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Optics Express

  • Editor: Michael Duncan
  • Vol. 13, Iss. 21 — Oct. 17, 2005
  • pp: 8565–8570
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Electric polarization induced by optical orientation of dipolar centers in non-polar piezoelectrics

Alexander I. Grachev and Alexei A. Kamshilin  »View Author Affiliations


Optics Express, Vol. 13, Issue 21, pp. 8565-8570 (2005)
http://dx.doi.org/10.1364/OPEX.13.008565


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Abstract

We predict new nonlinear optical phenomenon, static electric polarization induced in non-polar piezoelectric by linearly polarized light owing to orientation of the impurity centers with permanent dipole moment. Microscopic model of this effect for the crystals of cubic symmetry is proposed. Estimations of the model show that the induced polarization might be much higher than that caused by optical rectification. It is shown that a transient current Jd generated under non-steady-state illumination could be observed at extremely high modulation frequency. We expect that the effect could be applied for ultrafast optical signal processing.

© 2005 Optical Society of America

1. Introduction

Alignment of the impurity centers and complexes by the linearly polarized light is well-known phenomenon [1

1 . P. P. Feofilov , The Physical Basis of Polarized Emission ( Consultant’s Bureu, N.Y. , 1961 ).

,2

2 . F. Lüty , “ F A Centers in Alkali Halide Crystals ,” In: Physics of Color Centers Ed. by W. B. Fowler ( Academic Press, N.Y. , 1968 ).

]. It has been observed in centrosymmetric crystals such as alkali halides or KTaO3 for centers with lower symmetry than the crystal matrix [2–5

2 . F. Lüty , “ F A Centers in Alkali Halide Crystals ,” In: Physics of Color Centers Ed. by W. B. Fowler ( Academic Press, N.Y. , 1968 ).

]. Alignment of such centers results particularly in occurrence of induced anisotropy of light absorption. Axially symmetric centers or complexes often demonstrate noticeable permanent dipole moment (below we call such centers as dipolar). Alignment of these dipoles in a polar manner along axis determined by the plane of the light polarization (i.e., dipole orientation) is symmetrically forbidden in centrosymmetric crystals. However, in crystals lacking inversion symmetry this prohibition is lifted. Consequently, appearance of macroscopic polarization of the crystal due to optical orientation of dipolar centers (OODC) can be observed. This effect somehow resembles optical orientation of electron spins [6

6 . F. Meier and B. P. Zakharchenya , Eds., Optical Orientation ( Elsevier, Amsterdam , 1984 ).

]. To the best of our knowledge, possibility of such effect occurrence and its microscopic origin has not been discussed so far.

In this paper we consider the macroscopic electric polarization P induced by optical orientation of dipolar centers in non-polar piezoelectrics and a microscopic model of the effect. The experimental manifestation of the OODC, first of all transient electric current arising under non-stationary illumination, is also discussed. Estimated value of the current and its frequency-response function is shown to give an opportunity for ultrafast optoelectronic applications of the OODC effect.

2. Phenomenological description

Static electric polarization, P, of a noncentrosymmetric crystal exposed by an electromagnetic radiation may emerge as a consequence of the nonlinear optical effect, namely, optical rectification. As known, this effect is described by the second-order nonlinear susceptibility tensor χ ijk [7

7 . Y. R. Shen , The Principles of Nonlinear Optics ( Wiley, New York , 1984 ).

]. Since OODC also relates the dc polarization to the oscillating field of the optical wave, the lowest tensor phenomenologically describing this effect is of the third rank, as well:

Pi=dijk(ejek*+ej*ek)I
(1)

3. Microscopic model

There are two main microscopic mechanisms that can be realized for OODC occurrence like it takes place for the effect of optical alignment. The first mechanism is orientation-sensitive optical excitation of centers (including intra-center transitions) and their subsequent real reorientation with the reorientation time in the excited state being different from that in the ground state [2

2 . F. Lüty , “ F A Centers in Alkali Halide Crystals ,” In: Physics of Color Centers Ed. by W. B. Fowler ( Academic Press, N.Y. , 1968 ).

