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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 19 — Sep. 14, 2009
  • pp: 16415–16422
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Enhanced dispersion compensation capability of angular elements based on beam expansion

Rui Du, Runhua Jiang, and Ling Fu  »View Author Affiliations


Optics Express, Vol. 17, Issue 19, pp. 16415-16422 (2009)
http://dx.doi.org/10.1364/OE.17.016415


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Abstract

We demonstrate that beam size manipulation plays an important role in dispersion compensation. With expanded beam, the maximal negative group delay dispersion (GDD) provided by angular elements increases by an order of magnitude compared with original beam. Both calculation and experimental results show that a modest 2 × and 4 × expanded beams can improve dispersion compensation capability of prisms or acousto-optical deflectors: the restored minimal pulse width decreases by 50% and the corresponding distance between angular elements is shortened more than 70 cm. These findings will be helpful for designing dispersion compensation schemes for femtosecond pulse laser application systems such as multiphoton microscopy or laser micromachining.

© 2009 OSA

1. Introduction

Group delay dispersion (GDD) compensation plays an important role in ultrashort pulse generation [1

1. D. H. Sutter, G. Steinmeyer, L. Gallmann, N. Matuschek, F. Morier-Genoud, U. Keller, V. Scheuer, G. Angelow, and T. Tschudi, “Semiconductor saturable-absorber mirror assisted Kerr-lens mode-locked Ti:sapphire laser producing pulses in the two-cycle regime,” Opt. Lett. 24(9), 631–633 (1999). [CrossRef]

3

3. J. C. Diels, and W. Rudolph, Ultrashort Laser Pulse Phenomena (Academic, San Diego, Calif., 1996).

], chirped pulse amplification [4

4. M. Nisoli, S. De Silvestri, and O. Svelto, “Generation of high energy 10 fs pulses by a new pulse compression technique,” Appl. Phys. Lett. 68(20), 2793–2795 (1996). [CrossRef]

,5

5. I. Walmsley, L. Waxer, and C. Dorrer, “The role of dispersion in ultrafast optics,” Rev. Sci. Instrum. 72(1), 1–29 (2001). [CrossRef]

], laser micromaching [6

6. B. K. A. Ngoi, K. Venkatakrishnan, L. E. Lim, B. Tan, and L. E. N. Lim, “Angular dispersion compensation for acousto-optic devices used for ultrashort-pulsed laser micromachining,” Opt. Express 9(4), 200–206 (2001). [CrossRef] [PubMed]

], multiphoton imaging [7

7. W. Denk, J. H. Strickler, and W. W. Webb, “Two-photon laser scanning fluorescence microscopy,” Science 248(4951), 73–76 (1990). [CrossRef] [PubMed]

15

15. X. Lv, C. Zhan, S. Zeng, W. R. Chen, and Q. Luo, “Construction of multiphoton laser scanning microscope based on dual-axis acousto-optic deflector,” Rev. Sci. Instrum. 77(4), 046101 (2006). [CrossRef]

], etc. The conventional dispersion compensation methods employs four prisms [16

16. R. L. Fork, O. E. Martinez, and J. P. Gordon, “Negative dispersion using pairs of prisms,” Opt. Lett. 9(5), 150–152 (1984). [CrossRef] [PubMed]

] or four gratings [17

17. E. B. Treacy, “Optical pulse compression with diffraction gratings,” IEEE J. Quantum Electron. QE-5(9), 454–458 (1969). [CrossRef]

] to provide negative GDD to compensate for positive GDD induced by the propagation of ultrashort pulses through dispersive material, such as fibers and acousto-optical deflectors (AODs) [18

18. V. Iyer, B. E. Losavio, and P. Saggau, “Compensation of spatial and temporal dispersion for acousto-optic multiphoton laser-scanning microscopy,” J. Biomed. Opt. 8(3), 460–471 (2003). [CrossRef] [PubMed]

]. In some systems [19

19. M. Nakazawa, T. Nakashima, and H. Kubota, “Optical pulse compression using a TeO2 acousto-optical light deflector,” Opt. Lett. 13(2), 120–122 (1988). [CrossRef] [PubMed]

