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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 16, Iss. 5 — Mar. 3, 2008
  • pp: 3299–3304
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Bloch oscillations in chirped layered structures with metamaterials

Arthur R. Davoyan, Ilya V. Shadrivov, Andrey A. Sukhorukov, and Yuri S. Kivshar  »View Author Affiliations


Optics Express, Vol. 16, Issue 5, pp. 3299-3304 (2008)
http://dx.doi.org/10.1364/OE.16.003299


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Abstract

We analyze the Bloch oscillations of electromagnetic waves in chirped layered structures with alternating layers of negative-index metamaterial and conventional dielectric under the condition of the zero average refractive index. We consider the case when the chirp is introduced by varying the thickness of the layers linearly across the structure. We demonstrate that such structures can support three different types of the Bloch oscillations for electromagnetic waves associated with either propagating or evanescent guided modes. In particular, we predict a novel type of the Bloch oscillations associated with coupling between surface waves excited at the interfaces separating the layers of negative-index metamaterial and the layers of the conventional dielectric.

© 2008 Optical Society of America

1. Introduction

Periodic oscillations of electrons induced by a constant electric field were predicted by Bloch in 1928 [1

1. F. Bloch, “Uber die Quantenmechanik der Elektronen in Kristallgittern,” Z. Phys. A: Hadrons Nucl. 52, 555–600 (1928).

]. Such Bloch oscillations become possible due to beating of the localized eigenmodes of the structure corresponding to the equidistant eigenstates of the spectrum known as the Wannier-Stark ladder [2

2. G. H. Wannier, “Wave functions and effective Hamiltonian for Bloch electrons in an electric field,” Phys. Rev. 117, 432–439 (1960). [CrossRef]

]. Experimental verification of the theory was impossible at that time, since dephasing time of electrons in crystals is shorter than the period of the electron Bloch oscillation. Later, the electron Bloch oscillations were observed in semiconductor superlattices [3

3. C. Waschke, H. G. Roskos, R. Schwedler, K. Leo, H. Kurz, and K. Kohler, “Coherent submillimeter-wave emission from Bloch oscillations in a semiconductor superlattice,” Phys. Rev. Lett. 70, 3319–3322 (1993). [CrossRef] [PubMed]

] for which the periodwas reduced due to mini-bands of fabricated periodic structures.

Dephasing processes for electromagnetic waves are negligible, and this makes the observation of the Bloch oscillations in photonic systems much easier. The experimental observations of opticalWannier-Stark ladder and Bloch oscillations of electromagnetic waves were reported in Refs. [4

4. C. M. de Sterke, J. N. Bright, P. A. Krug, and T. E. Hammon, “Observation of an optical Wannier-Stark ladder,” Phys. Rev. E 57, 2365–2370 (1998). [CrossRef]

, 5

5. R. Morandotti, U. Peschel, J. S. Aitchinson, H. S. Eisenberg, and Y. Silberberg, “Experimental observation of linear and nonlinear optical Bloch oscillations,” Phys. Rev. Lett. 83, 4756–4759 (1999). [CrossRef]

, 6

6. T. Pertsch, P. Dannberg, W. Elflein, A. Brauer, and F. Lederer, “Optical Bloch Oscillations in Temperature Tuned Waveguide Arrays,” Phys. Rev. Lett. 83, 4752–4755 (1999). [CrossRef]

] for linearly chirped Bragg gratings and linearly growing effective refractive index. Later, several groups reported both theoretical and experimental studies of the Bloch oscillations of electromagnetic waves in various photonic structures [7

7. V. Agarwal, J. A. del Rio, G. Malpuech, M. Zamfirescu, A. Kavokin, D. Coquillat, D. Scalbert, M. Vladimirova, and B. Gil, “Photon Bloch Oscillations in Porous Silicon Optical Superlattices,” Phys. Rev. Lett. 92, 097401 (2004). [CrossRef] [PubMed]

, 8

8. P. B. Wilkinson, “Photonic Bloch oscillations and Wannier-Stark ladders in exponentially chirped Bragg gratings,” Phys. Rev. E 65, 056616 (2002). [CrossRef]

, 9

9. R. Sapienza, P. Costantino, D. Wiersma, M. Ghulinyan, C. J. Oton, and L. Pavesi, “Optical Analogue of Electronic Bloch Oscillations,” Phys. Rev. Lett. 91, 263902 (2003). [CrossRef]

, 10

10. G. Lenz, I. Talanina, and C. M. de Sterke, “Bloch Oscillations in an Array of Curved Optical Waveguides,” Phys. Rev. Lett. 83, 963–966 (1999). [CrossRef]

].

