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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 12 — Jun. 4, 2012
  • pp: 13302–13310
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Abruptly autofocusing vortex beams

Jeffrey A. Davis, Don M. Cottrell, and David Sand  »View Author Affiliations


Optics Express, Vol. 20, Issue 12, pp. 13302-13310 (2012)
http://dx.doi.org/10.1364/OE.20.013302


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Abstract

We generate abruptly autofocusing beams that produce vortices at the focus. We give explicit equations for the phase-only Fourier masks that generate these beams including explanations for controlling the focal distance and numerical aperture. We show experimental results for the focal distance, the vortex pattern and show that the diameter of the focused beam can be made smaller than the size of a comparable Airy beam from a lens. Finally we show how to move the focus spot in three dimensions by encoding additional optical elements onto the phase pattern.

© 2012 OSA

1. Introduction

Nondiffracting Airy and parabolic beams [1

1. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007). [CrossRef] [PubMed]

,2

2. M. A. Bandres, “Accelerating parabolic beams,” Opt. Lett. 33(15), 1678–1680 (2008). [CrossRef] [PubMed]

] demonstrate an invariant shape whose transverse position moves quadratically with distance over substantial ranges. They are most usually generated experimentally [3

3. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007). [CrossRef] [PubMed]

6

6. J. A. Davis, M. J. Mitry, M. A. Bandres, I. Ruiz, K. P. McAuley, and D. M. Cottrell, “Generation of accelerating Airy and accelerating parabolic beams using phase-only patterns,” Appl. Opt. 48(17), 3170–3176 (2009). [CrossRef] [PubMed]

] using Fourier masks with amplitude and phase or with phase-only information. There have been attempts to implant vortices onto these beams with limited success [7

7. H. T. Dai, Y. J. Liu, D. Luo, and X. W. Sun, “Propagation dynamics of an optical vortex imposed on an Airy beam,” Opt. Lett. 35(23), 4075–4077 (2010). [CrossRef] [PubMed]

,8

8. H. T. Dai, Y. J. Liu, D. Luo, and X. W. Sun, “Propagation dynamics of an optical vortex imposed on an Airy beam; experimental implementation,” Opt. Lett. 36(9), 1617–1619 (2011). [CrossRef] [PubMed]

].

By using radial symmetry, abruptly autofocusing beams have been predicted [9

9. N. K. Efremidis and D. N. Christodoulides, “Abruptly autofocusing waves,” Opt. Lett. 35(23), 4045–4047 (2010). [CrossRef] [PubMed]

] and experimentally generated [10

10. D. G. Papazoglou, N. K. Efremidis, D. N. Christodoulides, and S. Tzortzakis, “Observation of abruptly autofocusing waves,” Opt. Lett. 36(10), 1842–1844 (2011). [CrossRef] [PubMed]

12

12. I. Chremmos, P. Zhang, J. Prakash, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, “Fourier-space generation of abruptly autofocusing beams and optical bottle beams,” Opt. Lett. 36(18), 3675–3677 (2011). [CrossRef] [PubMed]

] that form a tightly focused beam. It is expected that these radial beams will have numerous applications because they can deliver high intensity with small spot size at substantial distances from the optical generating system. However the design parameters that define the focus distance and numerical aperture for these beams have not been discussed. In addition, the capability to encode vortices onto these beams would substantially increase the number of applications.

The capability for using abruptly autofocusing beams to generate high intensity vortex beams will be useful for optical trapping and machining applications. The ability to further adjust both the axial and transverse position of the focused spot will also be extremely useful.

2. Theory and Fourier mask design

The resulting Airy beam travels with a trajectory given [2

2. M. A. Bandres, “Accelerating parabolic beams,” Opt. Lett. 33(15), 1678–1680 (2008). [CrossRef] [PubMed]

] by x2=[1/(4k2κ3)]z2 where x2 is the transverse displacement in the region past the Fourier plane, and is the propagation distance. We can derive a more useful expression where we set β=κ3kx3/3 and obtain x2=[πW3/48λf3β]z2=αz2, and is the width of the LCD having pixels of size .