]. The second is repopulation of differently oriented centers resulting from orientation-sensitive release of electrons (holes) from the centers due to charge-transfer transitions and its following orientation-insensitive trapping (not necessarily by the same centers) [3

3 . R. L. Berney and D. L. Cowan , “ Photochromism of three photosensitive Fe centers in SrTiO 3 ” Phys. Rev. B 23 , 37 – 50 ( 1981 ). [CrossRef]

]. Although the latter mechanism does not involve real reorientation process it, nevertheless, results in preferred orientation of the dipole moment of the centers. In general, both mechanisms can be simultaneously realized with higher or smaller efficiency in a crystal. Therefore, below we examine a model of the OODC in which both mechanisms take part to a certain extent.

Typical dipolar centers in crystals are off-center impurity ions and impurity complexes arising as a consequence of the principle of the local charge compensation of excess impurity charge. The microscopic origin of OODC is illustrated here by the example of a crystal with zinc-blende structure (the symmetry class of 4̄3m) that contains dipolar complexes participating in charge-transfer optical transitions of electrons into the conduction band. Similar model of the acceptor-donor-pair complex was previously considered to describe the microscopic origin of the linear photogalvanic current [8

8 . V. V. Obukhovsky and A. V. Stoyanov , “ A model of photogalvanic centers in crystals ,” Ferroelectrics 43 , 137 – 141 ( 1982 ). [CrossRef]

]. Note that this current is also described by the third-rank material βijk [9

9 . B. I. Sturman and V. M. Fridkin , Photovoltaic and Photorefractive Effects in Noncentrosymmetric Materials ( Gordon Breach, Philadelphia, Pa. , 1992 ).

]. It was demonstrated that JLPG is generated only when the acceptor center (A) and donor center (D) occupy the adjacent positions in different sub-lattices as shown in Fig. 1 [8

8 . V. V. Obukhovsky and A. V. Stoyanov , “ A model of photogalvanic centers in crystals ,” Ferroelectrics 43 , 137 – 141 ( 1982 ). [CrossRef]

]. Therefore, the dipole moment μ of the donor-acceptor pair is parallel to any of the third-order symmetry axis (C3 ). Starting from this model, we additionally use the following simplifying assumptions:

  1. Complexes are capable to be reoriented due to thermally stimulated hopping of D-center between four possible tetrahedral positions around the A-center. In general, the time of D-center hopping, τ depends on the charge state of the A-center but, for simplicity, here we assume that τ is the same for different charge states.
  2. The sample illumination leads to photo-excitation of electrons from the ionized A-center having the charge state of -1.
  3. D-centers, with concentration NDN (here N is the total number of donor-acceptor pairs) are the main compensating impurity. It is supposed that these centers are ionized (i.e. keep the charge +1) both under illumination and in the dark.
  4. The lifetime of photo-excited electrons is mainly determined by the concentration of neutral A-centers, NA0NA (here NAN is concentration of A-centers).
  5. The dipole moment of the pairs with the neutral A-center is supposed to be much smaller than that of the pairs with the charged A-center, i.e. than μ.
Fig. 1. Dipolar complexes within the unit cell of a crystal with zink-blende structure. Four possible orientations of A and D centers occupied near-neighbor positions are indicated by a through d. Blue and red atoms refer to two sublattices of the crystal structure. Inset on the left shows propagation direction of the light and its polarization vector, e.

In the dark, the dipolar complexes on average are homogeneously distributed among four possible directions of μ (Fig. 1). This is provided, on the one hand, by orientation-independent thermal excitation of electrons from the complexes, and on the other hand by the orientation-independent relaxation process, which involves complex reorientation and capture of free electrons by neutral A-centers via nonradiative and/or radiative recombination. Therefore, the total dipole moment of the crystal as a whole is zero in the dark. Let now the linearly polarized light beam with the polarization vector e parallel to the [110] axis propagates along the [001] axis of the crystal (Fig. 1). It is well-known that the absorption cross-section of a dipolar center, σ, is strongly anisotropic [1

1 . P. P. Feofilov , The Physical Basis of Polarized Emission ( Consultant’s Bureu, N.Y. , 1961 ).

