Corrections to the knife-edge based reconstruction scheme of tightly focused light beams C. Huber,1,2 S. Orlov,1,2,∗ P. Banzer,1,2 and G. Leuchs1,2 2
1 Max Planck Institute for the Science of Light, G¨unther-Scharowsky-Str. 1, D-91058 Erlangen, Germany Institute of Optics, Information and Photonics, University Erlangen-Nuremberg, Staudtstr. 7B/2, D-91058 Erlangen, Germany
*[email protected]
Abstract: The knife-edge method is an established technique for profiling light beams. It was shown, that this technique even works for tightly focused beams, if the material and geometry of the probing knife-edges are chosen carefully. Furthermore, it was also reported recently that this method fails, when the knife-edges are made from pure materials. The artifacts introduced in the reconstructed beam shape and position depend strongly on the edge and input beam parameters, because the knife-edge is excited by the incoming beam. Here we show, that the actual beam shape and spot size of tightly focused beams can still be derived from knife-edge measurements for pure edge materials and different edge thicknesses by adapting the analysis method of the experimental data taking into account the interaction of the beam with the edge. © 2013 Optical Society of America OCIS codes: (140.3295) Laser beam characterization; (260.5430) Polarization; (050.6624) Subwavelength structures; (050.1940) Diffraction; (240.6680) Surface plasmons.
References and links 1. S. Quabis, R. Dorn, M. Eberler, O. Gl¨ockl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1-7 (2000). 2. R. Dorn, S. Quabis, and G. Leuchs, “The focus of light-linear polarization breaks the rotational symmetry of the focal spot,” J. Mod. Opt. 50, 1917–1926 (2003). 3. B. Richards and E. Wolf, “Electromagnetic diffraction in optical Systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. A 253, 358–379 (1959). 4. J. Kindler, P. Banzer, S. Quabis, U. Peschel, and G. Leuchs, “Waveguide properties of single subwavelength holes demonstrated with radially and azimuthally polarized light,” Appl. Phys. B 89, 517-520 (2007). 5. T. Z¨uchner, A. V. Failla, A. Hartschuh, and A. J. Meixner, “A novel approach to detect and characterize the scattering patterns of single Au nanoparticles using confocal microscopy,” J. Microsc. 229, 337-343 (2008). 6. P. Banzer, J. Kindler, S. Quabis, U. Peschel, and G. Leuchs, “Extraordinary transmission through a single coaxial aperture in a thin metal film,” Opt. Express 18, 10896–10904 (2010). 7. P. Banzer, U. Peschel, S. Quabis, and G. Leuchs, “On the experimental investigation of the electric and magnetic response of a single nano-structure,” Opt. Express 18, 10905-10923 (2010). 8. A. H. Firester, M. E. Heller, and P. Sheng, “Knife-edge scanning measurements of subwavelength focussed light beams,” Appl. Opt. 16, 197-1-1974 (1977). 9. M. B. Schneider and W. W. Webb, “Measurement of submicron laser beam radii,” Appl. Opt. 20, 1382–1388 (1981). 10. R. L. McCally, “Measurement of Gaussian beam parameters,” Appl. Opt. 23, 2227–2227 (1984).
#195753 - $15.00 USD Received 13 Aug 2013; accepted 1 Oct 2013; published 14 Oct 2013 (C) 2013 OSA 21 October 2013 | Vol. 21, No. 21 | DOI:10.1364/OE.21.025069 | OPTICS EXPRESS 25069
11. J. M. Khosrofian and B. A. Garetz, “Measurement of a Gaussian laser beam diameter through the direct inversion of knife-edge data”, Appl. Opt. 22, 3406–3410 (1983). 12. G. Brost, P. D. Horn, and A. Abtahi, “Convenient spatial profiling of pulsed laser beams,” Appl. Opt. 24, 38–40 (1985). 13. M. A. de Araujo, R. Silva, E. de Lima, D. P. Pereira, and P. C. de Oliveira, “Measurement of Gaussian laser beam radius using the knife-edge technique: improvement on data analysis,” Appl. Opt. 48, 393–396 (2009). 14. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003). 15. P. Marchenko, S. Orlov, C. Huber, P. Banzer, S. Quabis, U. Peschel, and G. Leuchs, “Interaction of highly focused vector beams with a metal knife-edge,” Opt. Express 19 7244–7261 (2011). 16. B. Sturman, E. Podivilov, and M. Gorkunov, “Eigenmodes for metal-dielectric light-transmitting nanostructures”, Phys. Rev. B 76, 125104 (2007). 17. S¸. E. Kocabas¸, G. Veronis, D. A. B. Miller, and S. Fan, “Modal analysis and coupling in metal-insulator-metal wavequides”, Phys. Rev. B 79, 035120 (2009).
1.
