## Influence of cubic nonlinearity on accuracy of polarization transformation by means of a quarter-wave plate |

Optics Express, Vol. 21, Issue 1, pp. 135-144 (2013)

http://dx.doi.org/10.1364/OE.21.000135

Acrobat PDF (2077 KB)

### Abstract

We consider the propagation of powerful laser radiation in an anisotropic medium with natural birefringence and cubic nonlinearity. By the example of a quarter-wave plate, we show theoretically and experimentally that, under the simultaneous influence of linear birefringence and nonlinearity, the accuracy of polarization transformation decreases in proportion to squared *В*-integral.

© 2013 OSA

## 1. Introduction

^{20}W/cm

^{2}and higher open a way to the rapidly advancing trend in laser physics – creation of compact accelerators of high-energy particles, such as electrons, protons (ions) and neutrons. The principle of operation of these accelerators is based on the interaction of a powerful laser pulse with a thin solid target (e.g., aluminum or gold foil). During several femtoseconds the target is ionized so that free electrons may be accelerated under the action of ponderomotive force of the laser pulse, giving rise to strong charge separation into electrons and heavier, hence, immobile ions. The resulting ambipolar electric field leads to acceleration of protons in the target plasma in different directions.

1. M. Borghesi, A. Schiavi, D. H. Campbell, M. G. Haines, O. Willi, A. J. Mackinnon, P. Patel, M. Galimberti, and L. A. Gizzi, “Proton imaging detection of transient electromagnetic fields in laser-plasma interactions,” Rev. Sci. Instrum. **74**(3), 1688–1694 (2003). [CrossRef]

2. S. V. Bulanov and V. S. Khoroshkov, “Feasibility of using laser ion accelerators in proton therapy,” Plasma Phys. Rep. **28**(5), 453–456 (2002). [CrossRef]

3. M. Roth, T. E. Cowan, M. H. Key, S. P. Hatchett, C. Brown, W. Fountain, J. Johnson, D. M. Pennington, R. A. Snavely, S. C. Wilks, K. Yasuike, H. Ruhl, F. Pegoraro, S. V. Bulanov, E. M. Campbell, M. D. Perry, and H. Powell, “Fast ignition by intense laser-accelerated proton beams,” Phys. Rev. Lett. **86**(3), 436–439 (2001). [CrossRef] [PubMed]

4. B. Shen and Zh. Xu, “Transparency of an overdense plasma layer,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **64**(5), 056406–056412 (2001). [CrossRef] [PubMed]

6. X. Q. Yan, C. Lin, Z. M. Sheng, Z. Y. Guo, B. C. Liu, Y. R. Lu, J. X. Fang, and J. E. Chen, “Generating high-current monoenergetic proton beams by a circularly polarized laser pulse in the phase-stable acceleration regime,” Phys. Rev. Lett. **100**(13), 135003 (2008). [CrossRef] [PubMed]

^{20}W/cm

^{2}. Consider the most broadly used method – the transformation of linear polarization to a circular one by means of a quarter-wave plate. In this case, it is important that polarization transformation should be accomplished in the region of intense radiation. Indeed, the CPA (Chirp Pulse Amplification) as well as OPCPA (Optical Parametric Chirped-Pulse Amplification) technology used for laser radiation amplification requires linear polarization, as diffraction gratings that are inherent components of the system operate with maximum efficiency with polarization of this type. Consequently, it is interesting to investigate the simultaneous influence of the properties of the wave plate material (natural birefringence and nonlinearity) on laser radiation polarization.

*В*-integral).

## 2. Nonlinear optical properties of crystalline quartz

*α*-modification crystals possessing trigonal crystalline structure [14]. The permittivity and gyration tensors for a structure of this type are written in the formThe expressions relating electric field vector

*k*= 2

*π*/

*λ*, and

*λ*is the wavelength of light in vacuum. The expression Eq. (2) permits characterizing the principal properties of quartz. For example, if a quartz plate is cut parallel to the optical axis, then the birefringent properties of quartz are shown due to the difference between

*ε*

_{11}and

*ε*

_{22}. If the plate is cut normally to the optical axis, then quartz possesses gyrotropic properties, which is used in manufacturing 90° polarization rotators. In the present work we will be interested in plates cut parallel to the optical axis of the crystal –

*Oz*.