Recent experimental realization of negative-index (or left-handed) metamaterials [11

11. D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and Negative Refractive Index,” Science 305, 788–792 (2004). [CrossRef] [PubMed]

] has opened up many unique opportunities to explore novel effects of the electromagnetic wave propagation in the structures with negative refractive index. In this paper we study, for the first time to our knowledge, the Bloch oscillations of electromagnetic waves in linearly chirped layered structures composed of alternating layers of negative-index and conventional dielectric materials. We choose the material parameters in such a way that the average refractive index across pair of the neighboring layers vanishes, thus fulfilling the condition for the existence of a novel type of the specific zero- bandgap [12

12. J. Li, L. Zhou, C. T. Chan, and P. Sheng, “Photonic Band Gap from a Stack of Positive and Negative Index Materials,” Phys. Rev. Lett. 90, 083901 (2003). [CrossRef] [PubMed]

, 13

13. I. V. Shadrivov, A. A. Sukhorukov, and Yu. S. Kivshar, “Beam shaping by a periodic structure with negative refraction,” Appl. Phys. Lett. 82, 3820–3822 (2003). [CrossRef]

]. To model the chirp, we change the layer thickness linearly in the structure and observe an analogue of the Wannier-Stark ladder in the eigenmode spectrum of electromagnetic waves, and the corresponding Bloch oscillations in the resonant transmission bands.We reveal that in such layered structures the Bloch oscillations of electromagnetic waves can be observed in the three different regimes. Compared to the optical Bloch oscillations in conventional dielectric structures, the structures with metamaterial can support a novel type of the Bloch oscillations associated with coupling of surface waves at the interfaces between negative-index and conventional dielectric layers.

2. Structure and bandgap diagram

We study one-dimensional chirped layered structures shown schematically in Fig. 1, where the slabs of a negative-index metamaterial (with the width bi) are separated by the layers of the conventional dielectric (with the width ai). Variation of the refractive index in the i-th pair of layers can be described as follows:

n(z)={nr=εrμrz(zi,zi+ai)nl=εlμlz(zi+ai,zi+Λi)
(1)

where nl and nr are the refractive indices of metamaterial and dielectric, respectively.We consider TE-polarized waves with the electric field having one component E=(Ex,0,0), and the waves propagating in the plane (y, z). In this case, the field distribution can be described by the scaler Helmholtz equation:

Fig. 1. Schematic of linearly chirped one-dimensional structure with alternating layers of negative-index metamaterial and conventional dielectric.
Δ2Ex(y,z)+n2(z)Ex(y,z)1μdμ(z)dzEx(y,z)z=0,
(2)

where Δ2 is the two-dimensional Laplacian, and the coordinates are normalized to c/ω.

First, we assume that the structure is periodic, i.e. no linear chirp is introduced. In this case, the electric field can be represented as a superposition of Bloch eigenmodes [14

14. P. Yeh, Optical Waves in Layered Media (John Wiley & Sons, New York, 1988).

], with the electric field envelopes U(z+Λ)=U(z), where Λ is the structure period. The dispersion relation for the Bloch waves can be found by the transfer matrix method [14

14. P. Yeh, Optical Waves in Layered Media (John Wiley & Sons, New York, 1988).

],

2cos(KBΛ)=2cos(kzra)cos(kzlb)(kzlμrkzrμl+kzrμlkzlμr)sin(kzra)sin(kzlb),
(3)

where KB is the Bloch wavenumber, kzl,zr=nl,r2ky2 and ky is the normalized propagation constant along the y axis. According to this relation an infinite layered structure containing metamaterials exhibits a non-resonant gap for ≡Λ-1Λ 0 n(z)dz=0, where is the average refractive index of the structure, i.e. nra=|nlb|. This condition is easy to fulfil for the metamaterials with the negative refractive index.

