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Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 17, Iss. 10 — May. 11, 2009
  • pp: 7800–7806
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The Bragg gap vanishing phenomena in one-dimensional photonic crystals

Hui Zhang, Xi Chen, Youquan Li, Yunqi Fu, and Naichang Yuan  »View Author Affiliations


Optics Express, Vol. 17, Issue 10, pp. 7800-7806 (2009)
http://dx.doi.org/10.1364/OE.17.007800


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Abstract

We theoretically deduce the Bragg gap vanishing conditions in one-dimensional photonic crystals and experimentally demonstrate the m=0 band-gap vanishing phenomena at microwave frequencies. In the case of mismatched impedance, the Bragg gap will vanish as long as the discrete modes appear in photonic crystals containing dispersive materials, while for the matched impedance cases, Bragg gaps will always disappear. The experimental results and the simulations agree extremely well with the theoretical expectation.

© 2009 Optical Society of America

1. Introduction

Recently, the study of electromagnetic properties of photonic crystals has attracted a great deal of attention. The emergence of left-handed materials (LHMs) [1–2

1. V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of permittivity and permeability,” Sov. Phys. Usp. 10, 509–514 (1968). [CrossRef]

] introduces many unique photonic band-gaps (PBG) to photonic crystals. LHMs, a class of artificially designed and structured materials with simultaneously negative permittivity and permeability, exhibit many different physical properties [3

3. J. B. Pendry, “Electromagnetic materials enter the negative age,” Physics World. 14, 47 (2001).

, 4

4. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental Verification of a Negative Index of Refraction,” Science 292, 77–79 (2001). [CrossRef] [PubMed]

] from the conventional right-handed materials (RHMs). The PBG of photonic crystals containing LHMs or RHMs generally originates from the Bragg scattering mechanism [5

5. M. W. Feise, I. V. Shadrivov, and Y. S. Kivshar, “Tunable transmission and bistability in left-handed band-gap structures,” Appl. Phys. Lett. 85, 1451 (2004). [CrossRef]

]. The Bragg gaps can be identified through the index m in the Bragg conditions

ψperoid=ψ1+ψ2=
(1)

where we suppose that photonic crystals are staked by alternating layers of two materials. The parameter ψperoid denotes the phase shift of a period, ψ 1 and ψ 2 respectively denote the phase shifts of two inclusion layers in a period, and m is the band-gap index which is integer, including zero, negative and positive numbers. The photonic crystals stacked by RHMs will only lead to the positive modes (m>0) of Bragg gaps, while the multilayered LHMs can just possess the negative modes (m<0) of Bragg gaps. However, the zero modes (m=0) of Bragg gaps can just exist in the alternating layers of LHMs and RHMs.

In this paper, we study the Bragg gap vanishing phenomena in one-dimensional photonic crystals containing LHMs or RHMs. If the Bragg gap is expected to appear, the vanishing conditions should be avoided. However, in some cases the Bragg gap is needed to be closed up to propagate the electromagnetic wave, for example, in the multilayered radome for high antenna gain [6

6. S. Enoch, G. Tayeb, P. Sabouroux, N. Guerin, and P. Vincent, “A Metamaterial for Directive Emission,” Phys. Rev. Lett. 89, 213902 (2002). [CrossRef] [PubMed]

] and the leaky wave antenna [7

7. L. Sungjoon, C. Caloz, and T. Itoh, “Metamaterial-based electronically controlled transmission-line structure as a novel leaky-wave antenna with tunable radiation angle and beamwidth,” IEEE Trans. Microwave Theory Tech. 53, 161–172 (2005). [CrossRef]

]. Thus, the Bragg gap vanishing conditions are essential guidelines on how to design Bragg gaps. Two photonic crystals containing LHMs and RHMs are fabricated to demonstrate the m=0 band-gap vanishing phenomena at microwave frequencies. The m=0 band-gap, also called the zero averaged refractive index gap (the zero- n̄ band-gap) [8–10

8. 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]

], is a special Bragg gap, because it is independent of the lattice constant. However, when the vanishing conditions are satisfied, the m=0 band-gap is completely closed up in experiment. The experimental results and the simulations agree extremely well with the theoretical expectation.

