in37 using a Gaussian beam, where values of γm,p ≈ 100% are only reached when the size of the nanohole is very small with respect to the wavelength.
Some other systems that have also been reported to produce such large helicity changes are q-plates56 and dielectric spheres49. Actually, similarly to the helicity change induced by dielectric spheres, the phenomenon is observed to be very dependent on the size of the structure. That is, as it is depicted in Figs 4, 5, 6, the value of γm,p for the increases from about 200% to 350% with an increase of ΔΦ ≈ 10 nm. This peak in γm,p is very unlikely to be an artifact. Notice that two independent measurements such as γ0,1 and γ0,−1 yield analogous results for the same nano-aperture, using completely different incident fields. Furthermore, to verify the strong size-dependence of the helicity transformation in this type of nano-apertures, we have carried out numerical simulations using the semi-analytical method described in10. The numerical simulations do not exactly reproduced the system described in Fig. 1, but they are close enough. The discrepancies in these simulations are mainly two: it has been assumed that the microscope slide was a semi-infinite medium, and the NA of MO2 was NA = 1. The results of these simulations for the incident beam are depicted in Fig. 7. Even if the peaks do not happen to occur at the same aperture size, Fig. 7 corroborates that this measurement is not an artifact, but rather an interesting phenomenon to be studied. A possible explanation for the peak could stem from its definition in eq. (6). Since γm,p is defined as a ratio, a peak could indicate a decrease of the direct helicity component or an increase of the crossed component . These effects could be related to duality conditions of the sample49,57. Nevertheless, a detailed explanation of the fundamental physical mechanism behind this γm,p change deserves a detailed study which is out of the scope of this work.
To conclude, we have shown that vortex beams can go through single nanoholes and have measured their far-field intensity profiles for the first time. We have split their transmitted field into two orthogonal helicity components, and we have observed that the two of them preserve the AM of the initial beam. The ratio between the two transmitted helicity components, γm,p, has been measured for six different incident beams. We have observed that their behavior can be grouped into three different sets, each with the same absolute value of the total AM, |m|. It is observed that the sets of beams with |m| = 0, 2 have a γm,p behavior very different from |m| = 1. Whereas the behavior of the beams with |m| = 1 is monotonically decreasing, the other two sets of beams have a non-trivial behavior. In particular, we have measured that the two beams with m = 0 can yield γm,p > 300% for certain sizes of nanoholes. Notice that vortex beams with m = 0 are already being used as an experimental toolbox for alignment-free quantum communication58.
How to cite this article: Zambrana-Puyalto, X. et al. Far-field measurements of vortex beams interacting with nanoholes. Sci. Rep. 6, 22185; doi: 10.1038/srep22185 (2016).
This work was funded by the Australian Research Council Discovery Project DP110103697 and DP160103332. G.M.-T. is the recipient of an Australian Research Council Future Fellowship (project number FT110100924).
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ACS Photonics (2018)