Introduction
When a linearly polarized Gaussian light beam is focused tightly, its focal electric energy density distribution does not exhibit a circular shape anymore but is elongated along the polarization axis of the input beam [1–3]. This symmetry break is predominantly caused by the appearance of longitudinal (parallel to the optical axis of the focusing system) components of the electric field in the focal plane, which can be noticeable depending on the numerical aperture of the focusing system. The formation of the aforementioned pattern is a result of interference of plane wave components the input beam can be decomposed into. The strength and shape of individual components and, hence, also the shape of the total electric energy density distribution in the focal plane depends sensitively on the quality of the focusing system and the alignment. As a direct consequence, it is crucial to experimentally analyze and profile tightly focused vectorial beams in a real-world setup before utilizing them for experiments in nano-optics (see [4–7] and others) etc. One well-known technique for beam-profiling in a given plane is the so-called knife-edge method (see for instance [8–13]). In this technique, a beam-block with a sharp edge made from an opaque material (such as a knife or a razor-blade) is line-scanned through the beam perpendicular to the optical axis while the transmitted intensity is measured with a detector. The measurement is repeated for several scanning directions. From the resulting photocurrent curves the so-called beam-projections onto the scan-line and finally the real beam shape can be tomographically reconstructed. This experimental method was also proven to work for tightly focused light beams [2, 14], if the material of the knife-edge, its thickness and other parameters are chosen carefully. In a more recent study, knife-edges made from pure materials (metals, etc.) were systematically studied and polarization dependent effects in the knife-edge profiling method were observed in general causing distortions in the reconstructed beam width, shape and position and preventing a proper reconstruction. These distortions are caused by the interaction of the incoming focused beam with the knife-edge [15]. In this paper, we now demonstrate, that the interaction between a knife-edge and a highly focused linearly polarized beam can be understood in terms of the moments of the beam (beam profile times a polynomial) and propose an adapted beam reconstruction and fitting technique for highly focused linearly polarized light beams when profiling with knife-edges made from pure materials. The experimental principle of the knife-edge method is depicted in Fig. 1. Two scanning directions of the knife-edge (s/p: electric field of the incoming beam perpendicular/parallel to the knife-edge) through the focused linearly polarized TEM00 -mode (in the xy-plane) are investigated (compare [15]). For experimental reasons the beam is profiled by two adjacent edges of a single rectangular metal pad. The photocurrent generated inside the photodiode is proportional to the power P detected by the photodiode and is recorded for each beam position
#195753 - $15.00 USD Received 13 Aug 2013; accepted 1 Oct 2013; published 14 Oct 2013 (C) 2013 OSA 21 October 2013 | Vol. 21, No. 21 | DOI:10.1364/OE.21.025069 | OPTICS EXPRESS 25070
d0
(a)
(b)
(c)
Fig. 1. Schematic depiction of the knife-edge method for a two-dimensional beam (a). Typical beam profiling data (photocurrent curves) (b) and their derivatives (beam-projections) (c). The state of polarization always refers to the orientation of the electric field of the input beam relative to the knife-edge in the xy-plane.
x0 with respect to the knife-edge (see Fig. 1(b)) P = P0
∞ −∞
0
dy
−∞
I (x − x0 , y, z = 0) dx,
(1)
where P0 is a proportionality coefficient and I is the electric field intensity. In the conventional knife-edge method the derivative ∂ P/∂ x0 of the photocurrent curve with respect to the beam position x0 (see Fig. 1(c)) reconstructs a projection of the intensity onto the xz-plane at z = 0 (projection onto the x-axis) [8]. In a next step, the two dimensional intensity distribution can be reconstructed from projections measured along different directions using the Radon backtransform [2, 14]. Two parameters ds and d p define the distance between the maxima (minima) of the reconstructed intensity projections for s- and p-polarized input beams respectively, see Fig. 1(c). For paraxial light beams (or plane waves), the term intensity can refer to the total electric energy density and the z-component of the Poynting-vector S both being proportional to each other because longitudinal electric field components are negligible in this limit. The latter is not necessarily true anymore in the case of tightly focused light beams (non-paraxial propagation), which can exhibit quite strong longitudinal electric field components resulting in different distributions of |E(x, y)|2 and Sz (x, y). Therefore, the question arises which distribution is meant by intensity in this case. It was believed, that the integral Eq. (1) borrowed/adopted from the con#195753 - $15.00 USD Received 13 Aug 2013; accepted 1 Oct 2013; published 14 Oct 2013 (C) 2013 OSA 21 October 2013 | Vol. 21, No. 21 | DOI:10.1364/OE.21.025069 | OPTICS EXPRESS 25071
Fig. 2. Dependence of the conventionally determined beamwidths w p , ws on the wavelength λ for various edge material of thickness h = 130 nm. The dashed lines represent the FWHM of the squared modulus of the electric field calculated with vectorial diffraction theory.
ventional knife-edge method for retrieving the beam projection of paraxial light beams allows for the reconstruction of the beam profile in terms of its total electric energy density distribution |E(x, y)|2 also in case of tightly focused vectorial beams. It was shown that this assumption holds true only if special edge materials, thicknesses and certain wavelengths are chosen [2]. Nevertheless, it was also shown just recently [15], that for pure knife-edge materials of different thicknesses and for different wavelengths of the input beam, the retrieved projections do not correspond to the expected projections of the electric energy density distributions. They appear shifted (ds = d p = d0 with d0 the width of the metal pad; see Fig. 1) and asymmetrically deformed also causing deviations in the retrieved beam diameters ws and w p due to interactions between the knife-edge and the beam, Fig. 1(c). For the reconstruction of light beams with diameters larger than several wavelengths such effects are negligible. Hence the derivatives of the photocurrent curves represent beam projections. The above mentioned distortions for the profiling of tightly focused vectorial beams are caused by the fact, that the knife-edge is not only blocking the beam while line-scanning but it is also excited by the beam. Furthermore, the power flow through the knife-edge is polarization dependent and proportional to the value of the projection of the electric energy density UE (±x0 ) onto the edge. Obviously, if one does not account for these effects, the standard scheme is not valid without corrections unless the knife-edge parameters are carefully chosen [1]. 2.