*λ*/4,

*λ*/2,

*λ*zero-order plates the phase difference between the extraordinary and ordinary waves is

*π*/2,

*π*, and 2

*π,*respectively. However, the thickness of the

*λ*plate would then be

*L*= 90 μm (at

*λ*= 795 nm), which is hard to achieve in practice. Hence, two multiple-order wave plates are used, the difference in thicknesses of which produces path difference of about 1/4 or 1/2 of wavelength, depending on the plate type. Optical axes of these multiple-order plates make an angle of 90° with each other (Fig. 1 ).This is the type of wave plate of interest to us.

*χ*:

_{ijkl}*z*corresponds to the optical axis of the crystal. The direction of the optical axis has been fixed, so now we have to define two other directions. From the expression Eq. (3) it follows that nonlinear properties of the crystal also depend on the way a wave plate is cut relative to the

*Ох*and

*Оу*axes. Note that there is no symmetry in tensor

*χ*with respect to subscripts

*x*and

*y*. For instance, the tensor component with subscripts

*yyyz*has no pair component with subscript

*xxxz.*Consequently, if the plate is cut along the

*Оу*axis, components with indices

*y*and

*z*play an important role in the system of differential equations. Quite the contrary, if the plate is cut along the

*Ox*axis, the components with indices

*х*and

*z*are of primary importance. It is worth mentioning that the choice of crystal cutting along the

*Оу*and

*Ox*axes does not affect the birefringent properties of the plate, which complicates identification of the direction cutting.

*В*for an anisotropic medium depends on polarization and direction of radiation propagation. Hence, in the present work we will determine

*В*for radiation with linear polarization coinciding with the polarization of an ordinary or extraordinary wave of crystalline quartz. The magnitudes of

*γ*corresponding to eigen modes of the crystal, according to [15], differ less than by 4%; so we set the largest

_{NL}*γ*= 3.2⋅10

_{NL}^{−7}cm

^{2}/GW.

## 3. System of differential equations

*Ох*axis, the

*Оу*axis will be taken to be the direction of radiation propagation and

*Ох*and

*Оz*will be directed along the optical axes in the first and second parts of a composite wave plate (Fig. 1). Then, the impact of cubic nonlinearity will be described by the

*χ*components, where

_{ijkl}*i*,

*j*,

*k*,

*l*are

*х*and

*z*. Whereas with the crystal cut parallel to the

*Oy*axis, formal substitution of

*x*for

*y*should be done in the equations taking into account the new terms containing components of permittivity tensor Eq. (3) with indices

*i*,

*j*,

*k*,

*l*are

*y*and

*z*.

*n*=

_{x}*n*= (

_{y}*ε*

_{11})

^{½}is the refractive index of an ordinary wave and

*n*= (

_{z}*ε*

_{22})

^{½}is the refractive index of an extraordinary wave. Dispersion characteristics of these waves of crystal quartz were found by [16,17].

*δ*(

*y*) =

*k*(

*n*–

_{z}*n*)

_{x}*y*in Eq. (6а) and

*δ*(

*x*) =

*k*(

*n*–

_{z}*n*)

_{y}*x*in Eq. (6b) are the phase differences of the extraordinary and ordinary waves. System Eq. (6а) corresponds to the case of the plate cut parallel to the

*Ох*axis, and Eq. (6b) parallel to the

*Оу*axis. From [15] it follows that

*χ*= 1.89⋅10

_{zzzz}^{−14}esu,

*χ*=

_{yyyy}*χ*= 1.82⋅10

_{xxxx}^{−14}esu. The magnitudes of

*χ*,

_{xxzz}*χ*and

_{yyzz}*χ*are unknown to the best of our knowledge.

_{yyyz}*χ*≅□

_{zzzz}*χ*=

_{yyyy}*χ*, from previous remark and system Eq. (6) it follows that it is the terms containing nondiagonal elements of tensor

_{xxxx}*χ*that will determine the change of polarization.