,20

20. Y. Kremer, J. F. Léger, R. Lapole, N. Honnorat, Y. Candela, S. Dieudonné, and L. Bourdieu, “A spatio-temporally compensated acousto-optic scanner for two-photon microscopy providing large field of view,” Opt. Express 16(14), 10066–10076 (2008). [CrossRef] [PubMed]

], only two angular elements are used to compensate for positive GDD. This configuration has the advantage of higher transmission efficiency and stability compared with the four angular elements configuration, because there is inevitably energy loss when a laser passes through the angular elements. However, GDD provided by prism pair is not always increased linearly with dispersion compensation distance and will be frozen to its maximal value [21

21. S. Zeng, D. Li, X. Lv, J. Liu, and Q. Luo, “Pulse broadening of the femtosecond pulses in a Gaussian beam passing an angular disperser,” Opt. Lett. 32(9), 1180–1182 (2007). [CrossRef] [PubMed]

,22

22. D. Li, S. Zeng, X. Lv, J. Liu, R. Du, R. Jiang, W. R. Chen, and Q. Luo, “Dispersion characteristics of acousto-optic deflector for scanning Gaussian laser beam of femtosecond pulses,” Opt. Express 15(8), 4726–4734 (2007). [CrossRef] [PubMed]

]. Thus, in the case that the absolute value of maximal GDD provided by prism pair is smaller than that of positive GDD, one cannot get enough negative GDD to compensate for the positive GDD. Considering the advantage of higher transmission efficiency and stability with the prism pair configuration, it is preferable to enhance the GDD of prism pair.

As is known to all, the amount of negative GDD can be controlled by adjusting the angular elements’ angular dispersion or changing the dispersion compensation distance. Actually, the beam size may also have effect on the GDD. Martinez has discussed the effect of beam size on the spectral lateral walk-off and found that larger beam size can weaken the spectral lateral walk-off and thus improve the compression ratio (see point 4 in his conclusion) [23

23. O. E. Martinez, “Grating and prism compressors in the case of finite beam size,” J. Opt. Soc. Am. B 3(7), 929–934 (1986). [CrossRef]

]. However, the effect of beam size on GDD has not been studied due to the well-collimation assumption applied in Martinez’s work [23

23. O. E. Martinez, “Grating and prism compressors in the case of finite beam size,” J. Opt. Soc. Am. B 3(7), 929–934 (1986). [CrossRef]

].

In this paper, we have explored the beam size’s effect on the GDD and found that a modest 2 × and 4 × beam expansion can improve the dispersion compensation capability of angular elements: decreasing the available minimal pulse width and shortening the dispersion compensation distance. This will make the pulse compressor more compact, stable and of higher transmission efficiency, which are particularly important in multiphoton microscopic imaging and micromachining, where high photon densities and high stability is required.

2. Theoretical analysis

We first take prism pair as an example to describe the effects of beam sizes on the GDD compensation. As shown in Fig. 1
Fig. 1 Schematic diagram of the prism pair with the configuration of the expanded beam for GDD compensation. The lenses F1 and F2, having the focal lengths of f 1 and f 2 respectively, constitute a beam expander with the magnification coefficient M = f 2 / f 1. L is the distance between BW1 and F1 .d is the distance between BW2 and the first prism. z is the inter-prism distance.
, the original Gaussian beam is expanded by the beam expander consisting of lenses F1 and F2. BW1 and BW2 are the beam waists of the original beam and the expanded beam, respectively. The position of BW2 is in the vicinity of the back focal plane of the lens F2 and the diameter of BW2 is:
w2=Mw11+(Lλπw12)2,
(1)
where w 1 is the diameter of BW1 and λ is the wavelength.

Based on Eqs. (1) and (2), the GDD provided by the prism pair as a function of z in case of different magnification coefficients can be analyzed. For simplicity, the ratio of the GDDa after the beam expansion to the GDDa with the original beam size is used to analyze how the beam size affects the negative GDD caused by angular elements. According actual situation of our experiments, the following parameters are used in the calculation: w 1 = 0.5 mm, λ = 800 nm; The incident angle upon the prism is 49°, the top angle of the prism is 60°, the refraction index of SF10 prism n = 1.7112 and dn /dλ = 4.98 × 10−5 nm−1; In the case of expanded beams, L = 1.745 m and d = 2.1 m; The distance between the beam waist and the prism is 2.1 m while using the original beam (The lenses F1 and F2 are moved out.).

