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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 21 — Oct. 21, 2013
  • pp: 25069–25076
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Corrections to the knife-edge based reconstruction scheme of tightly focused light beams

C. Huber, S. Orlov, P. Banzer, and G. Leuchs  »View Author Affiliations


Optics Express, Vol. 21, Issue 21, pp. 25069-25076 (2013)
http://dx.doi.org/10.1364/OE.21.025069


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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 OSA

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

1. S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000). [CrossRef]

3

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). [CrossRef]

]. 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

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). [CrossRef]

7

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). [CrossRef] [PubMed]

] 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

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). [CrossRef]

13

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). [CrossRef] [PubMed]

]). 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

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).

, 14

14. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003). [CrossRef] [PubMed]

], 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

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 197244–7261 (2011). [CrossRef] [PubMed]

].

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.

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.

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 conventional 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

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).

]. Nevertheless, it was also shown just recently [15

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 197244–7261 (2011). [CrossRef] [PubMed]

], 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 (dsdpd0 with d0 the width of the metal pad; see Fig. 1) and asymmetrically deformed also causing deviations in the retrieved beam diameters ws and wp 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 UEx0) 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

1. S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000). [CrossRef]

].

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 Ê onto the 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 magnetic (s-polarized) modes, see [15

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 197244–7261 (2011). [CrossRef] [PubMed]

]. The projection of the electric field Ê in the p-polarized case has only one non-vanishing component of the electric field Êy parallel to the knife-edge. The s-polarized solutions have two non-vanishing components of the electric field projections. The main component Êx is perpendicular (s-polarization) to the knife-edge while Êz depends on Êx as Êz = −(i/k0)∂Êx/∂x, where k0 is the wave vector of the carrier wave [15

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 197244–7261 (2011). [CrossRef] [PubMed]

17

17. Ş. E. Kocabaş, G. Veronis, D. A. B. Miller, and S. Fan, “Modal analysis and coupling in metal-insulator-metal wavequides”, Phys. Rev. B 79, 035120 (2009). [CrossRef]

].

In order to analyze the interaction of a highly focused incoming beam with an electric field distribution Eb with the knife-edge, we need to decompose the projection Êb into s- and p-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), the projection of the electric field Êb has only a non-vanishing Êy component, so it is decomposed only into p-modes. When the beam’s electric field is oriented perpendicularly (s-situation) 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=P00dxdkxU^E(kx,x0)T^(kx)eikxx
(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 ÛE (kx, x0)(kx). Here ÛE (kx, x0) is the Fourier-image of the signal, which we expect to measure (projection of the electric field energy density UE (x) onto xz-plane at the knife-edge). (kx) is a spectral representation of the polarization dependent knife-edge interaction operator. Let us introduce the Taylor expansion of (kx) with
T^(kx)=1+n=1kxnn!nT^(kx)kxn|kx=0.
(3)

We note that ÛE (kx, x0) = ÛE,0(kx)e∓ikxx0, where ÛE,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)nÛE (kx) eikxx, so the resulting expression reads
P=P00dx[UE(x±x0)+n=1AnnUE(x±x0)xn],
(4)
with An=(inn!)1nT^/kxn. Taking the derivative of Eq. (4) results in
PP0x0=UE,0(±x0)+n=1AnnUE,0(±x0)x0n.
(5)
So, the derivative of the photocurrent can be expressed as a sum containing the profile of the electric field energy density projection UE,0x0) (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.

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

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 197244–7261 (2011). [CrossRef] [PubMed]

]. The dependence of the conventionally evaluated beamwidths ws and wp 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

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). [CrossRef]

].

Fig. 2 Dependence of the conventionally determined beamwidths wp, 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.

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 Êz component of the s-polarized beam depends on Êx, 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 = dp = d0.

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.

Fig. 4 Dependence of the beamwidths wp, 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].

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.

Acknowledgments

We thank Stefan Malzer, Isabel Gäßner, Olga Rusina, Irina Harder and Daniel Ploß for their valuable support in preparing the samples.

References and links

1.

S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun. 179, 1–7 (2000). [CrossRef]

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). [CrossRef]

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). [CrossRef]

5.