The bandgap diagram of the layered structure is shown in Fig. 2 for the parameter plane (Λ,ky). Here we assume that dielectric is vacuum, εr=µr=1, and that it is two times thicker than the second layer, a/b=2. We choose the parameters of the left-handed media as follows: εl=-5 and µl=-0.8. This set of parameters allows surface waves to exist at the interfaces between metamaterial and vacuum [15

15. I. V. Shadrivov, A. A. Sukhorukov, Yu. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, “Nonlinear surface waves in left-handed materials,” Phys. Rev. E 69, 016617 (2004). [CrossRef]

]. As follows from Fig. 2, for the zero structure the bandgap spectrum differs substantially from the case of conventional periodic structures made of conventional dielectrics [14

14. P. Yeh, Optical Waves in Layered Media (John Wiley & Sons, New York, 1988).

]. Stack with the average zero refractive index possesses a complete gap with the transmission resonances [12

12. J. Li, L. Zhou, C. T. Chan, and P. Sheng, “Photonic Band Gap from a Stack of Positive and Negative Index Materials,” Phys. Rev. Lett. 90, 083901 (2003). [CrossRef] [PubMed]

, 13

13. I. V. Shadrivov, A. A. Sukhorukov, and Yu. S. Kivshar, “Beam shaping by a periodic structure with negative refraction,” Appl. Phys. Lett. 82, 3820–3822 (2003). [CrossRef]

] when the optical path of the wave in either layer of the period coincides with a half of the wavelength in the corresponding medium. Thus for the normal incidence (ky=0) transmission is observed only when nra=|nlb|=πm, where m is integer. For the slabs of equal thickness the regions of the transmission resonances in (Λ,ky) plane degenerate into infinitely thin lines.

Fig. 2. Bandgap diagram for the TE-polarized waves in structure with alternating dielectric (εr=µr=1) and metamaterial (εl=-5, µl=-0.8) in case of zero average refractive index. Black and white areas correspond to gaps and bands, respectively. Two spectra of the excited Bloch oscillations are shown on the left. The inset shows a magnified part of the spectrum.

3. Three types of Bloch oscillations

We study the propagation of electromagnetic waves in the layered structure shown in Fig. 1 with zero average refractive index in each pair of layers.We assume that the thickness of layers is chirped linearly, i.e. Λq0+Λ, where integer q numerates the layers. We are looking for localized electromagnetic waves in the structure, and for numerical simulations we consider a finite stack of the layers with perfect metal boundary conditions, E(z=0)=E(z=L)=0, where L is the total length of the structure. We start from the Gaussian field distribution in the plane y=0 across the layers, and in order to find the electromagnetic field distribution in the whole stack we look for its eigenmodes by solving the Helmholtz equation (2). Then we decompose the initial field using the eigenmode basis and find the solution in the structure.

To find the eigenmodes of Eq. (2), we employ the following discretization scheme [16

16. A. A. Sukhorukov, I. V. Shadrivov, and Yu. S. Kivshar, “Wave scattering by metamaterial wedges and interfaces,” Int. J. Num. Model. 19, 105–117 (2006). [CrossRef]

]:

2μm11+μm+11[Um+1Umμm+1UmUm1μm1]1h2+εm1+εm+1μm11+μm+11Um=ky2Um,
(4)

where zm=mh are the mesh points with the discretization step h and Ex(y,z)=U(z)ejkyy . Such a discretization scheme provides an algorithm convergence [16

16. A. A. Sukhorukov, I. V. Shadrivov, and Yu. S. Kivshar, “Wave scattering by metamaterial wedges and interfaces,” Int. J. Num. Model. 19, 105–117 (2006). [CrossRef]

], and it avoids excitation of spurious modes in the structure.