2. Bragg gap vanishing conditions

Fig. 1. A 1-D infinite periodic structure with alternating layers of two materials.

Consider a 1-D infinite periodic structure with alternating layers of two materials, as shown in Fig. 1. The parameters d 1 and d 2 are the widths of the two inclusion layers respectively, and a = d 1 + d 2 is the lattice constant. The dispersion relation [8

8. 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]

, 9

9. Y. Weng, Z. G. Wang, and H. Chen, “Band structures of one-dimensional subwavelength photonic crystals containing metamaterials,” Phys. Rev. E. 75, 046601 (2007). [CrossRef]

] can be determined using the Bloch-Floquet theorem:

cos(βa)=cos(α1d1+α2d2)12(F1F2+F2F12)sin(α1d1)sin(α2d2)
(2)

with αi=ωωiμiccosθi, Fi=εiμicosθi, cosθi=11εiμi(kcω)(i=1,2) for s-polarized waves and αi=ωωiμiccosθi, Fi=μiεicosθi, cosθi=11εiμi(kcω)(i=1,2) for p-polarized waves. Here, θ 1,2 are the angles between the propagating waves and normal to the interface in the two media, respectively. The parameter k is the parallel part of the wave vector and β is the Bloch propagation constant in the first Brillouin zone. In addition, the parameters Fi represent effective characteristic impedances of two media.

We suppose that the mth mode of Bragg gap occur at the frequency ω 0, then

ψperoidω0=(α1d1+α2d2)ω0=
(3)

In the case of matched impedance (F 1 = F 2), ∣cos(βa)∣ω 0∣ = 1 is obtained. If the constitutive materials are both nondispersive, the dispersion relation in Eq. (2) is given by

cos(βa)ω=cos(α1d1+α2d2)ω0=1
(4)

where the frequency ω is very close ω 0. Equation (4) indicates that the Bloch propagation constant β is real around ω 0, thus the mth gap is closed up. If one of the constitutive materials or both are dispersive, then

F1F2ωF1F2ω0=1
(5)

which make the dispersion relation in Eq. (4) valid around ω 0. This behavior exhibits the disappearance of the mth gap.

In a mismatched impedance case (F 1F 2), Eq. (2) at ω 0 becomes

cos(βa)ω0=1+12(F1F2+F2F12)sin2(α1d1)1
(6)

Except for the discrete solutions ψ 1 = α 1 d 1 = and ψ 2 = α 2 d 2 = (m - l)π, where l and m are integral numbers, Eq. (6) indicates that the Bloch propagation constant β has no real solutions around ω 0, which renders the appearance of the mth gap. These discrete solutions for the non-dispersive media will exhibit singular frequency propagations with no sidelobes in the Bragg gaps [8

8. 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]

]. For the dispersive materials, the discrete solutions make ∣cos(βa)∣ω 0∣ = 1 valid and the approximate relation at ω is obtained

α1d1ωdα1ω0d1(ωω0)+A+
α2d2ωdα2ω0d2(ωω0)+(ml)πB+(ml)π
(7)

where A and B are infinitesimal for ω very close to ω 0. Thus the dispersion relation in Eq. (2) becomes

cos(βa)ω112[A2+B2+(F1F2+F2F1)AB]<112(A+B)2
(8)

which implies that the Bloch propagation constant β is real around ω 0 for ∣cos(βa)∣ω∣<1. In other words, if one of two materials or both are dispersive, the passband of discrete modes will be enlarged to cover the Bragg gaps (the band-gaps entirely disappear).

Consequently, if the constitutive materials of photonic crystals have different impedances, Bragg gaps will vanish as long as the discrete modes appear in photonic crystals containing dispersive materials, while for matched impedance case, Bragg gaps will always disappear.

