1. Introduction

Doughnut beams with a dark central region are important in many areas in optics, for applications such as dark optical traps for atoms [1–2], and as erase beams in super-resolution fluorescence microscopy [3–5]. In the latter application the fluorescent sample is illuminated near-simultaneously with a pump beam and a doughnut-shaped erase beam, and various possible mechanisms, such as up-conversion fluorescence depletion [5] or stimulated emission [3] suppress the fluorescence process in the overlap region of the pump and erase beams. The rate of fluorescence depletion depends nonlinearly on the intensity of the erase beam [4, 6], thus the fluorescence spot size – that determines the resolution of the microscope – can be significantly smaller than the diffraction limit, provided that the central dark region of the focused doughnut beam has the smallest possible size. In such microscopes it is therefore desirable to create a sharply focused dark central region surrounded by light.

Laguerre-Gaussian helical beams are well-suited for this task. Such beams possess orbital angular momentum [7], characterized by their topological charge, t, which refers to the number of complete cycles of phase (2πt) upon going around the beam circumference [7]. The topological charge determines the order of the helical beam, so that beams with t = 1 are called first-order Laguerre-Gaussian beams. For this case, in the scalar approximation, each ray is out of phase by π relative to the ray situated symmetrically about the optical axis, and the destructive interference between all such pairs of rays leads to zero intensity at the focal point (and everywhere along the optical axis). However, in case of high numerical aperture (NA) focusing – which is needed to achieve a sharply focused spot – the scalar approximation is no longer valid, and polarization effects play a prominent part, as was demonstrated by several authors [8–9]. In this case the electric field vector near the focus can be calculated using the generalized Debye integral, expressed as [10–11]

where E is the electric field vector at location (r₂, ψ, z₂) expressed in cylindrical coordinates, λ is the illumination wavelength in the medium – having a refractive index n – in the focal region, k = 2π/λ, θ is the focusing angle (the angle between the optical axis and the given ray) so that NA = n∙sin(θ_max), and φ is the azimuth angle. A₁(θ, φ) is the amplitude distribution of the collimated input beam at the entrance pupil. For aberration-free first-order Laguerre-Gaussian beams A₁(θ, φ) = A₁(θ) = (r(θ)/w)∙exp(-r²(θ)/w²), where r is the distance from the optical axis, and w is a constant proportional to the radius of the bright ring. A₂(θ) is the so-called apodization factor of the focusing system, obtained from energy conservation and geometrical considerations [10–11]. For example, for the widely used aplanatic microscope objectives $A_2\left(\theta \right)=\sqrt{\cos \theta }$, and for Herschel-type systems A₂(θ) = 1 [12]. A₃(θ, φ) describes the aberrations introduced by the focusing system, and a(θ,φ) is the polarization unit vector that belongs to a given ray propagating toward the focus. Importantly, the three components of a(θ,φ) can be complex numbers – i.e. shifted in phase relative to each other -, as is the case e.g. for circular or elliptical incoming polarizations.

In the present paper, we investigate experimentally the effect of polarization on the focused intensity distribution of high NA first-order Laguerre Gaussian beams. Specifically, we consider three cases: linear, left-handed circular (LC) and right-handed circular (RC) polarization. Theoretically, linear polarization was shown to lead to a longitudinal component at the focus, whose intensity at NA = 1 reaches 48.8% of the maximum intensity [8]. Such a focal spot is unsuitable as the erase beam in fluorescence depletion microscopes, because fluorescence is radically attenuated at the center point, leading to low signal-to-noise ratio in the fluorescence signal. An additional drawback of linearly polarized light is its asymmetric focal intensity distribution at high NA, i.e. the focal spot is elongated along the direction of polarization [8]. On the other hand, circular polarization was theoretically shown to yield a perfect zero at the focus [9], provided that that the spin angular momentum of the photons -related to the handedness of the circular polarization – and the orbital angular momentum of the beam – related to the topological charge – have the same orientations (either both pointing along +z or along -z). Such circularly polarized first-order Laguerre-Gaussian beams are therefore, in theory, ideally suited for the task of erase beams in fluorescence depletion microscopy.