As shown in Fig. 2
Fig. 2 Enhancement of GDD as a function of z with different beam sizes. Enhancement of GDD is defined as the ratio of the GDDa after the beam expansion to the GDDa with the original beam size. The inset shows the actual GDDa with different beam sizes.
and its inset, -GDDa for 2 × and 4 × expanded beams are enhanced in comparison with the case of the original beam, implying that angular elements can provide more negative GDD at the same z after beam expansions. This is due to the fact that -GDDa is multiplied by two parts, in which one is 2 z and the other is η=(d+α2z)d+(πw22/λ)2(d+α2z)2+(πw22/λ)2. At the same z, the value of η increases as the beam size becomes larger. For example, at z = 2 m, the values of η are 0.22, 0.77 and 0.98 for original, 2 × and 4 × expanded beams, respectively, while the values of 2 z are the same for different beam sizes. As a result, the larger beam size leads to the increase of the -GDDa amount and the enhanced compensation capability of angular elements. It is noted that under the plane wave approximation, by setting w 2 → ∞ in Eq. (2), the negative GDD caused by angular elements reduces to - 2 z, which is consistent with the demonstrated Eq. (11) in Ref. 16

16. R. L. Fork, O. E. Martinez, and J. P. Gordon, “Negative dispersion using pairs of prisms,” Opt. Lett. 9(5), 150–152 (1984). [CrossRef] [PubMed]

.

It is also observed from Fig. 2 that the ratio of -GDDa for 2 × and 4 × expanded beams monotonically increased with z before they reach their maximal values, which are 4 and 16 times higher compared that with the original beam, respectively. In terms of quantity, the maximal GDD a provided by prism pair with original, 2 × and 4 × expanded beam can be calculated with Eqs. (1) and (2), which is −10000 fs2, −40000 fs2 and −160000 fs2, respectively. To illustrate this, analysis of Eq. (2) is necessary. The part - 2 z, which describes the negative GDD caused by angular elements under the plane wave approximation [16

16. R. L. Fork, O. E. Martinez, and J. P. Gordon, “Negative dispersion using pairs of prisms,” Opt. Lett. 9(5), 150–152 (1984). [CrossRef] [PubMed]

,17

17. E. B. Treacy, “Optical pulse compression with diffraction gratings,” IEEE J. Quantum Electron. QE-5(9), 454–458 (1969). [CrossRef]

], increases linearly with z, while η monotonically decreases as a function of z from its maximal value 1 at z = 0. Therefore, the multiplication of the increasing function 2 z and decreasing function η constitutes the formula that has a peak value. It is noted that the decrease of η with z turns slower for the larger beam size, resulting in the higher maximal value of -GDDa compared with the lower magnification coefficient. As shown in the inset of Fig. 2, the maximal negative GDD provided by the prism is approximately −10000 fs2 and the increasing of inter-prism distance cannot provide more negative GDD. This is due to the fact that η tends to zero as z increases [21

21. S. Zeng, D. Li, X. Lv, J. Liu, and Q. Luo, “Pulse broadening of the femtosecond pulses in a Gaussian beam passing an angular disperser,” Opt. Lett. 32(9), 1180–1182 (2007). [CrossRef] [PubMed]

,22

22. D. Li, S. Zeng, X. Lv, J. Liu, R. Du, R. Jiang, W. R. Chen, and Q. Luo, “Dispersion characteristics of acousto-optic deflector for scanning Gaussian laser beam of femtosecond pulses,” Opt. Express 15(8), 4726–4734 (2007). [CrossRef] [PubMed]

].

