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We demonstrate the formation of a sub-wavelength focal spot with a long depth of focus using a radially polarized, narrow-width annular beam. Theoretical analysis predicts that a tighter focal spot (approximately 0.4λ) and longer depth of focus (more than 4λ) can be formed by a longitudinal electric field when the width of the annular part of the beam is decreased. Experimental measurements using a radially polarized beam from a photonic crystal laser agree well with these predictions. Tight focal spots with long depths of focus have great potential for use in high-tolerance, high-resolution applications in optical systems.
©2010 Optical Society of America
1. Introduction
Recently, a number of applications have been proposed that require tailored optical beam modes. These include microscopy techniques that enable the observation of three-dimensional molecular orientation [1,2], optical trapping of opaque particles [3], and efficient material processing [4]. Many methods of producing tailored optical beams have been explored, including interferometric techniques [5], modifying the design of the inner laser cavity [6, 7], and the use of a liquid crystal phase modulator [8] or a polarization converter [9]. However, these methods all require additional optical elements that must also be aligned. If a range of optical beam modes could be made available in compact semiconductor lasers, it would become straightforward to realize the applications listed above. We have previously demonstrated that a range of tailored optical beam modes [10] can be produced by appropriate design of the lattice points of semiconductor photonic crystal (PC) surface-emitting lasers or by the introduction of phase shifts within the PC structure [11–13]. When the PC is comprised of circular air holes, doughnut-shaped beams with radial or azimuthal polarization can be obtained.
Among the various optical beam modes reported, the radially polarized doughnut beam has attracted much attention because of the unique possibility that it provides to create smaller spot sizes [14] and a longitudinal electric field [15] at the focal point. This longitudinal electric field would allow novel applications to be realized, such as particle acceleration [16] and Raman spectroscopy [17]. Furthermore, if the depth of focus could be extended while keeping the focal spot small, a longer interaction between the optical field and the object can be expected in the above applications, as well as simpler alignment procedures for many optical systems.
In this paper, we demonstrate that the depth of focus of a radially polarized beam can be extended and the spot size made smaller when the intensity distribution of the beam cross-section is changed. We have used vectorial diffraction theory [18] to theoretically evaluate the change in electric field distribution at the focus that occurs when the annular intensity profile of the beam cross-section is made narrower. Moreover, we have used a radially polarized beam supplied by a PC laser to experimentally investigate the intensity profile near the focal plane when the annular width of the beam is decreased.
2. Theoretical calculation
The electric field components in the vicinity of the focus of a radially polarized doughnut beam are formulated using vectorial diffraction theory [18, 19] as follows:
Here, l 0(θ) is the apodization function given in the form:We express the field components using cylindrical coordinates (ρ, ϕ, z) [see Fig. 1 (a) ], where the positive z-direction corresponds to the beam propagation direction and z=0 at the focal plane. In Eq. (1), A is the coefficient of power (A=1), β 0 is the ratio of the outer (pupil) radius to the beam waist, which we set at β 0=3/2, k is the wavenumber (k=2π/λ), and α is the maximum angle of the focus given by sin−1(NA/n), where NA is the numerical aperture. In this paper, we set NA to 0.9, the refractive index to n=1, and the wavelength to λ=980 nm. As described in Eq. (1), the radially polarized doughnut beam produces electric fields with radial (ρ) and longitudinal (z) components at the focus. Figure 1(a) shows a schematic picture of the beam focusing, where the electric field vectors are indicated by arrows; Fig. 1(b) shows the cross-sectional intensity profile at the focal plane (z/λ=0.0) along the ρ-axis. The length is normalized in units of wavelength. The longitudinal component produces strong intensity at the beam axis (ρ/λ=0.0), whereas the radial component produces a peak of intensity on each side of the axis. The total intensity profile at the focus takes the form of a single peak, which is much wider than that of the longitudinal component. Therefore, if the radial component can be reduced, making the longitudinal component dominant, we can expect a smaller focal spot.
Fig. 1 (a) Schematic sketch of the focusing of a radially polarized beam with δ=0.0. The red and blue arrows indicate the longitudinal and radial electric fields, respectively. (b) Cross-sectional intensity profiles of the electric field components at the focus. The sum of the radial (blue line) and longitudinal (red line) components gives the total intensity profile (black line).