2. Experimental setup

Because of the high NA geometry, ordinary photodetectors cannot be used to measure the intensity at the focal plane, due to their limited acceptance angle and generally insufficient resolution. The knife-edge method overcomes these difficulties [13–15], but it requires data processing, and yields an intensity graph only along one coordinate axis corresponding to the 1D scanning direction. Therefore in our experiments a different approach was chosen, which involves 2D scanning of the focal plane with a single 100nm diameter fluorescent microbead and measuring the fluorescence intensity. The advantage of this method is that it directly yields the 2D intensity profile of the beam in the measured plane, and in principle it does not require processing of the measured data.

The experimental setup is presented in Fig. 1. The linearly polarized collimated Gaussian beam of a dye laser (Spectra-Physics 375B, λ = 599nm) is first incident upon a spiral phase plate (SPP) [16]. The depth of the SPP varies step-wise along φ, so that the phase delay it introduces to the beam varies in 8 steps from 0 to (7/8)×2π along a complete circle. The role of the SPP is to add a topological charge t = 1 [16], and hence generate a first-order Laguerre-Gaussian beam. Next, the beam is incident on a quarter-wave-plate (QWP) which can be rotated to the appropriate angular position to create linear, LC or RC polarization. With the help of a beam splitter (BS) the beam is directed towards an aplanatic microscope objective (MO) (Olympus-HSRMFL100X), with NA = 0.9 and an entrance pupil diameter of 3.24mm, and is focused onto a sample placed on a 2D scanning stage (SS). The sample consists of a cover glass spin coated at 2500 rpm with a 100 times diluted solution of 100nm diameter Molecular Probes Inc., F8801 fluorescent microbeads (FMB), so that in the scanning area of 4μm×4μm, the average number of microbeads is ~1. The SS is controlled by a computer, and can be moved continuously in both transverse directions. The scanning speed is 4μm/s, and one complete scan takes approximately 2 minutes. The mechanical accuracy of the SS ~10nm in all 3 dimensions. During scanning, the dye molecules in the microbead are excited to a fluorescent state by the incoming light (represented with continuous arrows in Fig. 1). The reflected fluorescence signal (represented with dashed arrows) passes through a focusing lens (L), a holographic notch filter (HNF) – that eliminates the back-scattered pump light -, and is detected by a photomultiplier tube (PMT). The measured peak of fluorescence is at λ_f = 610nm. The fluorescence signal is stored in a computer in the form of a 100×100 pixel bitmap, yielding a pixel size of 40nm×40nm.

 

Fig. 1. Experimental setup for measuring the intensity distribution of high-NA first-order Laguerre Gaussian doughnut beams in the focal plane. The helical phase is added to the beam by a spiral phase plate (SPP), and the proper polarization state is created with a quarter-wave-plate (QWP). After reflection from a beam splitter (BS) the beam is focused by a high-NA microscope objective (MO) onto a 100nm diameter fluorescent microbead (FMB) placed on a 2D scanning stage (SS). The dye molecules in the microbead are excited to a fluorescent state by the incoming light (continuous arrows). The reflected fluorescence signal (dashed arrows) passes through a focusing lens (L), a holographic notch filter (HNF) and is detected by a photomultiplier tube (PMT). The aberrations of the illuminating pump beam are measured separately, using a movable mirror (MM) and a Shack-Hartmann sensor (SH).

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Between fluorescence measurements, to investigate the wavefront of the beam, a movable mirror (MM) was used to couple the pump beam into a Shack-Hartmann sensor (SH). The SH was used to measure the aberrations of the collimated pump beam before being focused. These aberrations are caused mainly by the imperfect laser mode and by the discrete, step-like nature of the SPP. The aberration of the MO, on the other hand, was measured separately using a Twyman-Green interferometer [17], and the first 36 Zernike coefficients were determined from the measured data. We found the peak-to-valley aberration of the MO to be ~0.16λ. These aberrations of the collimated pump beam and of the MO can be stored in a computer, and incorporated for the numerical simulations into A₁(θ, φ) and A₃(θ, φ) of eq. (1), respectively.