We can add a linear phase term having a linear period of d=nΔ pixels as ϕLIN=kx/d=kx/nΔ to the cubic phase term. Consequently the trajectory of the Airy beam is represented by x2=x02αz2 where x02=fλ/nΔ and α is defined above.

In this work, we simply formed a rotationally symmetric version of this mask following the same prescription as for the linear case. We begin with a radial cubic phase term as ϕAIRY=βr13. Here the radial mask is defined as 0<r1<1 and is normalized to half of the size of the LCD. This phase pattern is added to a radial phase term ϕLIN=2πr1/d=2πr1/nΔ (note that this pattern is actually a diffractive axicon). Consequently the total radial phase is written as ϕ=βr13+2πr1/nΔ. Finally, because this is a radially symmetric phase mask, we can add an azimuthal phase that is commonly used for creating vortex patterns as ϕVORTEX=2πϕ where represents the charge of the vortex and can be positive or negative and can have fractional values. Consequently the total phase that is encoded onto the LCD is given as

ϕ=βr13+2πr1/nΔ+2πϕ
(1)

It has been noted by a reviewer that this phase mask (without the vortex pattern) represents the phase of the argument of the Bessel function that is the Fourier transform of the Airy ring [12

12. I. Chremmos, P. Zhang, J. Prakash, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, “Fourier-space generation of abruptly autofocusing beams and optical bottle beams,” Opt. Lett. 36(18), 3675–3677 (2011). [CrossRef] [PubMed]

]. It is interesting that the phase-only version of this Bessel function mask is so effective.

Figure 1
Fig. 1 Steps in building the mask for a radially autofocusing vortex beam. (a) cubic phase for β = 50, (b) linear radial phase for period of n = 20 pixels, (c) sum of the two, (d) sum with azimuthal phase with charge of l = 10.
shows the details of this mask design. Figure 1(a) shows the 512x512 radial cubic phase term as ϕAIRY=βr13 formed with a rather small value of β=50 in order to clearly see the pattern. This pattern is multiplied by a circular aperture to remove the rectangular symmetry for the mask. Figure 1(b) shows a radial axicon phase term ϕLIN=2πr1/d=2πr1/nΔ – again with a rather large period of n=20 pixels. Figure 1(c) shows the sum of these two phase functions. Finally Fig. 1(d) shows the sum of an azimuthal phase having a charge of =10 with the sum from Fig. 1(c).

The Fourier transform of this phase term is formed using the optical system as before. In this case, the radial trajectory of the beam in the region past the Fourier plane can be described by the equation

ρ=ρ0αz2=fλ/nΔ[πW3/48λf3β]z2
(2)

Here, ρ is measured after the Fourier plane, the parameter α has been previously defined and ρ0=fλ/nΔ. We note that the initial radius ρ0 increases as the number of pixels in the period of the radial axicon pattern decreases. We also see that the acceleration increases as the parameter β decreases.

Using Eq. (2), we can define the distance zf from the output Fourier plane to the focus point where ρ=0 as

zf=ρ0/α=48f4λ2βπΔW3n=48f4λ2βπN3Δ4n
(3)

Equation (3) shows that the focal length of the Fourier transform lens, the parameter β, the size W=NΔ of the LCD and the pixel size drastically affect this focus distance.

We can also define an approximate numerical aperture for the beam at its focus by evaluating dρ/dz at the point z=zf. This gives an expression for the numerical aperture as

N.A.=2ρ0α=πW312Δf2nβ
(4)

The numerical aperture for these abruptly focusing beams (given in Eq. (4) can be compared with that for a conventional lens. In making this comparison, the focal length f1 of the lens is chosen to be equal to the distance zf for the abruptly focusing lens. Consequently, the ratio of the numerical apertures is given by

N.A.RadialN.A.LENS=2ρ0αW/2f1=2ρ0αW/2zf=2ρ0αW/(2ρ0/α)=4ρ0W=4fλWnΔ
(5)

This surprisingly simple result shows that the focus size of the abruptly focusing beam will decrease relative to that of a conventional lens as the focal length of the Fourier transform lens increases while the pixel size of the SLM and the period of the axicon decrease.