*Е*and

_{х}*Е*components as well as

_{у}*Е*at the output of the plate with preset

_{z}*δ*and

*В*. Such a change will be characterized by the degree of local (at each point of the cross-section) depolarization – the fraction in the output radiation

**E**

_{out}(

*B*≠ 0) of the intensity with polarization orthogonal to the polarization in the absence of nonlinearity

**E**

_{out⊥}(

*B*= 0) (i.e., circular polarization but with a different direction of rotation):Calculation of Γ using system Eq. (6) demonstrated that the impact of cubic nonlinearity reduces accuracy of polarization transformation by means of wave plates. In other words, if the radiation polarization at the input of a quarter-wave plate is such that clockwise circular polarization must be obtained at the output, then the depolarization (energy fraction contained in the counterclockwise circular polarization) at the plate output is proportional to squared

*В*-integral. Experimental verification of this result is presented in Section 4.

**E**

_{+},

**E**radiation components with clockwise and counterclockwise polarization respectively. Notice, that Σ

_{−}*=*−1 ( + 1) corresponds to the clockwise (counterclockwise) circular polarized radiation, Σ

*=*0 corresponds to the linear polarization. We assume that the vector

*E*of the linearly polarized incident radiation on the quarter-wave plate makes an angle of 45° with the optical axes of the plate. Therefore, the ellipticity of output radiation is defined by following expression:Without nonlinearity Γ = 0, and, hence, Σ = −1.

## 4. Experimental investigation of the influence of cubic nonlinearity on accuracy of transformation polarization by means of a quarter-wave plate

*λ*= 795 nm [18

18. A. A. Babin, A. M. Kiselev, A. M. Sergeev, and A. N. Stepanov, “Terawatt femtosecond Ti:sapphire laser system,” Quantum Electron. **31**(7), 623–626 (2001). [CrossRef]

*В*-integral.

### 4.1 Schematic of the experiment

**1**the main portion of the radiation passed through Galilean telescope

**2**-

**3**with magnification 1:2.5. This telescope permits increasing intensity up to the calculated level (~700 GW/cm

^{2}) at which the influence of cubic nonlinearity on polarization transformation in quarter-wave plate

**4**was calculated to be high enough. One more plate

**6**placed in lower-intensity radiation reflected from glass wedge

**5**was used to increase accuracy of depolarization measurements. Hence, cubic nonlinearity could not affect polarization transformation in plate

**6**. After that the principal radiation component with polarization corresponding to a large semi-axis of the polarization ellipse and the depolarized component with orthogonal polarization were separated in space by calcite wedge

**7**. The calcite wedge was adjusted so that its principal axes coincided with the principal axes of the polarization ellipse at

*В*= 0. The image was transferred from the plane of wave plate

**4**to the matrix of CCD-camera

**9**by lens

**8**. Thanks to the additional wave plate

**6**radiation polarization at the output of the experimental setup at

*В*= 0 coincides with the initial polarization. In this case, two-dimensional intensity distributions of the principal (bright spot) and depolarized (spot at the level of CCD-camera noise) components of radiation were recorded on the CCD-camera (Fig. 2(b), 2(c)). The brightness of the spot corresponding to the depolarized component increased at

*В*> 1 (Fig. 2(e)).

*exp*(−

*r*/

*R*

_{0})

^{2}, where

*R*

_{0}= 0.24 cm. The radiation time distribution is described well by the function 1/

*ch*

^{2}(

*t*/

*t*

_{0}), where

*t*

_{0}= 28 fs, which corresponds to the pulse duration of 75 fs FWHM. Pulse energy was measured within the 1-7 mJ range.

### 4.2 Experimental results

**4**and

**6**were added to the scheme. In this case the dependence of depolarization on intensity

*I*(

*В*-integral) is nonlinear. The value of Γ(

*B*= 0) equal to 0.1% is less than the analogous value in the absence of wave plates

**4**and

**6**, which may be attributed to reduced polarization ellipticity due to the plates. The maximum intensity magnitude of 700 GW/cm

^{2}achieved in experiment corresponds to Г = 1.1% and Σ = −0.87.