Theoretical considerations
We start with the introduction of a numerical technique to correct for the aforementioned artifacts introduced by the interaction of the edge with the focused light field, thus enabling the use of any kind of opaque material as a knife-edge. For that purpose, we investigate the discussed light-matter interaction between the focused light beam and the knife-edge as it is seen by a detector, i.e. we evaluate transformations of the electric field introduced in Eq. (1). First, the integration in Eq. (1) over the y-axis reduces the dimensionality of the Helmholtz equation by one, i.e. we do not have to consider the electric field E but its projection Eˆ onto the
#195753 - $15.00 USD Received 13 Aug 2013; accepted 1 Oct 2013; published 14 Oct 2013 (C) 2013 OSA 21 October 2013 | Vol. 21, No. 21 | DOI:10.1364/OE.21.025069 | OPTICS EXPRESS 25072
xz-plane. Next, due to the symmetry of the knife edge solutions of the Helmholtz equation consist of two independent classes: transverse electric (in our notation p-polarized) and transverse ˆ in the p-polarized magnetic (s-polarized) modes, see [15]. The projection of the electric field E case has only one non-vanishing component of the electric field Eˆy parallel to the knife-edge. The s-polarized solutions have two non-vanishing components of the electric field projections. The main component Eˆx is perpendicular (s-polarization) to the knife-edge while Eˆz depends on Eˆx as Eˆz = −(i/k0 )∂ Eˆx /∂ x, where k0 is the wave vector of the carrier wave [15–17]. In order to analyze the interaction of a highly focused incoming beam with an electric field ˆ b into s- and pdistribution Eb with the knife-edge, we need to decompose the projection E polarized modes. The highly focused linearly polarized TEM00 -mode (which we consider here) has two orientations of the main electric field component relative to the knife-edge, for which this decomposition is trivial. If the beam polarization is parallel to the knife-edge (p-situation), ˆ b has only a non-vanishing Eˆ y component, so it is dethe projection of the electric field E composed only into p-modes. When the beam’s electric field is oriented perpendicularly (ssituation) to the knife edge, it is decomposed only into s-modes. For the sake of brevity we consider further only those two orientations of the beam. We start with rewriting Eq. (1) as P = P0
0 −∞
∞
dx
−∞
dkxUˆ E (kx , x0 )Tˆ (kx )eikx x
(2)
where we have performed the integration over the y-axis and express the projection of the signal at the photodiode in the Fourier-domain as Uˆ E (kx , x0 )Tˆ (kx ). Here Uˆ E (kx , x0 ) is the Fourierimage of the signal, which we expect to measure (projection of the electric field energy density UE (x) onto xz-plane at the knife-edge). Tˆ (kx ) is a spectral representation of the polarization dependent knife-edge interaction operator. Let us introduce the Taylor expansion of Tˆ (kx ) with ∞ n kx ∂ n Tˆ (kx ) ˆ . (3) T (kx ) = 1 + ∑ ∂ kxn kx =0 n=1 n! We note that Uˆ E (kx , x0 ) = Uˆ E,0 (kx )e∓ikx x0 , where Uˆ E,0 (kx ) is the Fourier image of the electric energy density projection UE,0 (x) of the beam exactly on the knife-edge (x0 = 0). We substitute Eq. (3) into Eq. (2). We use the relation ∂ nUE (x)/∂ xn = dkx (ikx )n Uˆ E (kx ) eikx x , so the resulting expression reads 0 ∞ ∂ nUE (x ± x0 ) , (4) dx UE (x ± x0 ) + ∑ An P = P0 ∂ xn −∞ n=1 with An = (in n!)−1 ∂ n Tˆ /∂ kxn . Taking the derivative of Eq. (4) results in ∞ ∂ nUE,0 (±x0 ) ∂P = UE,0 (±x0 ) + ∑ An . P0 ∂ x0 ∂ x0n n=1
(5)
So, the derivative of the photocurrent can be expressed as a sum containing the profile of the electric field energy density projection UE,0 (±x0 ) (the classical knife-edge term) plus interaction terms expressed as a knife-edge and polarization dependent sum over higher order derivatives (which we call moments) of the beam projections. The physical meaning behind Eq. (5) is the following. The first term in the sum (n = 1) is due to the local response of the knife-edge to the s- or p-polarized electric field. The second term (n = 2) expresses the local response of the knife-edge to the projection of the electric field gradient and so on.