3. Theoretical considerations

The different expected behaviour of the three considered cases can be understood with the help of Figs. 2(a), 2(b) and 2(c). Figure 2 shows schematically the time dependence of the local electric field vector in the transverse plane of the collimated pump beam before focusing, viewed from the direction of propagation, for linear, LC and RC polarizations, respectively. The initial polarization of the laser light is parallel to the x axis. Also shown in Fig. 2 are the corresponding orientations of the fast-axis of the QWP. Since t = 1, at each point in the (x,y) plane the electric field vectors are delayed by π with respect to the symmetrically opposite location about the z axis. As seen, the phase-delay introduced by the SPP increases in the counter-clockwise direction, implying that for the LC case the spin and orbital angular momenta of the beam are oriented in the same direction (along +z), and for the RC case they are oriented oppositely (the direction of the spin angular momentum can be determined by the rotational direction of the local electric field, and by the right-hand rule).

 

Fig. 2. Schematic representation of the local electric field vectors (shown with arrow-heads) for first-order Laguerre-Gaussian beams having (a) linear, (b) left-handed circular (LC), and (c) right-handed circular (RC) polarizations. At the top of the Fig., the corresponding orientation of the fast axis (FA) of the QWP is shown for the three cases. The original polarization of the laser is along x.

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As can be deduced from Fig. 2(a), and was mentioned above, the linearly polarized beam, when focused with high NA optics, yields a rather strong z component in the focal point. On the other hand, as seen in Fig. 2(b), in the LC case all components of the field are exactly cancelled everywhere along the optical axis at any arbitrary time. Zero intensity at the focus is generated even at arbitrarily high NA, since the effect of rays whose polarization momentarily has an outward radial component, hence producing a longitudinal electric field in the +z direction at the focus, is cancelled by rays whose polarization at the same time has an inward radial component, thus producing an equal longitudinal electric field in the -z direction. The importance of the handedness of the circular polarization can be understood by comparing Figs. 2(b) and 2(c). For the RC case depicted in Fig. 2(c) the polarization state alternates between radial and azimuth polarizations. At high NA, radially polarized doughnut beams are known to create a very strong axial component at the focus [13], hence a similar behaviour can be expected for the RC polarized case.

 

Fig. 3. Calculated intensity contour maps in the focal plane, and corresponding calculated intensity cross sections along the x and y axis (red and blue curves, respectively), for the (a) linear, (b) LC, and (c) RC polarization case, predicting very large polarization dependence of the intensity distribution. Only the LC polarization case is expected to yield zero intensity in the geometrical focus.

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Figures 3(a), 3(b) and 3(c) present the calculated intensity contour maps in the focal plane and the calculated intensity cross sections along the x and y axis (solid and broken lines, respectively), for the linear, LC and RC polarization, respectively, obtained numerically, using eq. (1), with NA = 0.9 and λ = 599nm. The pixel size in the contour maps of Fig. 3 is 40nm. For the calculations we assume an ideal (aberration-free) Laguerre-Gaussian beam profile, and an aberration-free aplanatic objective. The constant w is chosen such that the radius of the bright ring of the Laguerre-Gaussian beam at the entrance pupil is 0.85mm, corresponding to the value in our experiments. As seen in Fig. 3, and as was discussed by previous authors [8–9], the three polarization cases behave very differently, and the difference is especially pronounced between the LC and RC polarizations.