In order to enhance the longitudinal component, which is mainly produced by the outer part of the annular input beam as shown in Fig. 1(a), it is important to make this outer part more intense. The characteristics of the modified beam obtained when the intensity is restricted to the outer part of the ring can be calculated by rewriting the apodization function l(θ) using a mask function T(θ) in the form:
whereHere, δ is the ratio of the inner focusing angle θ min to α, as shown in Fig. 2(a) . We used Eqs. (3) and (4) instead of Eq. (2) to calculate the cross-sectional intensity profile at the focal plane for the modified input beam; the result is shown in Fig. 2(b) for δ=0.8. The contribution of the radial electric field component has been decreased and the total intensity profile is now dominated by the longitudinal component. Therefore, the width of the total intensity decreases when the parameter δ increases. Modification of the intensity distribution of the input beam not only changes the profile of the cross-section at the focal plane (z/λ=0.0), but also along the z-axis. Figure 3 shows the intensity distribution through the focus in the propagation direction (z), (a) for the original input beam with δ=0.0 and (b) for the modified beam with δ = 0.8. For the original input beam, the depth of focus (the full-width at half maximum (FWHM) along the z-direction) is approximately 2λ, whereas the depth of focus is longer than 4λ for the modified beam. In the latter case the radial component has been strongly suppressed, whereas the z-component dominates the total intensity profile through the focus, as shown in Fig. 3(b). Figure 3(c) shows the depth of focus and the focal spot size (the FWHM along the ρ-axis) of the central peak as a function of δ. When δ>0.9, the depth of focus is more than 10λ and the focal spot size is approximately 0.4λ, which is smaller than the width (0.55λ) of focus obtained for a linearly polarized Gaussian beam under the same focusing conditions (NA=0.9, n=1).Fig. 2 (a) Schematic sketch of the focusing of a radially polarized, narrow-width annular beam with δ=0.8. (b) Cross-sectional intensity profiles of the electric field components at the focus. The central peak of the total intensity (black line) is dominated by the longitudinal component (red line); the contribution of the radial component (blue line) has been decreased.
Fig. 3 Intensity profiles of the electric fields through the focus (a) for δ=0.0 (the original doughnut beam) and (b) for δ=0.8 (a narrow-width annular beam). A long, sharp focal spot is formed for δ=0.8. The cross-sectional intensity profiles for δ=0.8 show that the longitudinal component (red line) dominates the total intensity (black line), with a much smaller contribution from the radial component (blue line). The focal plane lies at z/λ=0.0. (c) Focal spot size and depth of focus as a function of the ratio of inner and outer focusing angles δ.
Such an extension of the focal depth can be explained if we consider the interference of the outermost and innermost rays of the beam axis in the region around the focal point, as shown Fig. 4 . When the original radially polarized beam is focused, the incident angles of the two rays are significantly different (α » θ min). Constructive interference takes place only at the focal plane (z/λ=0.0) on the beam axis (ρ/λ=0.0) because the two rays are in phase only at this point, as shown in Fig. 4(a). Therefore, strong intensity appears only in close vicinity to the focal plane. In contrast, the two corresponding rays for the modified, narrower annular-width input beam interfere constructively on the beam axis even away from the focal plane; because the incident angles are almost identical, the phases of the rays match well, as shown in Fig. 4(b). The modified, narrower annular-width input beam thus possesses a longer depth of focus. Such an extension of the optical field at the focal point for optical rays with limited focusing angles has previously been explained for non-diffracting beams by Durnin [20].
Fig. 4 Schematic sketch of the interference of optical rays near the focal point. The blue and red arrows indicate rays originating from the outermost and innermost parts of the input beam, respectively. The dotted lines indicate the wave fronts of these rays. For the original beam (a), the rays are only in phase at the focal plane (z=0), whereas for the narrow-width annular beam (b), the rays are in phase on the beam axis even away from the focal plane.
3. Experiment
Two methods of measuring the focal spot size of a radially polarized beam have been reported so far: knife-edge scanning [21] and a photo-resist exposure method [22]. We employed the knife-edge scanning method in the current study because it gives spot sizes as a function of focal depth.
3.1.1. Detector
In the knife-edge method, we used a photo-diode to measure the photo-current change as a function of knife-edge position by cutting the focused beam spot with the sharp edge of a metal blade. We fabricated a special detector, where the knife-edge is formed very close to the photo-diode depletion layer, within the wavelength of the light used. This is necessary because the light ray passing the knife-edge has a large divergence angle due to the strong focusing, and it is important to measure the intensity of the non-propagating longitudinal field in the near-field region. Figure 5(a) shows a schematic structure of our detector. The knife-edge was formed at the edge of a metallic (Ti/Au) film of thickness 200 nm on top of a GaAs-based photo-diode, which consists of a p-GaAs layer (200 nm), an InGaAs depletion (photo-absorption) layer, and an n-GaAs substrate. Figure 5(b) shows a top view of the fabricated detector. A rectangular detection area (50 μm×25μm) was formed next to the knife-edge and the rest of the surface was covered by an insulator. The roughness of the knife-edge was evaluated to be less than 30nm using scanning electron microscopy, which was smaller than the knife-edge scanning step.
Fig. 5 (a) Cross-sectional structure of the detector. We used the edge of a metallic (Ti/Au) film of thickness 200 nm on top of a GaAs-based photo-diode as a knife-edge. The distance between the top of the detection area and the depletion (photo-absorption) layer was 200 nm. (b) Top view of the fabricated detector. The detection area was formed next to the knife-edge; the rest of the surface was covered by an insulator. (c) Experimental set-up. The beam emitted by the photonic-crystal laser was focused by an objective lens with 0.9 NA. The detector was placed at the focal plane and scanned using a high-resolution xyz stage. The cross-sectional beam profile was modified by placing a shading mask just before the objective lens. (d) In order to obtain the two-dimensional profile, several scans at different angles (ψ) are required.