In negative-index metamaterials the energy flow, defined as ∫[E×H]dz, can become negative, i.e. the energy can propagate in the opposite direction to the propagation constant k y [17

17. V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of epsilon and mu,” Usp. Fiz. Nauk92, 517–526 (1967) (in Russian) [English translation: Sov. Phys. Usp. 10, 509 (1968).

]. Consequently, we determine the direction of the energy flow of each eigenmode and choose the sign of the propagation constant such that the energy goes in the positive y-direction. Decomposition of the initial condition in the plane y=0 in the eigenmode basis is made using the least squares method.

To find the values of the propagation constant ky which correspond to the Bloch oscillations, we analyze the spectrum of eigenvalues of electromagnetic waves which can propagate in this layered structure. The Bloch oscillations are expected to appear in the regions of spectrum where the spectrum is equidistant. Practically for all gradients of a linear ramp we observe several sets of equidistant states. The equidistant eigenvalues of ky correspond to a spatial optical equivalent of the Wannier-Stark ladder which is associated with the Bloch oscillations.

Fig. 3. Field distribution in the case of surface-wave-assisted Bloch oscillations. The Wannier-Stark ladder appears for the propagation constants centered around k y0=2.47, normalized period is Ly=820.
Fig. 4. Field distribution for the case of guided waves. The Wannier-Stark ladder appears for the propagation constants centered around k y0=1.34, period is Ly≅100.

Spectrum of ky can be divided into three different regions. First, when ky<nr<|nl|, electromagnetic waves propagate in both left- and right-handed materials. In the second region, nr<ky<|nl|, waves propagate in metamaterial only being evanescent in the vacuum layers. In this regime, our structure can be considered as an array of coupled left-handed waveguides. When ky>|nl>|nr, only surface waves may propagate along the different interfaces.

We find that the Bloch oscillations can be observed in all three regimes of the wave propagation when the corresponding set of the equidistant values of the propagation constant is excited. For definiteness, we consider a stack containing 36 pairs of negative-index metamaterial and conventional dielectric layeres and the normalized period Λ varying from 3.7 to 6. First, we excite the eigenstates corresponding to the regime of surface waves with the center of the spectrum at k y0=2.47. Figure 3 presents the intensity distribution for the electric field which shows clearly spatially periodic oscillations of the beam position in the chirped layered structure. The corresponding spectrum of eigenmodes is shown on the left side of Fig. 2. We note that the beam reconstructs its shape after each period of oscillations. The field is highly confined to the interfaces between metamaterial and vacuum, demonstrating that such Bloch oscillations appear solely due to the coupling between surface waves exited at different interfaces of the structure. The distance between the Wannier-Stark eigenstates Δky defines the period of oscillations, Ly=2πky. For this case, we find Ly=820, and this value agrees well with Fig. 3.

Fig. 5. Field distribution for the case when waves propagate in both type of materials. The Wannier-Stark ladder appears with average propagation constant k y0=0.8, period of oscillations is Ly=210.

Example of the Bloch oscillations of the beam with the spectrum corresponding to the coupled waveguide regime, nr<ky<|nl|, are shown in Fig. 4. The equidistant spectrum of eigenstates corresponding to the Wannier-Stark ladder is also shown in Fig. 2 (top, left). We notice that oscillations are strongly anharmonic, but they are still periodic with the period defined well by the relation Ly=2πky; this period is less than the period of the Bloch oscillations associated with surface waves.

4. Conclusions

We have studied the propagation of electromagneticwaves in chirped layered structures with alternating layers of negative-index (left-handed) metamaterials and conventional dielectric with the zero average refractive index. We have revealed that linearly chirped structures the excitation spectra are equidistant manifesting a similarity with the so-called optical Wannier-Stark ladder. Using direct numerical numerical simulations, we have demonstrated that such chirped structures can support three types of Bloch oscillations for electromagnetic waves, including a novel type of unusual Bloch oscillations associated with the coupling of surface modes.

Acknowledgements

This work has been supported by the Australian Research Council through the Discovery and Federation Fellowship research projects.

References and links

1.

F. Bloch, “Uber die Quantenmechanik der Elektronen in Kristallgittern,” Z. Phys. A: Hadrons Nucl. 52, 555–600 (1928).

2.