3. Experimental setup

In our experiments, linearly polarized light from an Argon laser is spatially filtered, expanded, and collimated. The optical elements are encoded onto a parallel-aligned LCD manufactured by Seiko Epson with 640x480 pixels and pixel spacing of Δ=42μm [13

13. J. A. Davis, P. Tsai, D. M. Cottrell, T. Sonehara, and J. Amako, “Transmission variations in liquid crystal spatial light modulators caused by interference and diffraction effects,” Opt. Eng. 38(6), 1051–1057 (1999). [CrossRef]

,14

14. J. Wang, J. Bu, M. Wang, Y. Yang, and X. Yuan, “Generation of high quality Airy beams with blazed micron-optical cubic phase masks,” Appl. Opt. 50, 6626–6631 (2011).

] where the phase of each pixel can be controlled over a range of2π radians. The focal length of the Fourier transform lens was varied during the course of the experiments. The output images were recorded using a Wincam D camera witha pixel size of 6.7μm. The high resolution images were obtained using a microscope objective mounted in front of the camera.

4. Experimental results - focusing

We first control the focusing dynamics for these beams. Figure 2
Fig. 2 Images taken at distances from the focal plane of (columns) z=0,10,20,30,40cm for (top row) n=12,β=40rad, (middle row) n=6,β=40rad, (bottom row) n=6,β=95rad.
shows experimental results at various distances of z=0,10,20,30,40cm from the focal plane of the Fourier transform system where we vary the period of the radial pattern and the strength of the acceleration. In this case, the focal length of the Fourier transform lens was f=100cm and W=2cm corresponding to 480 pixels of the LCD (the maximum radius is 240 pixels). Each image shows a 512x512 array of detector pixels with no magnification. The top row shows results for a radial period of n=12 pixels and β=40rad. A circular pattern is formed at the focal plane of the lens with a diameter basically determined by the value of n and the beam forms a sharp focus at a distance of zf=20cm. The middle row shows results for a radial period of n=6 pixels and β=40rad. As expected, the initial diameter is larger and the beam now forms a sharp focus at a distance of zf=30cm. Finally the lower row shows results for a radial period again of n=6 pixels, but with a much larger value of β=95rad. Now the beam forms a sharp focus at a longer distance of zf=40cm. In each case, the image at the focal plane is not well defined and shows the various rays that are focused at different distances from the focal plane. All of these results are in excellent agreement with theory and show the versatility of the encoding procedure.

However, the success of this approach depends the capability of the SLM to encode the desired phase pattern and depends on several factors. First, we have a Nyquist limit of n=2 pixels/period for the axicon. For lower values of β and smaller values for the axicon period, the axicon pattern dominates. On the other hand, higher values of β can cause additional Nyquist encoding errors. The Nyquist limit for the cubic phase can be estimated by examining the case where the phase of the Nth pixel differs from the phase of the N1 pixel by π radians. Based on this analysis, we find that the Nyquist limit on the cubic phase is given by ϕN=πN/3 where N is the number of pixels for the maximum radius of the pattern. In our case where N=240 pixels, the maximum Nyquist phase is ϕN=250rad.