]. For the separate center, σ depends on the angle θ between the dipole moment μ and the vector e as following:

σ(θ)=σcos2θ+σsin2θ,
(2)

where σ and σ are the absorption cross-sections for e perpendicular and parallel to μ, respectively. For pairs with D-centers occupying positions a or b in Fig. 1 (we call these complexes as of the type I) e is perpendicular μ. If D-center is either in position c or d (complexes of the type II), the angle between e and μ satisfies the relation of cos2 θ = 2/3 . It is known that usually σ is much smaller than σ [1

1 . P. P. Feofilov , The Physical Basis of Polarized Emission ( Consultant’s Bureu, N.Y. , 1961 ).

]. Therefore in the geometry shown in Fig. 1, the light will be preferably absorbed by the complexes of the type II. Since the relaxation of the complexes does not depend on their orientation, the equilibrium between pairs of type I and II is broken towards predominance of the former complexes. This evidently results in appearance of the macroscopic dipole moment P directed along the [001] axis (see Fig. 1).

4. Effect estimation

The steady-state value of P and its kinetics can be derived from the balance equations for photo-excited electrons and for pairs of the type I and II. Assuming that the concentration of the photo-excited electrons, n, is much smaller than the concentration of neutral A-centers, these equations, supplemented by the relation for the total number of dipolar complexes under the illumination, are

dN1dt=σ1N1Ihv+γn(N2N1)2Γ(N1N2),
(3)
dN2dt=σ2N2Ihv+γn(N2N2)2+Γ(N1N2),
(4)
dndt=(σ1N1+σ2N2)IhvγnNA0,
(5)
N1+N2N(NA0+n).
(6)

Here N 1 and N 2 are the number of pairs with the ionized A-center of the type I and II, respectively, γ is the recombination coefficient, hv is the photon energy, and Γ = (3τ/2)-1 is the hopping rate accompanied by the reorientation of one type of the complexes to another. Absorption cross-sections σ 1 = σ and σ 2 = 2/3 σ +1/3 σ are assumed to satisfy the relation of Δσ = (σ 2-σ 1)≪σ 1,σ 2 that allows replacement of (σ 1 N 1 + σ 2 N 2) by σ 0 N with σ 0 = (σ 1 + σ 2)/2 in Eq. (5). The absorption cross-section of unpaired A-centers is assumed to be equal σ 5, as well. In equations (3–5) thermal excitation of electrons and their trapping by donor centers are neglected.

Considering that nNA0N and variations of N 1 + N 2 are negligible, Eq. (5) yields n = 2σ 0 NI / γNA0 hv and the system of Eqs. (3)–(6) is reduced to

Ndt=(N1+N2)ΔσI2hvNΔNσ0IhvNA02ΓΔN,
(7)

which results in the following expression for the steady-state value of ΔN:

ΔNSTNΔτrelIhv
(8)

with the relaxation time of

τrel=(2Γ+σ0NINA0hv)1.
(9)

When the light intensity I is much smaller than ITH = 2NA0Γhv/σ 0 N , the relaxation time is mainly determined by real reorientation of the complexes and the steady-state difference of complexes concentrations is ΔNST = ΔσNI /2Γhv. When the light intensity exceeds ITH , the concentration difference ΔNST is saturated at the level of ΔNSAT = ΔσNA0/σ 0. In this region kinetics ΔN is governed by the redistribution of electrons between A-centers constituting the complexes of different types which leads to predomination of complexes of the type I though the process of real reorientation of complexes is insignificant. One should take into account that at very high light intensity supposed relationship of nNA0 may be violated. To determine the validity limits of the proposed model and expected value of the light-induced polarization, numerical estimation is given below.

It is reasonable to accept the following values for the crystal parameters: N ≈ 1019 cm-3, N/NA0 ≈ 10 ,σ 0 =10-17cm2, and Δσ/σ ≈ 0.1. In frameworks of the conventional barrier-limited model, the hopping time is τ = τ 0 exp(-EA /kBT), where pre-exponential factor τ0 is in the range of 10-13÷10-11s; EA , the activation barrier; kB , the Boltzmann constant; and T, temperature. Taking τ 0 = 5×10-13 s and EA = 0.25 eV, we get Γ ≈ 108s-1 at the room temperature. Assuming simultaneous processes of radiative and nonradiative recombination, the value of γ is taken as about 10-9 cm-3 s-1, which corresponds to the electron lifetime of τn = (γNA0)-1 =10-9s. At last, the value of dipole moment, μ, is chosen to be about 1.6×10-29Cm.