G. H. Wannier, “Wave functions and effective Hamiltonian for Bloch electrons in an electric field,” Phys. Rev. 117, 432–439 (1960). [CrossRef]

3.

C. Waschke, H. G. Roskos, R. Schwedler, K. Leo, H. Kurz, and K. Kohler, “Coherent submillimeter-wave emission from Bloch oscillations in a semiconductor superlattice,” Phys. Rev. Lett. 70, 3319–3322 (1993). [CrossRef] [PubMed]

4.

C. M. de Sterke, J. N. Bright, P. A. Krug, and T. E. Hammon, “Observation of an optical Wannier-Stark ladder,” Phys. Rev. E 57, 2365–2370 (1998). [CrossRef]

5.

R. Morandotti, U. Peschel, J. S. Aitchinson, H. S. Eisenberg, and Y. Silberberg, “Experimental observation of linear and nonlinear optical Bloch oscillations,” Phys. Rev. Lett. 83, 4756–4759 (1999). [CrossRef]

6.

T. Pertsch, P. Dannberg, W. Elflein, A. Brauer, and F. Lederer, “Optical Bloch Oscillations in Temperature Tuned Waveguide Arrays,” Phys. Rev. Lett. 83, 4752–4755 (1999). [CrossRef]

7.

V. Agarwal, J. A. del Rio, G. Malpuech, M. Zamfirescu, A. Kavokin, D. Coquillat, D. Scalbert, M. Vladimirova, and B. Gil, “Photon Bloch Oscillations in Porous Silicon Optical Superlattices,” Phys. Rev. Lett. 92, 097401 (2004). [CrossRef] [PubMed]

8.

P. B. Wilkinson, “Photonic Bloch oscillations and Wannier-Stark ladders in exponentially chirped Bragg gratings,” Phys. Rev. E 65, 056616 (2002). [CrossRef]

9.

R. Sapienza, P. Costantino, D. Wiersma, M. Ghulinyan, C. J. Oton, and L. Pavesi, “Optical Analogue of Electronic Bloch Oscillations,” Phys. Rev. Lett. 91, 263902 (2003). [CrossRef]

10.

G. Lenz, I. Talanina, and C. M. de Sterke, “Bloch Oscillations in an Array of Curved Optical Waveguides,” Phys. Rev. Lett. 83, 963–966 (1999). [CrossRef]

11.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and Negative Refractive Index,” Science 305, 788–792 (2004). [CrossRef] [PubMed]

12.

J. Li, L. Zhou, C. T. Chan, and P. Sheng, “Photonic Band Gap from a Stack of Positive and Negative Index Materials,” Phys. Rev. Lett. 90, 083901 (2003). [CrossRef] [PubMed]

13.

I. V. Shadrivov, A. A. Sukhorukov, and Yu. S. Kivshar, “Beam shaping by a periodic structure with negative refraction,” Appl. Phys. Lett. 82, 3820–3822 (2003). [CrossRef]

14.

P. Yeh, Optical Waves in Layered Media (John Wiley & Sons, New York, 1988).

15.

I. V. Shadrivov, A. A. Sukhorukov, Yu. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, “Nonlinear surface waves in left-handed materials,” Phys. Rev. E 69, 016617 (2004). [CrossRef]

16.

A. A. Sukhorukov, I. V. Shadrivov, and Yu. S. Kivshar, “Wave scattering by metamaterial wedges and interfaces,” Int. J. Num. Model. 19, 105–117 (2006). [CrossRef]

17.

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of epsilon and mu,” Usp. Fiz. Nauk92, 517–526 (1967) (in Russian) [English translation: Sov. Phys. Usp. 10, 509 (1968).