4. Experimental results

The confirm the numerical predictions about the three polarizations, and to demonstrate the applicability of 2D fluorescence scanning for the measurement of high NA signals, a series of experiments was conducted using the setup of Fig. 1. Our results are presented in Fig. 4. Figures 4(a), 4(b) and 4(c) show the measured fluorescence intensity contour maps, and the corresponding intensity profiles along the x and y axis, for the linear, LC and RC polarized cases, respectively. To reduce the noise, the intensity graphs of Fig. 4 along x/y are obtained by averaging the central 5 rows/columns of the 100×100 matrix containing the measured fluorescence data. The difference between the three polarization states, including the different central depths of the focal spot, and its elongated shape in the linearly polarized case, appears very clearly in the fluorescence data, indicating that our 2D scanning method is very well suited for demonstrating high NA polarization effects experimentally. The measured full-width-at-half-maximum (FWHM) width of the central dark spot for the LC case is ~200nm, hence it is expected to provide a very suitable erase beam in high NA fluorescence depletion microscopy. Table 1 shows the numerical values of the central intensity relative to the maximum intensity, of the intensity profiles of Figs. 3 and 4, indicating good quantitative agreement as well.

 

Fig. 4. Measured intensity contour maps in the focal plane, and corresponding intensity cross sections along the x and y axis (red and blue curves, respectively, obtained by averaging the central 5 rows/columns of the corresponding intensity contour maps), for the (a) linear, (b) LC, and (c) RC polarization. The drastically different behaviour of the three polarizations is clearly demonstrated. Also apparent are the excellent properties of the LC polarized beam in achieving a very tightly focused spot with a dark center.

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Table 1. Numerical values of the central intensity relative to the maximum intensity (in %), of the intensity profiles of Figs. 3 and 4

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For the theoretical investigation of our experimental conditions in more detail, we recalculated Eq. (1), but this time the wavefront aberrations (obtained from the measured Shack-Hartmann data), and the aberrations of the MO (calculated analytically from 36 measured Zernike coefficients) were included in A₁(θ, φ) and A₃(θ, φ), respectively. The effect of the non-zero fluorescent microbead size (diameter = 100nm) and the quantized nature of the fluorescence data (pixel size = 40nm) were approximated by convolving Eq. (1) with a circular step function having a diameter of 120nm (= 3 pixels). The calculated intensity contour maps for the linear, LC and RC polarizations are presented in Figs. 5(a), 5(b) and 5(c), respectively, showing excellent qualitative agreement with the corresponding experimental contour maps of Fig. 4. Specifically, the tilted, asymmetrical nature of the intensity distributions and a slight increase in spot size (relative to the ideal case of Fig. 3) are apparent in Fig. 5. By calculating with the two aberrations separately, we found that the wavefront aberrations before focusing, and the aberrations introduced by the MO are approximately equally responsible for the asymmetry of the intensity distribution. We note that the calculations presented in Fig. 5 do not contain aberrations arising from the BS, from possible minor misalignment of the MO, and from polarization impurities. However, we expect these to be much smaller than the combined aberrations of the laser output wavefront, the step-like SPP, and the MO.

 

Fig. 5. Calculated intensity contour maps in the focal plane for the (a) linear, (b) LC, and (c) RC polarization case. The numerical calculations include the measured wavefront aberrations of the beams before focusing, the measured aberration of the MO, and convolution with a 120nm diameter circular step function (approximating the combined effect of the finite fluorescent bead size, and the finite pixel size of the measured data). Comparison with Fig. 3 (ideal case) shows that the tilted, asymmetrical nature of the measured intensity distributions and the slight increase in their spot size (see Fig. 4) are satisfactorily accounted for with the refined calculations.

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5. Conclusion

The relevance of polarization in high NA focusing of first-order Laguerre-Gaussian beams was demonstrated experimentally, using the example of linear, left-handed circular and right-handed circular polarizations, and a simple setup based on 2D scanning with a single fluorescent microbead. Specifically, the central intensity minimum, which is expected from scalar theory and is present at low NA independently of polarization, was shown to disappear almost completely at NA = 0.9 for the circularly polarized case in which the spin and orbital angular momenta carried by the beam are anti-parallel. On the other hand, if the spin and orbital angular momenta are oriented in the same direction, a doughnut beam very low central intensity can be experimentally observed even at NA = 0.9. Linear polarization performs between these two extreme cases. Our results demonstrate that 100nm diameter fluorescent microbeads are small enough to yield sufficient resolution for visible light and high NA.