With chosen parameters the intensity ITH is about 4×105 Wcm-2 at hv = 1.25eV. Consequently, at small intensity (IITH ) the steady-state polarization is PST = μΔNST /√3= d 14 I, where d 14 = 1.5×10-16 V-1 s is the only non-vanishing component of the tensor dijk in a crystal of the symmetry class 4̄3m . For comparison in typical nonlinear crystals, the components of the optical rectification tensor are much smaller: of the order of 10-19 V-1s [7

7 . Y. R. Shen , The Principles of Nonlinear Optics ( Wiley, New York , 1984 ).

]. Saturation of the concentration difference, ΔNSAT = 1017 cm-3, which generates polarization of 10-6 Cm-2, is reached at intensity of IITH . It is worth noting that inequality nNA0 is well satisfied even at the intensity I equal to ITH .

Non-steady-state illumination of the crystal naturally leads to generation of a transient current density J d = dP/dt. To separate this current from conventional photovoltaic current, one can use the polarization modulation technique. The OODC current is generated because switching the linear polarization of the light beam to the orthogonal one leads to the change of the sign of P and of J d . However, the linear photogalvanic current JLPG possessing the same polarization feature [9

9 . B. I. Sturman and V. M. Fridkin , Photovoltaic and Photorefractive Effects in Noncentrosymmetric Materials ( Gordon Breach, Philadelphia, Pa. , 1992 ).

] can compete with Jd in such experiment. The problem of separating Jd and JLPG , which was ignored so far, can be solved studying the difference of the frequency response function (FRF) of these currents. Dispersion of tensor βijk (Ω) responsible for generation of JLPG is βijk (Ω)∝ (1 +iΩτP )-1 , where τP is the relaxation time of the electron’s momentum, τP ~10-14÷10-12 s [9

9 . B. I. Sturman and V. M. Fridkin , Photovoltaic and Photorefractive Effects in Noncentrosymmetric Materials ( Gordon Breach, Philadelphia, Pa. , 1992 ).

]. Thus the current JLPG is independent on the frequency up to Ω ≈ τP1 and then drops off as Ω-1. Dispersion of dijk (Ω) is evidently similar to βijk (Ω) with substitution of τP by τP and, correspondingly, Jd (Ω)∝Ωτrel /(1 + iΩτrel ). Therefore in contrast to JLPG (Ω), the current of OODC, Jd (Ω), increases linearly with the frequency up to Ω ≈ τrel1and reaching further a plateau.

To compare the order of values of Jd (Ω) and JLPG (Ω) we use the following expression for the latter current [9

9 . B. I. Sturman and V. M. Fridkin , Photovoltaic and Photorefractive Effects in Noncentrosymmetric Materials ( Gordon Breach, Philadelphia, Pa. , 1992 ).

]:

JLPGeNσ0l0ζexIhv,
(10)

with e, the electron charge; l 0, electron free length; and ζex , the parameter of photo-excitation asymmetry of electrons from the dipolar centers [9

9 . B. I. Sturman and V. M. Fridkin , Photovoltaic and Photorefractive Effects in Noncentrosymmetric Materials ( Gordon Breach, Philadelphia, Pa. , 1992 ).

]. Taking other parameters as above, for medium values of l0 ~; 3×10-6 cm, ξex ~ 10-3, we obtain (at IITH ) the ratio Jd (Ω)/JLPG (Ω)≈ Ωτrel / [1 + (Ω τrel )2]1/2. This implies that in the plateau region Jd can reach the same order of value as JLPG . Note that in the frameworks of the proposed model both currents have the same sign.

For experimental observation of the OODC-effect one should take special care of separation Jd from other photo-voltaic effects. Abovementioned polarization-modulation technique is a good tool for separation from convenient photo-voltaic currents. To achieve further separation from .JLPG -current one can analyze RFR-function of the current measured by the polarization-modulation technique. Existence of a linearly growing part of the function would confirm presence of sought OODC-effect. These currents can be also separated in a time-resolved experiment, for example by measuring current relaxation after fast switching-off the illumination light. The relaxation time of JLPG is expected to be much shorter than that of Jd since the former is defined by the momentum relaxation of an excited free-electron, while the latter is characterized by either recombination time of an electron at the center or the center reorientation time.