T. Züchner, 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). [CrossRef] [PubMed]

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). [CrossRef] [PubMed]

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). [CrossRef] [PubMed]

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). [CrossRef]

9.

M. B. Schneider and W. W. Webb, “Measurement of submicron laser beam radii,” Appl. Opt. 20, 1382–1388 (1981). [CrossRef] [PubMed]

10.

R. L. McCally, “Measurement of Gaussian beam parameters,” Appl. Opt. 23, 2227–2227 (1984). [CrossRef] [PubMed]

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). [CrossRef] [PubMed]

12.

G. Brost, P. D. Horn, and A. Abtahi, “Convenient spatial profiling of pulsed laser beams,” Appl. Opt. 24, 38–40 (1985). [CrossRef] [PubMed]

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). [CrossRef] [PubMed]

14.

R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett. 91, 233901 (2003). [CrossRef] [PubMed]

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 197244–7261 (2011). [CrossRef] [PubMed]

16.

B. Sturman, E. Podivilov, and M. Gorkunov, “Eigenmodes for metal-dielectric light-transmitting nanostructures”, Phys. Rev. B 76, 125104 (2007). [CrossRef]

17.

Ş. E. Kocabaş, G. Veronis, D. A. B. Miller, and S. Fan, “Modal analysis and coupling in metal-insulator-metal wavequides”, Phys. Rev. B 79, 035120 (2009). [CrossRef]

OCIS Codes
(050.1940) Diffraction and gratings : Diffraction
(240.6680) Optics at surfaces : Surface plasmons
(260.5430) Physical optics : Polarization
(140.3295) Lasers and laser optics : Laser beam characterization
(050.6624) Diffraction and gratings : Subwavelength structures

ToC Category:
Diffraction and Gratings

History
Original Manuscript: August 13, 2013
Manuscript Accepted: October 1, 2013
Published: October 14, 2013

Citation
C. Huber, S. Orlov, P. Banzer, and G. Leuchs, "Corrections to the knife-edge based reconstruction scheme of tightly focused light beams," Opt. Express 21, 25069-25076 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-21-25069


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References

  1. S. Quabis, R. Dorn, M. Eberler, O. Glöckl, and G. Leuchs, “Focusing light to a tighter spot,” Opt. Commun.179, 1–7 (2000). [CrossRef]
  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. A253, 358–379 (1959). [CrossRef]
  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. B89, 517–520 (2007). [CrossRef]
  5. T. Züchner, 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). [CrossRef] [PubMed]
  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. Express18, 10896–10904 (2010). [CrossRef] [PubMed]
  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. Express18, 10905–10923 (2010). [CrossRef] [PubMed]
  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). [CrossRef]
  9. M. B. Schneider and W. W. Webb, “Measurement of submicron laser beam radii,” Appl. Opt.20, 1382–1388 (1981). [CrossRef] [PubMed]
  10. R. L. McCally, “Measurement of Gaussian beam parameters,” Appl. Opt.23, 2227–2227 (1984). [CrossRef] [PubMed]
  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). [CrossRef] [PubMed]
  12. G. Brost, P. D. Horn, and A. Abtahi, “Convenient spatial profiling of pulsed laser beams,” Appl. Opt.24, 38–40 (1985). [CrossRef] [PubMed]
  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). [CrossRef] [PubMed]
  14. R. Dorn, S. Quabis, and G. Leuchs, “Sharper focus for a radially polarized light beam,” Phys. Rev. Lett.91, 233901 (2003). [CrossRef] [PubMed]
  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. Express197244–7261 (2011). [CrossRef] [PubMed]
  16. B. Sturman, E. Podivilov, and M. Gorkunov, “Eigenmodes for metal-dielectric light-transmitting nanostructures”, Phys. Rev. B76, 125104 (2007). [CrossRef]
  17. Ş. E. Kocabaş, G. Veronis, D. A. B. Miller, and S. Fan, “Modal analysis and coupling in metal-insulator-metal wavequides”, Phys. Rev. B79, 035120 (2009). [CrossRef]

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