OCIS Codes
(050.2770) Diffraction and gratings : Gratings
(350.5500) Other areas of optics : Propagation
(350.3618) Other areas of optics : Left-handed materials

ToC Category:
Metamaterials

History
Original Manuscript: December 18, 2007
Revised Manuscript: February 18, 2008
Manuscript Accepted: February 20, 2008
Published: February 26, 2008

Citation
Arthur R. Davoyan, Ilya V. Shadrivov, Andrey A. Sukhorukov, and Yuri S. Kivshar, "Bloch oscillations in chirped layered structures with metamaterials," Opt. Express 16, 3299-3304 (2008)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-5-3299


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References

  1. F. Bloch, "Uber die Quantenmechanik der Elektronen in Kristallgittern," Z. Phys. A: Hadrons Nucl. 52, 555-600 (1928).
  2. G. H. Wannier, "Wave functions and effective Hamiltonian for Bloch electrons in an electric field," Phys. Rev. 117, 432-439 (1960). [CrossRef]
  3. C. Waschke, H. G. Roskos, R. Schwedler, K. Leo, H. Kurz, and K. Kohler, "Coherent submillimeter-wave emission from Bloch oscillations in a semiconductor superlattice," Phys. Rev. Lett. 70, 3319-3322 (1993). [CrossRef] [PubMed]
  4. C. M. de Sterke, J. N. Bright, P. A. Krug, and T. E. Hammon, "Observation of an optical Wannier-Stark ladder," Phys. Rev. E 57, 2365-2370 (1998). [CrossRef]
  5. R. Morandotti, U. Peschel, J. S. Aitchinson, H. S. Eisenberg, Y. Silberberg, "Experimental observation of linear and nonlinear optical Bloch oscillations," Phys. Rev. Lett. 83, 4756-4759 (1999) [CrossRef]
  6. T. Pertsch, P. Dannberg, W. Elflein, A. Brauer, and F. Lederer, "Optical Bloch Oscillations in Temperature Tuned Waveguide Arrays," Phys. Rev. Lett. 83, 4752-4755 (1999). [CrossRef]
  7. V. Agarwal, J. A. del Rio, G. Malpuech, M. Zamfirescu, A. Kavokin, D. Coquillat, D. Scalbert, M. Vladimirova, and B. Gil, "Photon Bloch Oscillations in Porous Silicon Optical Superlattices," Phys. Rev. Lett. 92, 097401 (2004). [CrossRef] [PubMed]
  8. P. B. Wilkinson, "Photonic Bloch oscillations and Wannier-Stark ladders in exponentially chirped Bragg gratings," Phys. Rev. E 65, 056616 (2002). [CrossRef]
  9. R. Sapienza, P. Costantino, D. Wiersma, M. Ghulinyan, C. J. Oton, and L. Pavesi, "Optical Analogue of Electronic Bloch Oscillations," Phys. Rev. Lett. 91, 263902 (2003). [CrossRef]
  10. G. Lenz, I. Talanina, and C. M. de Sterke, "Bloch Oscillations in an Array of Curved Optical Waveguides," Phys. Rev. Lett. 83, 963-966 (1999). [CrossRef]
  11. D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, "Metamaterials and Negative Refractive Index," Science 305, 788-792 (2004). [CrossRef] [PubMed]
  12. J. Li, L. Zhou, C. T. Chan, and P. Sheng, "Photonic Band Gap from a Stack of Positive and Negative Index Materials," Phys. Rev. Lett. 90, 083901 (2003). [CrossRef] [PubMed]
  13. I. V. Shadrivov, A. A. Sukhorukov, and Yu. S. Kivshar, "Beam shaping by a periodic structure with negative refraction," Appl. Phys. Lett. 82, 3820-3822 (2003). [CrossRef]
  14. P. Yeh, Optical Waves in Layered Media (John Wiley & Sons, New York, 1988).
  15. I. V. Shadrivov, A. A. Sukhorukov, Yu. S. Kivshar, A. A. Zharov, A. D. Boardman, and P. Egan, "Nonlinear surface waves in left-handed materials," Phys. Rev. E 69, 016617 (2004). [CrossRef]
  16. A. A. Sukhorukov, I. V. Shadrivov, and Yu. S. Kivshar, "Wave scattering by metamaterial wedges and interfaces," Int. J. Num. Model. 19, 105-117 (2006). [CrossRef]
  17. V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of epsilon and mu," Usp. Fiz. Nauk 92, 517-526 (1967) (in Russian) [English translation: Sov. Phys. Usp. 10, 509 (1968)].

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