#195753 - $15.00 USD Received 13 Aug 2013; accepted 1 Oct 2013; published 14 Oct 2013 (C) 2013 OSA 21 October 2013 | Vol. 21, No. 21 | DOI:10.1364/OE.21.025069 | OPTICS EXPRESS 25073
Fig. 3. Depiction of the adapted knife-edge method for a tightly focused and linearly polarized beam. The derivatives of the experimentally measured photocurrents (gray circles) and the fitted curve (black) with the beam profile (red) and its first four derivatives (moments) are shown for λ = 700 nm and a knife-edge made from gold with a thickness of h = 130 nm. The corresponding states of polarization are shown in the graphs. The position of the knife-edge is schematically depicted by the gray bar.
3.
Experimental results and adapted fit algorithm
In order to verify the validity of this approach, we have performed a number of experiments using knife-edges produced from various materials (Au, Ti, Ge) with an exemplary thickness of h = 130 nm on gallium-arsenide photodiodes at wavelengths from 530 nm to 700 nm. The details of our experimental setup and measurement technique can be found in [15]. The dependence of the conventionally evaluated beamwidths ws and w p on the wavelength λ for various samples is depicted in Fig. 2. As it was already mentioned, we see that conventionally determined beamwidths do not fit to the expectations from the estimations based on the Debye integrals [3]. Thus, we have implemented a least-square fitting algorithm, based on function (5), with up to the fourth order derivative of the electric field energy density projection UE (x0 ). With a high degree of accuracy we can assume a Gaussian distribution of the x (y) components of the s-(p-)polarized beams in the plane of projection. The Eˆz component of the s-polarized beam depends on Eˆx , thus a different ansatz is used. Before fitting, we predetermine the exact position of the knife-edge to reduce the number of free parameters in the fitting-routine and to fix the coordinate frame. For that purpose, we measure the real distances d0 between both edges using a scanning electron microscope (SEM), find the center xc between the peak values of both projections in one scan and finally set the actual positions of both knife-edges to be at xc ± d0 /2. An example of such a fitting procedure is presented in Fig. 3 for two polarization states (s and p respectively). It turns out, that for all investigated experimental samples, we were able to successfully determine the real diameters of the beam projections by simultaneously ensuring ds = d p = d0 . Exemplarily, we show the result for the actual beam projection widths ws and w p reconstructed with this technique, see Fig. 4. Here, we compare the theoretically expected values for the beam width calculated using vectorial diffraction theory [3] with the values retrieved from the fitted data. We see, that after applying our fitting procedure the beam diameters can #195753 - $15.00 USD Received 13 Aug 2013; accepted 1 Oct 2013; published 14 Oct 2013 (C) 2013 OSA 21 October 2013 | Vol. 21, No. 21 | DOI:10.1364/OE.21.025069 | OPTICS EXPRESS 25074
Fig. 4. Dependence of the beamwidths w p , ws on the wavelength λ reconstructed with the adapted method for Au, Ti and Ge samples of thickness h = 130 nm. The dashed lines represent the FWHM of the squared electric field estimated from the Debye integrals [3].
be accurately derived from the experimental data, compare Figs. 2 and 4. A very good overlap with the theoretical values from vectorial diffraction theory is observed allowing for an accurate determination of the beam size. The reconstruction of the beam projection for the cases, when the polarization of the input beam is neither parallel nor perpendicular, is performed in a similar fashion, however it requires an additional polarizer in order to decompose the incoming beam into s- and p-polarized projections. In a next step, after correcting respective ansatz functions for the s- and p-polarized projections of the beam, the technique presented here, can be also applied to each projection separately. To prove the robustness and flexibility of our approach, we have performed additional experiments in different scenarios (smaller focusing NA) and successfully reconstructed beamwidths using this fitting procedure. The proposed fitting procedure also works for other edge parameters, e.g. knife-edge thicknesses. In any case, the retrieved expansion coefficients An , Eq. (5), depend on the edge parameters (thickness, material and optical properties). Hence, once determined, they can be used as fitting parameters also in situations in which, for instance, the focusing NA is smaller or larger than in the case for which they have been retrieved. Furthermore, also for input beams of other geometries, such as cylindrical vector beams etc., a proper retrieval of the beam parameters, such as the beam size in the focal plane can be achieved when choosing appropriate ansatz functions in Eq. (5). A more detailed study on these topics will be presented elsewhere soon. 4.
Conclusions
In conclusion, we have presented a very straight-forward and easy to implement method to retrieve the beam parameters such as the beam width in the focus of high numerical aperture lenses using corrected knife-edge data. At the same time, this method allows for the correction of the shifts of the beam projections observed in knife-edge measurements. This adapted analysis method is highly flexible and robust.
#195753 - $15.00 USD Received 13 Aug 2013; accepted 1 Oct 2013; published 14 Oct 2013 (C) 2013 OSA 21 October 2013 | Vol. 21, No. 21 | DOI:10.1364/OE.21.025069 | OPTICS EXPRESS 25075
Acknowledgments We thank Stefan Malzer, Isabel G¨aßner, Olga Rusina, Irina Harder and Daniel Ploß for their valuable support in preparing the samples.