It should be noted that the effect of optical generation of macroscopic electric polarization in polar crystals due to impurity light absorption was studied in the seventies [10

10 . D. H. Auston , A. M. Glass , and A. A. Ballman , “ Optical rectification by impurities in polar crystals ,” Phys. Rev. Lett. 28 , 897 – 900 ( 1972 ). [CrossRef]

,11

11 . D. H. Auston and A. M. Glass , “ Optical generation of intense picosecond electrical pulses ,” Appl. Phys. Lett. 20 , 398 – 399 ( 1972 ). [CrossRef]

]. The mechanism of this effect involves optical excitation of dipolar impurities with different dipole moments in the ground and excited states without any orientation of impurities. In contrast with the proposed OODC effect, the exciting light in this case may be non-polarized since the most of dipole moments of the centers are oriented along the polar axis of the crystal.

The FRF of the current caused by macroscopic polarization change in polar crystals (excluding relatively slow pyroelectric component) is similar to Jd (Ω) [10

10 . D. H. Auston , A. M. Glass , and A. A. Ballman , “ Optical rectification by impurities in polar crystals ,” Phys. Rev. Lett. 28 , 897 – 900 ( 1972 ). [CrossRef]

,11

11 . D. H. Auston and A. M. Glass , “ Optical generation of intense picosecond electrical pulses ,” Appl. Phys. Lett. 20 , 398 – 399 ( 1972 ). [CrossRef]

]. The generation of picosecond current pulses up to 4A was experimentally observed in LiTaO3:Cu crystal irradiated by laser pulses with duration of few picoseconds [11

11 . D. H. Auston and A. M. Glass , “ Optical generation of intense picosecond electrical pulses ,” Appl. Phys. Lett. 20 , 398 – 399 ( 1972 ). [CrossRef]

]. Considering similarity of FRF, we expect that optical generation of ultrashort electric pulses might be also efficient in non-polar piezoelectrics due to proposed OODC effect. When the laser pulse is much shorter than τrel , the relaxation terms in Eq. (7) can be neglected that results in following evolution of ΔN(t) and, consequently, P(t) on the time scale of the pulse length, Δtp :

dPdt(ΔσNμ23hv)I(t)
(11)

As a result, the polarization current of Jd (t) ∝ dP/dtI(t) practically repeats the laser pulse shape. Note, that for generation of Jd in the time scale of t ≪ τrel the difference of dipole moments in ground and excited states is the necessary condition. Considering that exhausting of electron population of A-centers occurs at the intensity of 1010 W/cm2 for ΔtP ≈ 10-12 s (same intensity was used in Ref 11), we estimate at this intensity the peak value of generated current density of about 4×102 A/cm2.

5. Conclusion

In conclusion, we have predicted a new nonlinear optical phenomenon of static electric polarization of non-polar piezoelectrics irradiated by linearly polarized light, which arises due to population change of differently oriented dipolar centers. Estimation based on the proposed macroscopic model of the effect shows that expected value of the induced polarization is much higher than that caused by optical rectification. The frequency response function of the transient current generated under non-steady-state illumination and its amplitude allow us to consider the predicted effect as the base for efficient generation of utrashort electric pulses.

Acknowledgments

Authors acknowledge financial support of the Academy of Finland under the research project No. 108010.

References

1 .

P. P. Feofilov , The Physical Basis of Polarized Emission ( Consultant’s Bureu, N.Y. , 1961 ).

2 .

F. Lüty , “ F A Centers in Alkali Halide Crystals ,” In: Physics of Color Centers Ed. by W. B. Fowler ( Academic Press, N.Y. , 1968 ).

3 .

R. L. Berney and D. L. Cowan , “ Photochromism of three photosensitive Fe centers in SrTiO 3 ” Phys. Rev. B 23 , 37 – 50 ( 1981 ). [CrossRef]

4 .