3. Experimental results

To verify the effectiveness of beam expansion on dispersion compensation capability described by Eq. (2), pulse widths are measured for different beam sizes when a prism pair with various dispersion compensation distances is used to compensate for the positive GDD of the Tellurium oxide (TeO2) material with total thickness 36 mm. The experimental setup is the same as Fig. 1 except that the TeO2 material is inserted between BW1 and F1. The femtosecond laser generated from the Ti: Sapphire laser (Mai Tai, Spectra Physics) at wavelength λ = 800 nm has a pulse width of τ 0 = 82 fs, radius of original beam waist of w 1 = 0.5 mm, and a repetition rate of 80 MHz. The TeO2 material introduces the total positive GDD of approximately 16000 fs2. The beam is expanded by the combination of two lenses. For the 2 × beam expansion, the expander is composed of an f 1 = 38.1 mm and an f 2 = 75 mm lenses; An f 1 = 38.1 mm and an f 2 = 150 mm lenses are used for the 4 × expansion. The material dispersion of the beam expander is about 1500 fs2. When using expanded beams, L is 1.745 m and d is 0.22 m. The distance between BW1 and the prism is 2.1 m for the original beam. The expanded beam passed through the prism pair with an incident angle of 49°. As mentioned in above paragraph, the refraction index of SF10 prism n = 1.7112 and dn /dλ = 4.98 × 10−5 nm−1. The pulse widths of the emerged beams after the prisms are measured with the autocorrelator based on two-photon absorption (Carpe, APE, Germany).

Under the above experimental conditions, measured and calculated pulse widths as a function of path length between the prism pair for different beam sizes have been depicted in Fig. 3
Fig. 3 Measured and calculated pulse widths as a function of z when to compensate for the positive GDD introduced by the Tellurium oxide (TeO2) with different beam sizes.
. The theoretical curves of pulse width evolution are achieved by substituting GDD of Eq. (2) into Eq. (9) in Ref. 22

22. D. Li, S. Zeng, X. Lv, J. Liu, R. Du, R. Jiang, W. R. Chen, and Q. Luo, “Dispersion characteristics of acousto-optic deflector for scanning Gaussian laser beam of femtosecond pulses,” Opt. Express 15(8), 4726–4734 (2007). [CrossRef] [PubMed]

. As shown in Fig. 3, the experimental results have good coincidence with the theory data. In the case of the original beam without expanding, it is found that the minimal pulse obtained is 203 fs, with the inter-prism distance of 1.5 m, and the increase of the inter-prism distance cannot lead to the shorter pulse width. This is due to the fact that the maximal available negative GDD for the original beam size is approximately −10000 fs2 (see the blue line in the inset of Fig. 2), which is insufficient to completely compensate the TeO2-induced positive GDD of 16000 fs2. It is also noted that the negative GDD cannot be increased by increasing the inter-prism distance due the fact that that the η turns to zero as z increase [21

21. S. Zeng, D. Li, X. Lv, J. Liu, and Q. Luo, “Pulse broadening of the femtosecond pulses in a Gaussian beam passing an angular disperser,” Opt. Lett. 32(9), 1180–1182 (2007). [CrossRef] [PubMed]

,22

22. D. Li, S. Zeng, X. Lv, J. Liu, R. Du, R. Jiang, W. R. Chen, and Q. Luo, “Dispersion characteristics of acousto-optic deflector for scanning Gaussian laser beam of femtosecond pulses,” Opt. Express 15(8), 4726–4734 (2007). [CrossRef] [PubMed]

]. In contrast, for the 2 × and 4 × expanded beams, the minimum pulse widths are 140 fs and 106 fs, with the shorter inter-prism distance of 0.7 m and 0.8 m, respectively. Additionally, as the inter-prism distance increases, the pulse widths are further broadened after reaching their minima. This fact suggests that the positive GDD induced by TeO2 can be completely compensated by using the expanded beams. Actually, the calculated GDD a is −16030 fs2 at 0.7m for the 2 × expanded beam and −18020 fs2 at 0.8 m for the 4 × expanded beam, resulting in the sufficient compensation for the positive GDD induced by TeO2. Since the positive GDD of the prism’s material in the 4 × case is bigger than that of the 2 × case, (larger beam size experiences longer path length in the prism) its corresponding inter-prism distance for the minimal pulse width is slightly longer. Therefore, it is demonstrated that beam expansion can improve the dispersion compensation capability of the prism pair: decreasing the achievable minimum pulse width and shortening the inter-prism distance to obtain the minimal pulse width.