Some of these effects are shown in Fig. 3
Fig. 3 Effective focal length as a function of 1/n for values ofβ=50rad (squares), β=100rad (diamonds) and β=200rad (circles).
where we plot the effective focal distance zf for values of n=2.28,5.10,15,20,25,30,35,40 (from right to left) and for different values of β as a function of1/n. The circles and solid line show results for β=200rad and the agreement between theory and experiment is excellent. The diamonds and the dashed line show results for β=100rad. Here the agreement is good until the lower values for the axicon period. Finally the squares and lower dashed line show results for β=50rad. Nowthe experimental results deviate significantly from theory at lower values for the axicon period. We see these same results using a smaller focal length of f=50cm for the Fourier transform lens. As mentioned above, we attribute these results to the fact that the axicon can dominate the cubic phase term at lower values for the cubic phase and smaller axicon periods. Some of these problems could be alleviated by encoding these cubic phase masks using submicron etching techniques (14).

In addition, several experimental factors are important. First, the Fourier transform optical system must be correctly aligned where input and output planes are separated from the Fourier lens by the distance . Otherwise the experimental results in Fig. 3 will not agree with theory. For example, if the first distance is longer than , then the slope in Fig. 3 will be higher while, if the first distance is less than , then the slope will be lower. These effects have been experimentally confirmed and similar effects have been reported elsewhere [6

6. J. A. Davis, M. J. Mitry, M. A. Bandres, I. Ruiz, K. P. McAuley, and D. M. Cottrell, “Generation of accelerating Airy and accelerating parabolic beams using phase-only patterns,” Appl. Opt. 48(17), 3170–3176 (2009). [CrossRef] [PubMed]

].

Secondly, the diameter of the Fourier transform lens should be large enough so that low pass spatial filtering does not occur. This is particularly important when using large values for the Fourier lens focal length and for small values of the axicon parameter n and the pixel size Δ in Eq. (3).

5. Experimental results – vortex focus points

Next, we examined the image in the plane where z=zf when we encoded various circular harmonics onto the mask as shown in Fig. 1(d). In this case, we used a radial period of n=2.28 pixels, a value of β=195rad, a Fourier transform lens with a focal length of f=100cm and an aperture of W=2cm. These values predicted a focal distance of zf=100cm in agreement with experiment. Results from left to right in Fig. 4
Fig. 4 Images taken at z=zf=100cm for circular harmonic values of =0,10,20,30 using n=2.28,β=195rad.
show circular harmonic values of =0,10,20,30. In each case, the image represents an area of 172x172pixels of the detector with no magnification. As expected the diameter of the vortex spot increases as the value of the circular harmonic increases. At distances before this focal plane, the beams look similar to those shown in Fig. 2.

6. Numerical aperture

Obvious improvements would be to use a LCD with a smaller pixel size or to use submicron-etching techniques for the cubic phase masks (14). It would be interesting to see how small the focused spot size can be made.

7. Manipulation of the focus position

The normal ray matrix (15) for a Fourier transform system is given by the following

(0f1/f0)
(6)

Note that the "A" matrix element has a value of zero corresponding to the Fourier transform condition. Now, we consider the matrix system corresponding to the case where a lens having a focal length of F is placed in front of the Fourier transform system and followed by a translation matrix.

(1z01)(0f1/f0)(101/F1)=(z/ff/Ff1/f0)
(7)

The Fourier transform condition that A=0 is satisfied when z=f2/F.

Figure 6
Fig. 6 Movement of the axial position of the focused spot by encoding an additional lens function having a focal length of F onto the cubic phase mask.
shows experimental results where an additional lens is encoded onto the cubic phase pattern. Note that small axial shift distances require large focal lengths encoded onto the cubic phase and this is no problem for the LCD [15

15. J. A. Davis and D. M. Cottrell, “Ray matrix analysis of the fast Fresnel transform with applications towards liquid crystal displays,” Appl. Opt. 51(5), 644–650 (2012). [CrossRef] [PubMed]

]. Here we again use f=100cm for the Fourier lens and encode lenses having focal lengths of F=±100m,±33.3m,±20m onto the patterns on the LCD. The axial movement of the focal plane is in excellent agreement with the theory.