#195753 - $15.00 USD Received 13 Aug 2013; accepted 1 Oct 2013; published 14 Oct 2013 (C) 2013 OSA 21 October 2013 | Vol. 21, No. 21 | DOI:10.1364/OE.21.025069 | OPTICS EXPRESS 25076
1 Max Planck Institute for the Science of Light, G¨unther-Scharowsky-Str. 1, D-91058 Erlangen, Germany Institute of Optics, Information and Photonics, University Erlangen-Nuremberg, Staudtstr. 7B/2, D-91058 Erlangen, Germany
*[email protected]
Abstract: The knife-edge method is an established technique for profiling light beams. It was shown, that this technique even works for tightly focused beams, if the material and geometry of the probing knife-edges are chosen carefully. Furthermore, it was also reported recently that this method fails, when the knife-edges are made from pure materials. The artifacts introduced in the reconstructed beam shape and position depend strongly on the edge and input beam parameters, because the knife-edge is excited by the incoming beam. Here we show, that the actual beam shape and spot size of tightly focused beams can still be derived from knife-edge measurements for pure edge materials and different edge thicknesses by adapting the analysis method of the experimental data taking into account the interaction of the beam with the edge. © 2013 Optical Society of America OCIS codes: (140.3295) Laser beam characterization; (260.5430) Polarization; (050.6624) Subwavelength structures; (050.1940) Diffraction; (240.6680) Surface plasmons.
References and links 1. S. Quabis, R. Dorn, M. Eberler, O. Gl¨ockl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1-7 (2000). 2. R. Dorn, S. Quabis, and G. Leuchs, “The focus of light-linear polarization breaks the rotational symmetry of the focal spot,” J. Mod. Opt. 50, 1917–1926 (2003). 3. B. Richards and E. Wolf, “Electromagnetic diffraction in optical Systems. II. Structure of the image field in an aplanatic system,” Proc. R. Soc. A 253, 358–379 (1959). 4. J. Kindler, P. Banzer, S. Quabis, U. Peschel, and G. Leuchs, “Waveguide properties of single subwavelength holes demonstrated with radially and azimuthally polarized light,” Appl. Phys. B 89, 517-520 (2007). 5. T. Z¨uchner, A. V. Failla, A. Hartschuh, and A. J. Meixner, “A novel approach to detect and characterize the scattering patterns of single Au nanoparticles using confocal microscopy,” J. Microsc. 229, 337-343 (2008). 6. P. Banzer, J. Kindler, S. Quabis, U. Peschel, and G. Leuchs, “Extraordinary transmission through a single coaxial aperture in a thin metal film,” Opt. Express 18, 10896–10904 (2010). 7. P. Banzer, U. Peschel, S. Quabis, and G. Leuchs, “On the experimental investigation of the electric and magnetic response of a single nano-structure,” Opt. Express 18, 10905-10923 (2010). 8. A. H. Firester, M. E. Heller, and P. Sheng, “Knife-edge scanning measurements of subwavelength focussed light beams,” Appl. Opt. 16, 197-1-1974 (1977). 9. M. B. Schneider and W. W. Webb, “Measurement of submicron laser beam radii,” Appl. Opt. 20, 1382–1388 (1981). 10. R. L. McCally, “Measurement of Gaussian beam parameters,” Appl. Opt. 23, 2227–2227 (1984).
#195753 - $15.00 USD Received 13 Aug 2013; accepted 1 Oct 2013; published 14 Oct 2013 (C) 2013 OSA 21 October 2013 | Vol. 21, No. 21 | DOI:10.1364/OE.21.025069 | OPTICS EXPRESS 25069
11. J. M. Khosrofian and B. A. Garetz, “Measurement of a Gaussian laser beam diameter through the direct inversion of knife-edge data”, Appl. Opt. 22, 3406–3410 (1983). 12. G. Brost, P. D. Horn, and A. Abtahi, “Convenient spatial profiling of pulsed laser beams,” Appl. Opt. 24, 38–40 (1985). 13. M. A. de Araujo, R. Silva, E. de Lima, D. P. Pereira, and P. C. de Oliveira, “Measurement of Gaussian laser beam radius using the knife-edge technique: improvement on data analysis,” Appl. Opt. 48, 393–396 (2009). 14. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003). 15. P. Marchenko, S. Orlov, C. Huber, P. Banzer, S. Quabis, U. Peschel, and G. Leuchs, “Interaction of highly focused vector beams with a metal knife-edge,” Opt. Express 19 7244–7261 (2011). 16. B. Sturman, E. Podivilov, and M. Gorkunov, “Eigenmodes for metal-dielectric light-transmitting nanostructures”, Phys. Rev. B 76, 125104 (2007). 17. S¸. E. Kocabas¸, G. Veronis, D. A. B. Miller, and S. Fan, “Modal analysis and coupling in metal-insulator-metal wavequides”, Phys. Rev. B 79, 035120 (2009).
1.