*χ*unknown for crystalline quartz. Let us take

*I*= 700 GW/cm

^{2}and plot a one-dimensional Γ(

*χ*) and a two-dimensional Γ(

_{xxzz}*χ*,

_{yyzz}*χ*) functions for each type of wave plate cutting, respectively (Figs. 3(b), 4(b)). The resulting dependences will be used to determine values of the sought components of tensor

_{yyyz}*χ*, for which Г(

*I*= 700 GW/cm

^{2}) = 1.1%. Further, we will search among the selected values of

*χ*components the ones at which the theoretical and experimental curves for Г(

*I*) coincide at all points (Fig. 3(a), 4(a)). Note, we assume that the both parts (multiple-order plates) of the quarter-wave plates used in experiments have the identical direction cutting.

*χ*:

*χ*= 0.76

_{xxzz}*χ*and

_{xxxx}*χ*= −0.28

_{xxzz}*χ*were found for the plate cut parallel to the

_{xxxx}*Ох*axis (Fig. 3(b)). The curves in Fig. 3(b) are plotted for different values of radiation polarization ellipticity at the input, i.e., for the depolarization Г(

*В*= 0) = 0%, 0.01% and 0.1%. Thus, we have found that the uncertainty of determining Г(

*В*= 0) in the given range of values leads to 8% error in calculations of

*χ*. Note that the obtained values of

_{xxzz}*χ*demonstrate a good agreement of experimental and theoretical data for the clockwise polarized radiation (black solid and dashed curves in Fig. 3(а)). For the case of the counterclockwise polarized radiation, good agreement between experiment and theory is attained only at one point

_{xxzz}*I*= 700 GW/cm

^{2}(Fig. 3(а), grey solid line).

*Оу*axis, similarly to the considered case of clock- and counterclockwise polarized radiation, there exist several sets of pairs

*χ*,

_{yyzz}*χ*, at which experiment and theory agree well. Values of tensor

_{yyyz}*χ*components corresponding to the clockwise polarized radiation are shown by the solid curve in Fig. 4(b), where the dashed curve mark pairs

*χ*,

_{yyzz}*χ*for the counterclockwise polarization.

_{yyyz}*В*-integral larger than unity, rather than determination of unknown components of tensor

*χ*. We have found the region of possible values of parameters

*χ*,

_{xxzz}*χ*and

_{yyzz}*χ*at which the experimental data agree with the theoretical calculations.

_{yyyz}*L*= 0.18 cm have the following shape

*А*

_{0}(

*t*) = 1/

*ch*

^{2}(

*t*/

*t*

_{0}), where

*t*

_{0}= 28 fs. For the radiation with linear polarization coinciding with one of the eigen polarizations of the plate, a phase additive appears at the plate output defined by the nonlinear part of the refractive index:

*А*(

*t*) = 1/

*ch*

^{2}(

*t*/

*t*

_{0})⋅exp(–

*ikγ*

_{NL}I_{0}(

*А*(

_{0}*t*))

^{2}

*L*), where

*I*

_{0}= 700 GW/cm

^{2}is the maximum value obtained in the experiment. The spectra corresponding to the pulses

*А*

_{0}(

*t*) and

*А*(

*t*) are plotted in Fig. 5 .

_{0}versus

*λ*specified by

*δ*(

*λ*) =

*k*(

*n*(

_{z}*λ*) –

*n*(

_{y}*λ*))

*L*. This function crosses zero at the point

*λ*= 795 nm as the plate is fabricated to for radiation at this central wavelength. Integral of depolarization Γ

_{0}(

*λ*) over the spectrum

*А*(

_{λ}*λ*) is much less than the Γ measured in the experiment (0.1-1%). Hence, the radiation spectrum broadening due to cubic nonlinearity may be neglected even at

*I*

_{0}= 700 GW/cm

^{2}.

## 5. Conclusion

^{2}and higher was investigated. A specific feature in this case is that, despite the small thickness of the wave plate, we have to consider the influence of ellipse polarization rotation caused by cubic nonlinearity on the polarization transformation.

*λ*/4 plate 0.18 cm thick was conducted. It was found that the depolarization degree Γ is proportional to squared

*В*-integral (nonlinear phase incursion). The maximum measured value of Γ was 1.1% at the laser radiation intensity of 700 GW/cm

^{2}. Good agreement between the experimental and the theoretical data was obtained.

*λ*/4 plates of arbitrary thickness may be cut from them. Study of the specific features of using such wave plates will be the next step in the work on this problem.