Finally, the position of the focused spot can be moved in the lateral direction by changing the center location of the added lens. Figure 7
Fig. 7 Movement of the transverse position of the focused spot by moving the center of the additional lens function by+5pixels,0,5pixels. The focused spot moves by +220μm,0,220μm.
shows experimental results where the encoded lens is offset by 5,0,+5 pixels. Again we use parameters of n=2.28, and β=195rad with f=100cm and F=20m. We show a detector size of 128x128 pixels with no magnification. The lateral position of the focused spot changes by ±220μm.

These results show that we can dynamically control the position of the focused spot in three dimensions.

8. Conclusions

We show explicit equations that show the relationship between a cubic phase mask and the focusing properties of the abruptly focusing Airy beam. We show that vortex patterns can be encoded onto these beams to produce focus spots having angular momentum. We show that the focused spot size can be made smaller than the Airy disc spot size for a conventional lens. Finally, we can adjust the position of the focused spot in 3 dimensions by adding an additional lens function onto the cubic phase pattern. We expect that the best performance can be obtained by a submicron etched cubic phase mask combined with a LCD for control of the vortex charge and for 3-D control of the location of the focused spot.

Acknowledgments

We thank Tomio Sonehara of Seiko Epson Corporation for the use of the LCD.

References and links

1.

G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007). [CrossRef] [PubMed]

2.

M. A. Bandres, “Accelerating parabolic beams,” Opt. Lett. 33(15), 1678–1680 (2008). [CrossRef] [PubMed]

3.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007). [CrossRef] [PubMed]

4.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett. 33(3), 207–209 (2008). [CrossRef] [PubMed]

5.

J. A. Davis, M. J. Mintry, M. A. Bandres, and D. M. Cottrell, “Observation of accelerating parabolic beams,” Opt. Express 16(17), 12866–12871 (2008). [CrossRef] [PubMed]

6.

J. A. Davis, M. J. Mitry, M. A. Bandres, I. Ruiz, K. P. McAuley, and D. M. Cottrell, “Generation of accelerating Airy and accelerating parabolic beams using phase-only patterns,” Appl. Opt. 48(17), 3170–3176 (2009). [CrossRef] [PubMed]

7.

H. T. Dai, Y. J. Liu, D. Luo, and X. W. Sun, “Propagation dynamics of an optical vortex imposed on an Airy beam,” Opt. Lett. 35(23), 4075–4077 (2010). [CrossRef] [PubMed]

8.

H. T. Dai, Y. J. Liu, D. Luo, and X. W. Sun, “Propagation dynamics of an optical vortex imposed on an Airy beam; experimental implementation,” Opt. Lett. 36(9), 1617–1619 (2011). [CrossRef] [PubMed]

9.

N. K. Efremidis and D. N. Christodoulides, “Abruptly autofocusing waves,” Opt. Lett. 35(23), 4045–4047 (2010). [CrossRef] [PubMed]

10.

D. G. Papazoglou, N. K. Efremidis, D. N. Christodoulides, and S. Tzortzakis, “Observation of abruptly autofocusing waves,” Opt. Lett. 36(10), 1842–1844 (2011). [CrossRef] [PubMed]

11.

I. Chremmos, N. K. Efremidis, and D. N. Christodoulides, “Pre-engineered abruptly autofocusing beams,” Opt. Lett. 36(10), 1890–1892 (2011). [CrossRef] [PubMed]

12.

I. Chremmos, P. Zhang, J. Prakash, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, “Fourier-space generation of abruptly autofocusing beams and optical bottle beams,” Opt. Lett. 36(18), 3675–3677 (2011). [CrossRef] [PubMed]

13.

J. A. Davis, P. Tsai, D. M. Cottrell, T. Sonehara, and J. Amako, “Transmission variations in liquid crystal spatial light modulators caused by interference and diffraction effects,” Opt. Eng. 38(6), 1051–1057 (1999). [CrossRef]

14.

J. Wang, J. Bu, M. Wang, Y. Yang, and X. Yuan, “Generation of high quality Airy beams with blazed micron-optical cubic phase masks,” Appl. Opt. 50, 6626–6631 (2011).