3.1.2. Measurement setup
Figure 5(c) shows our experimental set-up. We used the radially polarized doughnut beam emitted by a PC laser, which consisted of a semi-conductor laser structure with a PC laser cavity near the active layer [10]. The beam was focused by an objective lens with 0.9 NA (W. D. =1.0 mm) after passing through a collimating lens and a non-polarizing beam splitter. The detector described above was placed at the focal plane and scanned using a high-resolution xyz stage with a positional precision of 10 nm. Part of the collimated beam was divided using a beam splitter and passed to a CCD camera to check the quality of the input beam. For precise alignment, the confocal image of the objective lens was monitored by another CCD camera, allowing the knife-edge position relative to the focal spot to be observed. The knife-edge was scanned in 50 nm steps by a distance of 20 μm through the focal spot using the high-resolution stage. The photo-current and the position of the knife-edge were measured at every step. The changes in photo-current and the corresponding knife-edge position give the intensity profile along the scanning direction. The intensity along the knife-edge was integrated in each measurement. In order to obtain a two-dimensional (2D) profile, the same procedure was carried out for several different scanning angles, as shown in Fig. 5(d). The data that we acquired allowed the 2D beam intensity distribution at the focus to be reconstructed using the Radon back-transformation [23]. Although the intensity of the input beam in the current study is symmetrical over most of the cross-section perpendicular to the beam axis, a small region of the cross-section is asymmetric and we thus used data from a single scan performed along the most symmetrical direction to reconstruct the 2D profile.
In order to modify the cross-sectional intensity profile of the beam to obtain a narrower annular shape, we placed a shading mask with a circular opaque region on the glass such that the inner part of the input beam was blocked. The radii of the opaque area and the objective lens pupil then corresponded to the inner and outer radii of the modified beam, respectively. We changed the value of δ by varying the radius of the opaque area.
3.2. Results
Figure 6 shows the experimentally determined cross-sectional intensity profiles of the focal spot for the original radially polarized beam with δ=0.0 and for the modified beam with δ=0.8. The profiles were measured in the focal plane (z=0 μm) and in planes away from the focus (z=±1, ±2 μm). Plus signs indicate positions away from the objective lens with respect to the focal plane. For the original beam, shown in Fig. 6(a), the center of the intensity profile changes drastically from a single peak to a double peak when the observation plane is shifted by ± 1 μm from the focus; the peak intensity is less than half of that at z=0 μm. At z=2 μm the cross-sectional intensity profile could not be measured because it became widely dispersed with intensity changes smaller than the noise level. In contrast, the modified beam with δ=0.8 gave a central peak profile that was almost single-peaked for all observation planes from z=−2 μm to z=2 μm; the intensity of the central peak was maintained at more than half of the z=0 μm value between z=−1 μm and z=2 μm, as shown in Fig. 6(b). These results clearly show that the depth of focus is expanded when δ is enlarged. Furthermore, the modified beam with δ=0.8 produces a smaller spot at the focal plane (approximately 600 nm) than the original beam with δ=0.0 (approximately 1200 nm). The minor discrepancies between the calculated and experimental results, for example the dip in the central peak of the experimental profile at z = −1 μm in Fig. 6(b) and the larger side-lobe of Fig. 6(b) than Fig. 3(b), are probably due to scattering from the knife-edge or to the effect of coupling to a surface plasmon mode of the thin metal film, and will be reported in a future article.
Fig. 6 Normalized cross-sectional intensity profiles at the focal plane (z=0) and its vicinity. (a) Intensity profiles for the original beam (δ=0.0) show significant changes from a single peak to a double peak for a deviation of ±1 μm from the focal plane. The intensity profile at z=2 μm could not be measured due to the small signal intensity. (b) Intensity profiles for the narrower-width annular beam (δ=0.8), show a central peak for all observation planes from −2 μm to 2 μm.
4. Conclusion
We have studied the spot size and depth of focus of a radially polarized beam generated by a PC laser. Theoretical calculations showed that if the intensity profile of the doughnut beam cross-section is modified to give a narrower ring, the focal spot size can be decreased to 0.4λ and the depth of focus can be extended to more than 4λ. Corresponding experiments showed good agreement with the calculations. Because the resultant focal spot arises mainly from the longitudinal electric field component, we envisage further research related to the longitudinal field. We believe that it should be possible to obtain narrower-width, radially polarized annular beams from PC lasers by tailoring the PC design. Tighter focal spots and longer depths of focus with strong longitudinal polarization would then be realized without any power loss, contributing to the development and expansion of a range of optical technologies.
Acknowledgments
This work was partly supported by the Japan Society of Promotion and Science, the Global COE program of Kyoto University, and Special Coordination Funds for Promoting Science and Technology (SCF) commissioned by the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan. We thank Y. Kurosaka and W. Kunishi for providing the PC lasers. We also thank K. Ishizaki for helpful discussions concerning the calculations.
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