3. Experiments of the m=0 band-gap vanishing phenomena

To experimentally demonstrate the Bragg gap vanishing phenomena, we respectively fabricate two photonic crystals containing LHMs and RHMs for mismatched/matched impedance cases. LHMs do not exist in nature, but the artificial structures have been realized by using periodic structures. Two main approaches to realize LHMs have been reported: resonant structures made of the array of wires and split-ring resonators [2

2. D. R. Smith, W. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “A composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000). [CrossRef] [PubMed]

] and non-resonant transmission line (TL) structures made of lumped elements [11

11. C. Caloz and T. Itoh, “Transmission line approach of Left-Handed (LH) materials and microstrip implementation of an artificial LH transmission Line,” IEEE Trans. Antennas Propag. 52, 1159 (2004). [CrossRef]

, 12

12. G. V. Eleftheriades, A.K. Iyer, and P.C. Kremer,”Planar negative refractive index media using periodically L-C loaded transmission lines,” IEEE Trans. Microwave Theory Tech. 50, 2702–2712 (2002). [CrossRef]

]. In particular, the TL method to LHMs, which is known as composite right/left-handed transmission line (CRLH-TL), presents the advantage of lower losses over a broader bandwidth [13

13. A. Sanada, C. Caloz, and T. Itoh, “Characteristics of the composite right/left-handed transmission lines,” IEEE Microwave Wireless Components Lett. 14, 68–71, 2004. [CrossRef]

]. The two photonic crystals, which are implemented by TL method, are both made on a Teflon substrate of the thickness h=0.5 mm, relative permittivity εr =2.65 and relative permeability μr = 1.

Fig. 2. The circuit model of (a) CRLH-TL unit by loading series capacitors CL and shunt inductors LL on the transmission line and (b) CRH unit by loading shunt capacitors CR and series inductors LR on the transmission line.

One photonic crystal for the mismatched impedance case is implemented by the proposed CRLH-TL and composite right-handed transmission line (CRH-TL). The equivalent circuit model of CRLH-TL is shown in Fig. 2 (a). The host microstrip transmission line of the CRLH-TL is designed with the width w1=1.37 mm for Z 1 = 50Ω and the length d1=6 mm, and the lumped-element components are chosen as CL =3.3 pF and LL = 8.2 nH. In Fig. 2 (b) , the host transmission line of the CRH-TL are designed with the width w2 = 5mm for Z 2 = 18Ω and the length d2=7 mm, and the lumped-element components are chosen as CR =9.1 pF and LR =2.7 nH. The CRH-TL structures are used here to realize RH phase shift ψ 2 = π with extraordinary shorter length of the transmission line.

In the lossless case, the ABCD matrix of the CRLH-TL and the CRH-TL can be obtained from the equivalent circuit models in Fig. 2 and the resulting dispersion relations [11

11. C. Caloz and T. Itoh, “Transmission line approach of Left-Handed (LH) materials and microstrip implementation of an artificial LH transmission Line,” IEEE Trans. Antennas Propag. 52, 1159 (2004). [CrossRef]

, 12

12. G. V. Eleftheriades, A.K. Iyer, and P.C. Kremer,”Planar negative refractive index media using periodically L-C loaded transmission lines,” IEEE Trans. Microwave Theory Tech. 50, 2702–2712 (2002). [CrossRef]

] are determined

cos(βCRLHd1)=cos(k1d1)(114ω2LLCL)+sin(k1d1)(12ωCLZ1+Z12ωLL)14ω2LLCL
(9)
cos(βCRHd2)=cos(k2d2)(1ω2LRCR4)+sin(k2d2)(ωCRZ22+ωLR2Z2)ω2LRCR4
(10)

where ki and Zi; (i.e. i=1, 2) are the wave numbers and the characteristic impedances of the host transmission lines for the CRLH-TL and the CRH-TL, respectively. The parameters βCRLH and βCRH are the Bloch propagation constants of the CRLH-TL and the CRH-TL, respectively.