## References and links

1. | M. Borghesi, A. Schiavi, D. H. Campbell, M. G. Haines, O. Willi, A. J. Mackinnon, P. Patel, M. Galimberti, and L. A. Gizzi, “Proton imaging detection of transient electromagnetic fields in laser-plasma interactions,” Rev. Sci. Instrum. |

2. | S. V. Bulanov and V. S. Khoroshkov, “Feasibility of using laser ion accelerators in proton therapy,” Plasma Phys. Rep. |

3. | M. Roth, T. E. Cowan, M. H. Key, S. P. Hatchett, C. Brown, W. Fountain, J. Johnson, D. M. Pennington, R. A. Snavely, S. C. Wilks, K. Yasuike, H. Ruhl, F. Pegoraro, S. V. Bulanov, E. M. Campbell, M. D. Perry, and H. Powell, “Fast ignition by intense laser-accelerated proton beams,” Phys. Rev. Lett. |

4. | B. Shen and Zh. Xu, “Transparency of an overdense plasma layer,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

5. | A. Macchi, “A femtosecond neutron source,” Appl. Phys. B |

6. | X. Q. Yan, C. Lin, Z. M. Sheng, Z. Y. Guo, B. C. Liu, Y. R. Lu, J. X. Fang, and J. E. Chen, “Generating high-current monoenergetic proton beams by a circularly polarized laser pulse in the phase-stable acceleration regime,” Phys. Rev. Lett. |

7. | D.-H. Kwon, K. Lee, S. H. Park, and Y. U. Jeong, “The effect of the transparency of an overdense plasma on proton beam generation by an intense ultra-short laser pulse,” J. Korean Phys. Soc. |

8. | S. N. Vlasov, V. I. Kryzhanovskiĭ, and V. E. Yashin, “Use of circularly polarized optical beams to suppress selffocusing instability in a nonlinear cubic medium with repeaters,” Sov. J. Quantum Electron. |

9. | Y. B. Zel'dovich and Y. P. Raizer, “Self-focusing of light. Role of Kerr effect and striction,” JETP Lett. |

10. | P. D. Maker, R. W. Terhune, and C. M. Savage, “Intensity-dependent changes in the refractive index of liquids,” Phys. Rev. Lett. |

11. | A. L. Berkhoer and V. E. Zakharov, “Self excitation of waves with different polarizations in nonlinear media,” Sov. Phys. JETP |

12. | D. Auric and A. Labadens, “On the use of circulary polarized beam to reduce the self-focusing effect in a glass rod amplifier,” Opt. Commun. |

13. | M. S. Kuzmina, M. A. Martyanov, A. K. Poteomkin, E. A. Khazanov, and A. A. Shaykin, “Theoretical and experimental study of laser radiation propagating in a medium with thermally induced birefringence and cubic nonlinearity,” Opt. Express |

14. | R. J. Pressley, |

15. | R. L. Suthrland, |

16. | H. R. Phillip, |

17. | S. S. Ballard, K. A. McCarthy, and W. L. Wolfe, |

18. | A. A. Babin, A. M. Kiselev, A. M. Sergeev, and A. N. Stepanov, “Terawatt femtosecond Ti:sapphire laser system,” Quantum Electron. |

19. | N. G. Bondarenko, I. V. Eremina, and A. I. Makarov, “Measurement of nonlinear susceptibility tensor χ |

**OCIS Codes**

(190.0190) Nonlinear optics : Nonlinear optics

(140.3295) Lasers and laser optics : Laser beam characterization

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: October 31, 2012

Revised Manuscript: December 10, 2012

Manuscript Accepted: December 10, 2012

Published: January 2, 2013

**Citation**

M. S. Kuzmina, E. A. Khazanov, A. A. Shaykin, A. N. Stepanov, and Yu. Malkov, "Influence of cubic nonlinearity on accuracy of polarization transformation by means of a quarter-wave plate," Opt. Express **21**, 135-144 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-1-135