Introduction
When a linearly polarized Gaussian light beam is focused tightly, its focal electric energy density distribution does not exhibit a circular shape anymore but is elongated along the polarization axis of the input beam [1–3]. This symmetry break is predominantly caused by the appearance of longitudinal (parallel to the optical axis of the focusing system) components of the electric field in the focal plane, which can be noticeable depending on the numerical aperture of the focusing system. The formation of the aforementioned pattern is a result of interference of plane wave components the input beam can be decomposed into. The strength and shape of individual components and, hence, also the shape of the total electric energy density distribution in the focal plane depends sensitively on the quality of the focusing system and the alignment. As a direct consequence, it is crucial to experimentally analyze and profile tightly focused vectorial beams in a real-world setup before utilizing them for experiments in nano-optics (see [4–7] and others) etc. One well-known technique for beam-profiling in a given plane is the so-called knife-edge method (see for instance [8–13]). In this technique, a beam-block with a sharp edge made from an opaque material (such as a knife or a razor-blade) is line-scanned through the beam perpendicular to the optical axis while the transmitted intensity is measured with a detector. The measurement is repeated for several scanning directions. From the resulting photocurrent curves the so-called beam-projections onto the scan-line and finally the real beam shape can be tomographically reconstructed. This experimental method was also proven to work for tightly focused light beams [2, 14], if the material of the knife-edge, its thickness and other parameters are chosen carefully. In a more recent study, knife-edges made from pure materials (metals, etc.) were systematically studied and polarization dependent effects in the knife-edge profiling method were observed in general causing distortions in the reconstructed beam width, shape and position and preventing a proper reconstruction. These distortions are caused by the interaction of the incoming focused beam with the knife-edge [15]. In this paper, we now demonstrate, that the interaction between a knife-edge and a highly focused linearly polarized beam can be understood in terms of the moments of the beam (beam profile times a polynomial) and propose an adapted beam reconstruction and fitting technique for highly focused linearly polarized light beams when profiling with knife-edges made from pure materials. The experimental principle of the knife-edge method is depicted in Fig. 1. Two scanning directions of the knife-edge (s/p: electric field of the incoming beam perpendicular/parallel to the knife-edge) through the focused linearly polarized TEM00 -mode (in the xy-plane) are investigated (compare [15]). For experimental reasons the beam is profiled by two adjacent edges of a single rectangular metal pad. The photocurrent generated inside the photodiode is proportional to the power P detected by the photodiode and is recorded for each beam position
#195753 - $15.00 USD Received 13 Aug 2013; accepted 1 Oct 2013; published 14 Oct 2013 (C) 2013 OSA 21 October 2013 | Vol. 21, No. 21 | DOI:10.1364/OE.21.025069 | OPTICS EXPRESS 25070
d0
(a)
(b)
(c)
Fig. 1. Schematic depiction of the knife-edge method for a two-dimensional beam (a). Typical beam profiling data (photocurrent curves) (b) and their derivatives (beam-projections) (c). The state of polarization always refers to the orientation of the electric field of the input beam relative to the knife-edge in the xy-plane.
x0 with respect to the knife-edge (see Fig. 1(b)) P = P0
∞ −∞
0
dy
−∞
I (x − x0 , y, z = 0) dx,
(1)
where P0 is a proportionality coefficient and I is the electric field intensity. In the conventional knife-edge method the derivative ∂ P/∂ x0 of the photocurrent curve with respect to the beam position x0 (see Fig. 1(c)) reconstructs a projection of the intensity onto the xz-plane at z = 0 (projection onto the x-axis) [8]. In a next step, the two dimensional intensity distribution can be reconstructed from projections measured along different directions using the Radon backtransform [2, 14]. Two parameters ds and d p define the distance between the maxima (minima) of the reconstructed intensity projections for s- and p-polarized input beams respectively, see Fig. 1(c). For paraxial light beams (or plane waves), the term intensity can refer to the total electric energy density and the z-component of the Poynting-vector S both being proportional to each other because longitudinal electric field components are negligible in this limit. The latter is not necessarily true anymore in the case of tightly focused light beams (non-paraxial propagation), which can exhibit quite strong longitudinal electric field components resulting in different distributions of |E(x, y)|2 and Sz (x, y). Therefore, the question arises which distribution is meant by intensity in this case. It was believed, that the integral Eq. (1) borrowed/adopted from the con#195753 - $15.00 USD Received 13 Aug 2013; accepted 1 Oct 2013; published 14 Oct 2013 (C) 2013 OSA 21 October 2013 | Vol. 21, No. 21 | DOI:10.1364/OE.21.025069 | OPTICS EXPRESS 25071
Fig. 2. Dependence of the conventionally determined beamwidths w p , ws on the wavelength λ for various edge material of thickness h = 130 nm. The dashed lines represent the FWHM of the squared modulus of the electric field calculated with vectorial diffraction theory.
ventional knife-edge method for retrieving the beam projection of paraxial light beams allows for the reconstruction of the beam profile in terms of its total electric energy density distribution |E(x, y)|2 also in case of tightly focused vectorial beams. It was shown that this assumption holds true only if special edge materials, thicknesses and certain wavelengths are chosen [2]. Nevertheless, it was also shown just recently [15], that for pure knife-edge materials of different thicknesses and for different wavelengths of the input beam, the retrieved projections do not correspond to the expected projections of the electric energy density distributions. They appear shifted (ds = d p = d0 with d0 the width of the metal pad; see Fig. 1) and asymmetrically deformed also causing deviations in the retrieved beam diameters ws and w p due to interactions between the knife-edge and the beam, Fig. 1(c). For the reconstruction of light beams with diameters larger than several wavelengths such effects are negligible. Hence the derivatives of the photocurrent curves represent beam projections. The above mentioned distortions for the profiling of tightly focused vectorial beams are caused by the fact, that the knife-edge is not only blocking the beam while line-scanning but it is also excited by the beam. Furthermore, the power flow through the knife-edge is polarization dependent and proportional to the value of the projection of the electric energy density UE (±x0 ) onto the edge. Obviously, if one does not account for these effects, the standard scheme is not valid without corrections unless the knife-edge parameters are carefully chosen [1]. 2.