Fig. 3. The calculated dispersion relations of the CRLH-TL and the CRH-TL.

According to Eqs. (9) and (10), the calculated dispersion characteristics of the CRLH-TL and the CRH-TL are shown in Fig. 3. The CRLH-TL exhibits the LH passband in the lower frequency range, while the RH transmission properties are dominant in the upper frequency range. The LH property is attributed to shunt inductors LL and series capacitors CL. As the balanced condition Z1=LL/CL=50Ω is satisfied, there are no gaps between the LH passband and the RH passband, and the transition frequency is about 2.25 GHz. The dispersion diagram of the CRH-TL only shows a RH passband in the lower frequency range.

Fig. 4. Photograph of the fabricated periodic structure (CRLH3CRH3)7.

In Fig. 3, the LH passband of the CRLH-TL intersects the RH passband of the CRH-TL at 0.84 GHz with the Bloch phase shift 60°. Therefore, by cascading three CRLH-TL cells for ψ 1 = -π and three CRH-TL cells for ψ 2 = π as a period (CRLH3CRH3), the periodic structure will exhibit a discrete mode in the m=0 gap at 0.84 GHz. A photograph of the fabricated (CRLH3CRH3)7 using real lumped-element components is illustrated in Fig. 4, where the subscript “7” represents the period number. This periodic structure is a typical photonic crystal, because they are made of two different effectively homogeneous media (the average electrical length of a unit cell CRLH or CRH is much smaller than the guide wavelength λg in the certain range of frequencies).

Fig. 5. The simulated and measured results of (a) the transmission properties of the photonic crystal (CRLH3CRH3)7 (b) the simulated unit cell phase delay of CRLH3 (-ψ 1), CRH3(-ψ 2) and a period CRLH3CRH3 (-ψ).
Fig. 6. The calculated dispersion relation of infinite periodic structure CRLH3CRH3-TL

The photonic crystal is simulated by using the advanced design systems (ADS) of agilent and measured by the vector network analyzer. Figure 5(b) draws the simulated phase delay of effective media CRLH3 (-ψ 1) and CRH3 (-ψ 2) compared with a period CRLH3CRH3 (-ψ). The zero phase shifts of CRLH-TL and CRH-TL occur respectively at the balanced point (2.25 GHz) and zero frequency point (0 GHz). Therefore, at the frequency 0.84 GHz, the phase shift of unit cell CRLH3 happens to be ψ 1 =-π, while CRH3 is ψ 2 = π, corresponding to the phase shift of a period CRLH3CRH3 ψ to be zero. Figure 5(a) shows the simulated and measured transmission properties of the photonic crystals (CRLH3CRH3)7. Both the experimental results and the simulation show that there is a broad passband around the frequency of 0.84 GHz. The calculated dispersion relation of infinite periodic structure CRLH3CRH3-TL is provided in Fig. 6 and the m=0 band-gap is entirely closed up. This behaviors can be explained by that the dispersive property of CRLH-TL and CRH-TL enlarge the passband of discrete modes to cover the m=0 band-gap.

Two neighboring Bragg gaps are also observed in both the transmission spectrum (Fig. 5) and the dispersion curve (Fig. 6). The Bragg gap at the frequency 0.59 GHz is in the effective left-handed passband, corresponding to the phase shift ψ = -π for the band-gap index m=-1, while the band-gap at the frequency 1.20 GHz is in the effective right-handed passband, corresponding to the phase shift ψ = π for the index m=+1.