This idea also works for other angular devices, and is demonstrated with an AOD pair, which is actually a dynamic grating pair [19

19. M. Nakazawa, T. Nakashima, and H. Kubota, “Optical pulse compression using a TeO2 acousto-optical light deflector,” Opt. Lett. 13(2), 120–122 (1988). [CrossRef] [PubMed]

,20

20. Y. Kremer, J. F. Léger, R. Lapole, N. Honnorat, Y. Candela, S. Dieudonné, and L. Bourdieu, “A spatio-temporally compensated acousto-optic scanner for two-photon microscopy providing large field of view,” Opt. Express 16(14), 10066–10076 (2008). [CrossRef] [PubMed]

]. The experimental condition is identical as above, except that the two prisms are substituted by an AOD pair activated at 96 MHz with counter-propagating acoustic waves. As shown in Fig. 4
Fig. 4 Measured and calculated pulse widths as a function of the distance between the acousto-optic deflector pair with different beam sizes.
, both measured and calculated pulse widths exhibit the similar behavior as that in Fig. 3. The minimal pulse widths for 2 × and 4 × expanded beams are 109 fs and 96 fs respectively, which are shorter than the minimal pulse width of 120 fs achieved by using the original beam. The inter-AOD distance for the expanded beams to reach their minimal pulse widths is 0.8 m, shorter than that for the original beam. As a result, the expanded beam can also improve the dispersion capability of gratings.

4. Discussion

We have shown that the beam expansion can enhance dispersion capability of angular elements. It is noted that, the pulse width of the expanded beam (140 fs and 106 fs) shown in Fig. 3 cannot be recovered to its original value (82 fs). It results from the spectral lateral walk-off of the emerged beam from the prisms [23

23. O. E. Martinez, “Grating and prism compressors in the case of finite beam size,” J. Opt. Soc. Am. B 3(7), 929–934 (1986). [CrossRef]

-25

25. X. Gu, S. Akturk, and R. Trebino, “Spatial chirp in ultrafast optics,” Opt. Commun. 242(4-6), 599–604 (2004). [CrossRef]

]. This is common in such systems using a prism pair, and can be fixed with four prisms configuration. Thus, the configuration of a prism pair becomes impractical in the case that the original transform-limited pulse width is needed. However, in various applications, such as in multiphoton imaging, the original transform-limited pulse width is not strictly demanded. Thus, the prism pair with the beam expansion can be applied in many practical systems.

Beam expander and dispersion compensation system are routinely used in multiphoton microscopy and micromachining systems. This paper helps to clarify the role of beam expander in the dispersion compensation system. The beam expander is not only to manipulate beam size to fit optical aperture, to enlarge the effective numerical aperture and obtain better spatial resolution, but also to affect dispersion capability of angular elements if placed before the two-element pulse compressor. Thus, the expander’s magnification coefficient and its position in optical path should be deliberately chosen according to practical experimental conditions.

Beam size affects both the spectral lateral walk-off and the GDD, which are two factors determine the compression ratio. However, in the case of using the prism pair compressor with expanded beam, which factor contributes more to the improvement of compression ratio is unknown. To state the question, the compression ratio in the four cases of t0 = t (U 4, GDDa 4), t1 = t (U 0, GDDa 4), t2 = t (U 4, GDDa 0), and t3 = t (U 0, GDDa 0) are depicted in Fig. 5
Fig. 5 Compression ratio as a function of inter-prism distance with different U and GDDa parameters.
(parameters is the same as in Fig. 2, material dispersion GDD m = 19000 fs2). t (U, GDDa) is the compression ratio as a function of the spectral lateral walk-off U and the GDD of angular elements GDDa (See the Eq. (9) in Ref. 22

22. D. Li, S. Zeng, X. Lv, J. Liu, R. Du, R. Jiang, W. R. Chen, and Q. Luo, “Dispersion characteristics of acousto-optic deflector for scanning Gaussian laser beam of femtosecond pulses,” Opt. Express 15(8), 4726–4734 (2007). [CrossRef] [PubMed]

). U 0 and GDDa 0 are the spectral lateral walk-off and GDD of prism pair with unexpanded beam, respectively. U 4 and GDDa 4 are the spectral lateral walk-off and GDD of prism pair with 4 × expanded beam, respectively. t0 and t3 is the compression ratio with the 4 × expanded beam and unexpanded beam, respectively.