Theoretical considerations
We start with the introduction of a numerical technique to correct for the aforementioned artifacts introduced by the interaction of the edge with the focused light field, thus enabling the use of any kind of opaque material as a knife-edge. For that purpose, we investigate the discussed light-matter interaction between the focused light beam and the knife-edge as it is seen by a detector, i.e. we evaluate transformations of the electric field introduced in Eq. (1). First, the integration in Eq. (1) over the y-axis reduces the dimensionality of the Helmholtz equation by one, i.e. we do not have to consider the electric field E but its projection Eˆ onto the
#195753 - $15.00 USD Received 13 Aug 2013; accepted 1 Oct 2013; published 14 Oct 2013 (C) 2013 OSA 21 October 2013 | Vol. 21, No. 21 | DOI:10.1364/OE.21.025069 | OPTICS EXPRESS 25072
xz-plane. Next, due to the symmetry of the knife edge solutions of the Helmholtz equation consist of two independent classes: transverse electric (in our notation p-polarized) and transverse ˆ in the p-polarized magnetic (s-polarized) modes, see [15]. The projection of the electric field E case has only one non-vanishing component of the electric field Eˆy parallel to the knife-edge. The s-polarized solutions have two non-vanishing components of the electric field projections. The main component Eˆx is perpendicular (s-polarization) to the knife-edge while Eˆz depends on Eˆx as Eˆz = −(i/k0 )∂ Eˆx /∂ x, where k0 is the wave vector of the carrier wave [15–17]. In order to analyze the interaction of a highly focused incoming beam with an electric field ˆ b into s- and pdistribution Eb with the knife-edge, we need to decompose the projection E polarized modes. The highly focused linearly polarized TEM00 -mode (which we consider here) has two orientations of the main electric field component relative to the knife-edge, for which this decomposition is trivial. If the beam polarization is parallel to the knife-edge (p-situation), ˆ b has only a non-vanishing Eˆ y component, so it is dethe projection of the electric field E composed only into p-modes. When the beam’s electric field is oriented perpendicularly (ssituation) to the knife edge, it is decomposed only into s-modes. For the sake of brevity we consider further only those two orientations of the beam. We start with rewriting Eq. (1) as P = P0
0 −∞
∞
dx
−∞
dkxUˆ E (kx , x0 )Tˆ (kx )eikx x
(2)
where we have performed the integration over the y-axis and express the projection of the signal at the photodiode in the Fourier-domain as Uˆ E (kx , x0 )Tˆ (kx ). Here Uˆ E (kx , x0 ) is the Fourierimage of the signal, which we expect to measure (projection of the electric field energy density UE (x) onto xz-plane at the knife-edge). Tˆ (kx ) is a spectral representation of the polarization dependent knife-edge interaction operator. Let us introduce the Taylor expansion of Tˆ (kx ) with ∞ n kx ∂ n Tˆ (kx ) ˆ . (3) T (kx ) = 1 + ∑ ∂ kxn kx =0 n=1 n! We note that Uˆ E (kx , x0 ) = Uˆ E,0 (kx )e∓ikx x0 , where Uˆ E,0 (kx ) is the Fourier image of the electric energy density projection UE,0 (x) of the beam exactly on the knife-edge (x0 = 0). We substitute Eq. (3) into Eq. (2). We use the relation ∂ nUE (x)/∂ xn = dkx (ikx )n Uˆ E (kx ) eikx x , so the resulting expression reads 0 ∞ ∂ nUE (x ± x0 ) , (4) dx UE (x ± x0 ) + ∑ An P = P0 ∂ xn −∞ n=1 with An = (in n!)−1 ∂ n Tˆ /∂ kxn . Taking the derivative of Eq. (4) results in ∞ ∂ nUE,0 (±x0 ) ∂P = UE,0 (±x0 ) + ∑ An . P0 ∂ x0 ∂ x0n n=1
(5)
So, the derivative of the photocurrent can be expressed as a sum containing the profile of the electric field energy density projection UE,0 (±x0 ) (the classical knife-edge term) plus interaction terms expressed as a knife-edge and polarization dependent sum over higher order derivatives (which we call moments) of the beam projections. The physical meaning behind Eq. (5) is the following. The first term in the sum (n = 1) is due to the local response of the knife-edge to the s- or p-polarized electric field. The second term (n = 2) expresses the local response of the knife-edge to the projection of the electric field gradient and so on.