15.

J. A. Davis and D. M. Cottrell, “Ray matrix analysis of the fast Fresnel transform with applications towards liquid crystal displays,” Appl. Opt. 51(5), 644–650 (2012). [CrossRef] [PubMed]

OCIS Codes
(050.1940) Diffraction and gratings : Diffraction
(350.5500) Other areas of optics : Propagation
(050.4865) Diffraction and gratings : Optical vortices

ToC Category:
Physical Optics

History
Original Manuscript: April 12, 2012
Revised Manuscript: May 17, 2012
Manuscript Accepted: May 20, 2012
Published: May 29, 2012

Citation
Jeffrey A. Davis, Don M. Cottrell, and David Sand, "Abruptly autofocusing vortex beams," Opt. Express 20, 13302-13310 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-12-13302


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References

  1. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett.32(8), 979–981 (2007). [CrossRef] [PubMed]
  2. M. A. Bandres, “Accelerating parabolic beams,” Opt. Lett.33(15), 1678–1680 (2008). [CrossRef] [PubMed]
  3. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett.99(21), 213901 (2007). [CrossRef] [PubMed]
  4. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett.33(3), 207–209 (2008). [CrossRef] [PubMed]
  5. J. A. Davis, M. J. Mintry, M. A. Bandres, and D. M. Cottrell, “Observation of accelerating parabolic beams,” Opt. Express16(17), 12866–12871 (2008). [CrossRef] [PubMed]
  6. J. A. Davis, M. J. Mitry, M. A. Bandres, I. Ruiz, K. P. McAuley, and D. M. Cottrell, “Generation of accelerating Airy and accelerating parabolic beams using phase-only patterns,” Appl. Opt.48(17), 3170–3176 (2009). [CrossRef] [PubMed]
  7. H. T. Dai, Y. J. Liu, D. Luo, and X. W. Sun, “Propagation dynamics of an optical vortex imposed on an Airy beam,” Opt. Lett.35(23), 4075–4077 (2010). [CrossRef] [PubMed]
  8. H. T. Dai, Y. J. Liu, D. Luo, and X. W. Sun, “Propagation dynamics of an optical vortex imposed on an Airy beam; experimental implementation,” Opt. Lett.36(9), 1617–1619 (2011). [CrossRef] [PubMed]
  9. N. K. Efremidis and D. N. Christodoulides, “Abruptly autofocusing waves,” Opt. Lett.35(23), 4045–4047 (2010). [CrossRef] [PubMed]
  10. D. G. Papazoglou, N. K. Efremidis, D. N. Christodoulides, and S. Tzortzakis, “Observation of abruptly autofocusing waves,” Opt. Lett.36(10), 1842–1844 (2011). [CrossRef] [PubMed]
  11. I. Chremmos, N. K. Efremidis, and D. N. Christodoulides, “Pre-engineered abruptly autofocusing beams,” Opt. Lett.36(10), 1890–1892 (2011). [CrossRef] [PubMed]
  12. I. Chremmos, P. Zhang, J. Prakash, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, “Fourier-space generation of abruptly autofocusing beams and optical bottle beams,” Opt. Lett.36(18), 3675–3677 (2011). [CrossRef] [PubMed]
  13. J. A. Davis, P. Tsai, D. M. Cottrell, T. Sonehara, and J. Amako, “Transmission variations in liquid crystal spatial light modulators caused by interference and diffraction effects,” Opt. Eng.38(6), 1051–1057 (1999). [CrossRef]
  14. J. Wang, J. Bu, M. Wang, Y. Yang, and X. Yuan, “Generation of high quality Airy beams with blazed micron-optical cubic phase masks,” Appl. Opt.50, 6626–6631 (2011).
  15. J. A. Davis and D. M. Cottrell, “Ray matrix analysis of the fast Fresnel transform with applications towards liquid crystal displays,” Appl. Opt.51(5), 644–650 (2012). [CrossRef] [PubMed]

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