To separate the effect of spectral lateral walk-off and negative GDD on compression ratio, t1 is obtained only by substituting GDDa 4 with GDDa 0 in t3, and t2 is acquired only by substituting U 4 with U 0 in t3. As shown in Fig. 5, t1 indicates the improvement of compression ratio, which is similar as t0, while t2 still suffers larger compression ratio. Please note that the significant improvement of compensation ratio can be achieved, as indicated by t1, only by the enhancement of negative GDD (-GDDa 4 is bigger than-GDDa 0 as shown in Fig. 2). As a result, the main cause for compression ratio improvement with expanded beam is the enhancement of negative GDD. The negligible difference between t0 and t1 is due to the spectral lateral walk-off. When the GDD is zero, due to the weakness of spectral lateral walk-off [23

23. O. E. Martinez, “Grating and prism compressors in the case of finite beam size,” J. Opt. Soc. Am. B 3(7), 929–934 (1986). [CrossRef]

], larger beam size can slightly decrease the minimal pulse width, as have been shown in both Fig. 3 and Fig. 4. Therefore, the weakness of spectral lateral walk-off is the minor cause for the compression ratio improvement.

5. Conclusion

The mechanism of how beam size affects the GDD and the dispersion compensation has been analyzed theoretically and experimentally. It is found that beam expansion could improve the dispersion compensation capability of the prism and AOD pair. These results suggest that the pair configuration pulse compressor with large beam size is able to make the compressor more compact and stable, and improve the power transmission efficiency by using less number of devices. These results are very useful for designing practical femtosecond pulse laser application systems such as multiphoton microscopy and micromachining systems.

Appendix

The calculation of α and β for prism.

The deflecting angle θ emerging from angular element for a femtosecond laser is depend on the incident angle γ and the spectral component ω . The difference between the angle of a spectral component and the angle of the central spectral component can be expressed as [23

23. O. E. Martinez, “Grating and prism compressors in the case of finite beam size,” J. Opt. Soc. Am. B 3(7), 929–934 (1986). [CrossRef]

]

Δθ=αΔγ+βΔω, (A1)

where α = dθ / dγ and β = dθ / dω.

sinθ=sinφn2sin2γcosφsinγ. (A2)

By taking the first derivative of Eq. (2) with respect to γ and ω, respectively, the coefficient α and β can be expressed as

α=(sinφsinγcosγn2sin2γ+cosφcosγ)/cosθ, (A3)

β=nsinφ1n2sin2φsin2γcos2φ+sin2φsinγn2sin2γn2sin2γλ22πcdndλ.

(A4)

In the case of Brewster angle incidence, i.e., n = tg (γ),α=1,β=λ2πcdndλ, which is consistent with equations (13) and (14) in Ref. 23

23. O. E. Martinez, “Grating and prism compressors in the case of finite beam size,” J. Opt. Soc. Am. B 3(7), 929–934 (1986). [CrossRef]

.

Acknowledgments

The authors thank the support by the National Natural Science Foundation of China (No.60708025, 60828009) and the National High-Tech Research and Development Program of China (No.2006AA020801).

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K. Varjú, A. P. Kovács, K. Osvay, and G. Kurdi, “Angular dispersion of femtosecond pulses in a Gaussian beam,” Opt. Lett. 27(22), 2034–2036 (2002). [CrossRef]

25.

X. Gu, S. Akturk, and R. Trebino, “Spatial chirp in ultrafast optics,” Opt. Commun. 242(4-6), 599–604 (2004). [CrossRef]

OCIS Codes
(230.1040) Optical devices : Acousto-optical devices
(320.5520) Ultrafast optics : Pulse compression
(320.7090) Ultrafast optics : Ultrafast lasers

ToC Category:
Ultrafast Optics

History
Original Manuscript: May 18, 2009
Revised Manuscript: July 8, 2009
Manuscript Accepted: July 10, 2009
Published: August 31, 2009

Citation
Rui Du, Runhua Jiang, and Ling Fu, "Enhanced dispersion compensation capability of angular elements based on beam expansion," Opt. Express 17, 16415-16422 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-19-16415


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