#195753 - $15.00 USD Received 13 Aug 2013; accepted 1 Oct 2013; published 14 Oct 2013 (C) 2013 OSA 21 October 2013 | Vol. 21, No. 21 | DOI:10.1364/OE.21.025069 | OPTICS EXPRESS 25073
Fig. 3. Depiction of the adapted knife-edge method for a tightly focused and linearly polarized beam. The derivatives of the experimentally measured photocurrents (gray circles) and the fitted curve (black) with the beam profile (red) and its first four derivatives (moments) are shown for λ = 700 nm and a knife-edge made from gold with a thickness of h = 130 nm. The corresponding states of polarization are shown in the graphs. The position of the knife-edge is schematically depicted by the gray bar.
3.
Experimental results and adapted fit algorithm
In order to verify the validity of this approach, we have performed a number of experiments using knife-edges produced from various materials (Au, Ti, Ge) with an exemplary thickness of h = 130 nm on gallium-arsenide photodiodes at wavelengths from 530 nm to 700 nm. The details of our experimental setup and measurement technique can be found in [15]. The dependence of the conventionally evaluated beamwidths ws and w p on the wavelength λ for various samples is depicted in Fig. 2. As it was already mentioned, we see that conventionally determined beamwidths do not fit to the expectations from the estimations based on the Debye integrals [3]. Thus, we have implemented a least-square fitting algorithm, based on function (5), with up to the fourth order derivative of the electric field energy density projection UE (x0 ). With a high degree of accuracy we can assume a Gaussian distribution of the x (y) components of the s-(p-)polarized beams in the plane of projection. The Eˆz component of the s-polarized beam depends on Eˆx , thus a different ansatz is used. Before fitting, we predetermine the exact position of the knife-edge to reduce the number of free parameters in the fitting-routine and to fix the coordinate frame. For that purpose, we measure the real distances d0 between both edges using a scanning electron microscope (SEM), find the center xc between the peak values of both projections in one scan and finally set the actual positions of both knife-edges to be at xc ± d0 /2. An example of such a fitting procedure is presented in Fig. 3 for two polarization states (s and p respectively). It turns out, that for all investigated experimental samples, we were able to successfully determine the real diameters of the beam projections by simultaneously ensuring ds = d p = d0 . Exemplarily, we show the result for the actual beam projection widths ws and w p reconstructed with this technique, see Fig. 4. Here, we compare the theoretically expected values for the beam width calculated using vectorial diffraction theory [3] with the values retrieved from the fitted data. We see, that after applying our fitting procedure the beam diameters can #195753 - $15.00 USD Received 13 Aug 2013; accepted 1 Oct 2013; published 14 Oct 2013 (C) 2013 OSA 21 October 2013 | Vol. 21, No. 21 | DOI:10.1364/OE.21.025069 | OPTICS EXPRESS 25074
Fig. 4. Dependence of the beamwidths w p , ws on the wavelength λ reconstructed with the adapted method for Au, Ti and Ge samples of thickness h = 130 nm. The dashed lines represent the FWHM of the squared electric field estimated from the Debye integrals [3].
be accurately derived from the experimental data, compare Figs. 2 and 4. A very good overlap with the theoretical values from vectorial diffraction theory is observed allowing for an accurate determination of the beam size. The reconstruction of the beam projection for the cases, when the polarization of the input beam is neither parallel nor perpendicular, is performed in a similar fashion, however it requires an additional polarizer in order to decompose the incoming beam into s- and p-polarized projections. In a next step, after correcting respective ansatz functions for the s- and p-polarized projections of the beam, the technique presented here, can be also applied to each projection separately. To prove the robustness and flexibility of our approach, we have performed additional experiments in different scenarios (smaller focusing NA) and successfully reconstructed beamwidths using this fitting procedure. The proposed fitting procedure also works for other edge parameters, e.g. knife-edge thicknesses. In any case, the retrieved expansion coefficients An , Eq. (5), depend on the edge parameters (thickness, material and optical properties). Hence, once determined, they can be used as fitting parameters also in situations in which, for instance, the focusing NA is smaller or larger than in the case for which they have been retrieved. Furthermore, also for input beams of other geometries, such as cylindrical vector beams etc., a proper retrieval of the beam parameters, such as the beam size in the focal plane can be achieved when choosing appropriate ansatz functions in Eq. (5). A more detailed study on these topics will be presented elsewhere soon. 4.
Conclusions
In conclusion, we have presented a very straight-forward and easy to implement method to retrieve the beam parameters such as the beam width in the focus of high numerical aperture lenses using corrected knife-edge data. At the same time, this method allows for the correction of the shifts of the beam projections observed in knife-edge measurements. This adapted analysis method is highly flexible and robust.
#195753 - $15.00 USD Received 13 Aug 2013; accepted 1 Oct 2013; published 14 Oct 2013 (C) 2013 OSA 21 October 2013 | Vol. 21, No. 21 | DOI:10.1364/OE.21.025069 | OPTICS EXPRESS 25075
Acknowledgments We thank Stefan Malzer, Isabel G¨aßner, Olga Rusina, Irina Harder and Daniel Ploß for their valuable support in preparing the samples.
#195753 - $15.00 USD Received 13 Aug 2013; accepted 1 Oct 2013; published 14 Oct 2013 (C) 2013 OSA 21 October 2013 | Vol. 21, No. 21 | DOI:10.1364/OE.21.025069 | OPTICS EXPRESS 25076