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### References

- M. Borghesi, A. Schiavi, D. H. Campbell, M. G. Haines, O. Willi, A. J. Mackinnon, P. Patel, M. Galimberti, and L. A. Gizzi, “Proton imaging detection of transient electromagnetic fields in laser-plasma interactions,” Rev. Sci. Instrum.74(3), 1688–1694 (2003). [CrossRef]
- S. V. Bulanov and V. S. Khoroshkov, “Feasibility of using laser ion accelerators in proton therapy,” Plasma Phys. Rep.28(5), 453–456 (2002). [CrossRef]
- M. Roth, T. E. Cowan, M. H. Key, S. P. Hatchett, C. Brown, W. Fountain, J. Johnson, D. M. Pennington, R. A. Snavely, S. C. Wilks, K. Yasuike, H. Ruhl, F. Pegoraro, S. V. Bulanov, E. M. Campbell, M. D. Perry, and H. Powell, “Fast ignition by intense laser-accelerated proton beams,” Phys. Rev. Lett.86(3), 436–439 (2001). [CrossRef] [PubMed]
- B. Shen and Zh. Xu, “Transparency of an overdense plasma layer,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.64(5), 056406–056412 (2001). [CrossRef] [PubMed]
- A. Macchi, “A femtosecond neutron source,” Appl. Phys. B82(3), 337–340 (2006). [CrossRef]
- X. Q. Yan, C. Lin, Z. M. Sheng, Z. Y. Guo, B. C. Liu, Y. R. Lu, J. X. Fang, and J. E. Chen, “Generating high-current monoenergetic proton beams by a circularly polarized laser pulse in the phase-stable acceleration regime,” Phys. Rev. Lett.100(13), 135003 (2008). [CrossRef] [PubMed]
- D.-H. Kwon, K. Lee, S. H. Park, and Y. U. Jeong, “The effect of the transparency of an overdense plasma on proton beam generation by an intense ultra-short laser pulse,” J. Korean Phys. Soc.49, 347–353 (2006).
- S. N. Vlasov, V. I. Kryzhanovskiĭ, and V. E. Yashin, “Use of circularly polarized optical beams to suppress selffocusing instability in a nonlinear cubic medium with repeaters,” Sov. J. Quantum Electron.12(1), 7–10 (1982). [CrossRef]
- Y. B. Zel'dovich and Y. P. Raizer, “Self-focusing of light. Role of Kerr effect and striction,” JETP Lett.3, 86–89 (1966).
- P. D. Maker, R. W. Terhune, and C. M. Savage, “Intensity-dependent changes in the refractive index of liquids,” Phys. Rev. Lett.12(18), 507–509 (1964). [CrossRef]
- A. L. Berkhoer and V. E. Zakharov, “Self excitation of waves with different polarizations in nonlinear media,” Sov. Phys. JETP31, 903–911 (1970).
- D. Auric and A. Labadens, “On the use of circulary polarized beam to reduce the self-focusing effect in a glass rod amplifier,” Opt. Commun.21(2), 241–242 (1977). [CrossRef]
- M. S. Kuzmina, M. A. Martyanov, A. K. Poteomkin, E. A. Khazanov, and A. A. Shaykin, “Theoretical and experimental study of laser radiation propagating in a medium with thermally induced birefringence and cubic nonlinearity,” Opt. Express19(22), 21977–21988 (2011). [CrossRef] [PubMed]
- R. J. Pressley, Handbook of Lasers With selected Data on Optical Technology (Chemical Rubber Co, Cleveland, 1979).
- R. L. Suthrland, Handbook of nonlinear optics, Marcel Dekker (New York, 2003).
- H. R. Phillip, Handbook of Optical Constants of Solids (Academic Press, 1985).
- S. S. Ballard, K. A. McCarthy, and W. L. Wolfe, Optical Materials for Infrared Instrumentation (IRIA-University of Michigan, Report #2389–11-S, 1959).
- A. A. Babin, A. M. Kiselev, A. M. Sergeev, and A. N. Stepanov, “Terawatt femtosecond Ti:sapphire laser system,” Quantum Electron.31(7), 623–626 (2001). [CrossRef]
- N. G. Bondarenko, I. V. Eremina, and A. I. Makarov, “Measurement of nonlinear susceptibility tensor χijkl for DKDP crystals,” in Proceedings of the Five All-Union Conference on Laser Optics (Gosudarstvenny Optichesky Institut, Leningrad, 1987), pp. 37–40.

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