H.- J. Reyher , B. Faust , M. Käding , H. Hesse , E. Ruza , and M. Wühlecke , “ Optical alignment of axial Fe centers in KTaO 3 ,” Phys. Rev. B 51 , 6707 – 6710 ( 1995 ). [CrossRef]

5 .

S.A. Basun , V. E. Bursian , A. G. Razdobarin , and L. S. Sochava , “ Anisotropic photoionization as a mechanism of axial iron center alignment in KTaO 3 ,” Phys. Sol. State 43 , 1059 – 1071 ( 2001 ). [CrossRef]

6 .

F. Meier and B. P. Zakharchenya , Eds., Optical Orientation ( Elsevier, Amsterdam , 1984 ).

7 .

Y. R. Shen , The Principles of Nonlinear Optics ( Wiley, New York , 1984 ).

8 .

V. V. Obukhovsky and A. V. Stoyanov , “ A model of photogalvanic centers in crystals ,” Ferroelectrics 43 , 137 – 141 ( 1982 ). [CrossRef]

9 .

B. I. Sturman and V. M. Fridkin , Photovoltaic and Photorefractive Effects in Noncentrosymmetric Materials ( Gordon Breach, Philadelphia, Pa. , 1992 ).

10 .

D. H. Auston , A. M. Glass , and A. A. Ballman , “ Optical rectification by impurities in polar crystals ,” Phys. Rev. Lett. 28 , 897 – 900 ( 1972 ). [CrossRef]

11 .

D. H. Auston and A. M. Glass , “ Optical generation of intense picosecond electrical pulses ,” Appl. Phys. Lett. 20 , 398 – 399 ( 1972 ). [CrossRef]

OCIS Codes
(190.4720) Nonlinear optics : Optical nonlinearities of condensed matter
(190.7110) Nonlinear optics : Ultrafast nonlinear optics
(320.7130) Ultrafast optics : Ultrafast processes in condensed matter, including semiconductors

ToC Category:
Research Papers

History
Original Manuscript: September 7, 2005
Revised Manuscript: October 6, 2005
Published: October 17, 2005

Citation
Alexander Grachev and Alexei Kamshilin, "Electric polarization induced by optical orientation of dipolar centers in non-polar piezoelectrics," Opt. Express 13, 8565-8570 (2005)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-21-8565


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References

  1. P. P. Feofilov, The Physical Basis of Polarized Emission (Consultant`s Bureu, N.Y., 1961).
  2. F. Lüty, �??FA Centers in Alkali Halide Crystals,�?? In: Physics of Color Centers Ed. by W. B. Fowler (Academic Press, N.Y., 1968).
  3. R. L. Berney and D. L. Cowan, �??Photochromism of three photosensitive Fe centers in SrTiO3,�?? Phys. Rev. B 23, 37-50 (1981). [CrossRef]
  4. H.- J. Reyher, B. Faust, M. Käding, H. Hesse, E. Ruza, and M. Wöhlecke, �??Optical alignment of axial Fe centers in KTaO3,�?? Phys. Rev. B 51, 6707-6710 (1995). [CrossRef]
  5. S. A. Basun, V. E. Bursian, A. G. Razdobarin, and L. S. Sochava, �??Anisotropic photoionization as a mechanism of axial iron center alignment in KTaO3,�?? Phys. Sol. State 43, 1059-1071 (2001). [CrossRef]
  6. F. Meier and B. P. Zakharchenya, Eds., Optical Orientation (Elsevier, Amsterdam, 1984).
  7. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984).
  8. V. V. Obukhovsky and A. V. Stoyanov, �??A model of photogalvanic centers in crystals,�?? Ferroelectrics 43, 137-141 (1982). [CrossRef]
  9. B. I. Sturman and V. M. Fridkin, Photovoltaic and Photorefractive Effects in Noncentrosymmetric Materials (Gordon Breach, Philadelphia, Pa., 1992).
  10. D. H. Auston, A. M. Glass, and A. A. Ballman, �??Optical rectification by impurities in polar crystals,�?? Phys. Rev. Lett. 28, 897-900 (1972). [CrossRef]
  11. D. H. Auston and A. M.Glass, �??Optical generation of intense picosecond electrical pulses,�?? Appl. Phys. Lett. 20, 398-399 (1972). [CrossRef]

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