Another photonic crystal is a periodic structure in the matched impedance case, as shown in Fig. 7(a). When the balanced CRLH-TL is cascaded with a matched RH-TL in a period, the transmission line can be adjusted together for RHMs, while lumped elements (series capacitors and shunt inductors) can be approximately considered as purely LHMs. The circuit model of the LH unit cell and RH-TL are provided in Figs. 7(b) and 7(c). The RH-TL is a section of microstrip transmission line with the width wL=137 mm for Z 1 = 50Ω and length d1=6 mm, whereas the LH unit cell is designed with series capacitors CL =3.3 pF and shunt inductors LL =8.2 nH for the balanced condition Z1=L/C=50Ω. Therefore, the two materials are with matched impedances but inverse refractive index, and LHMs are dispersive.

Fig. 7. The photonic crystal in the matched impedance case. (a) Photograph of the fabricated periodic structure (RH/LH)12, (b)the circuit model of LH unit with the loading lumped elements: series capacitors CL and shunt inductors LL , and (c) the circuit model of RH-TL.
Fig. 8. The simulated and measured transmission properties of the photonic crystals (LH/RH)12, and the simulated unit cell phase delay of LH/RH (-ψ) with dash line.

Figure 8 shows the simulated and measured transmission properties of the photonic crystal (LH/RH)12, as well as the simulated phase delay of a period (-ψperoid). It is seen that there is no gap around the frequencies 2.25 GHz with the phase shift ψperoid = 0. The experimental results and the simulations agree extremely well with the theoretical expectation for the matched impedance case.

4. Conclusion

To summarize, we have studied the Bragg gap vanishing phenomenon in detail. The vanishing conditions, as a guideline to design Bragg gaps, are applicable to one-dimensional photonic crystals of two materials which include LHMs or RHMs. Although the Bragg gap vanishing phenomena are experimentally observed in microwave frequency, the vanishing conditions are also suitable to infrared or optic bands. For the mismatched cases, as long as discrete modes of the Bragg gap appear in photonic crystals containing dispersive materials, the gap will vanish, while for the matched impedance cases, Bragg gaps will always vanish.

Acknowledgments

This work was partially supported by the National Science Foundation of China (Grant No.60871069) and Aeronautical Science Foundation of China (Grant No.20070188002).

References and links

1.

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of permittivity and permeability,” Sov. Phys. Usp. 10, 509–514 (1968). [CrossRef]

2.

D. R. Smith, W. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “A composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84, 4184–4187 (2000). [CrossRef] [PubMed]

3.

J. B. Pendry, “Electromagnetic materials enter the negative age,” Physics World. 14, 47 (2001).

4.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental Verification of a Negative Index of Refraction,” Science 292, 77–79 (2001). [CrossRef] [PubMed]

5.

M. W. Feise, I. V. Shadrivov, and Y. S. Kivshar, “Tunable transmission and bistability in left-handed band-gap structures,” Appl. Phys. Lett. 85, 1451 (2004). [CrossRef]

6.

S. Enoch, G. Tayeb, P. Sabouroux, N. Guerin, and P. Vincent, “A Metamaterial for Directive Emission,” Phys. Rev. Lett. 89, 213902 (2002). [CrossRef] [PubMed]

7.

L. Sungjoon, C. Caloz, and T. Itoh, “Metamaterial-based electronically controlled transmission-line structure as a novel leaky-wave antenna with tunable radiation angle and beamwidth,” IEEE Trans. Microwave Theory Tech. 53, 161–172 (2005). [CrossRef]

8.

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]

9.

Y. Weng, Z. G. Wang, and H. Chen, “Band structures of one-dimensional subwavelength photonic crystals containing metamaterials,” Phys. Rev. E. 75, 046601 (2007). [CrossRef]

10.

H. T. Jiang, H. Chen, H. Q. Li, Y. W. Zhang, and S. Y. Zhu, “Omnidirectional gap and defect mode of one-dimensional photonic crystals containing negative-index materials,” Appl. Phys. Lett. 83, 5386 (2003). [CrossRef]

11.

C. Caloz and T. Itoh, “Transmission line approach of Left-Handed (LH) materials and microstrip implementation of an artificial LH transmission Line,” IEEE Trans. Antennas Propag. 52, 1159 (2004). [CrossRef]

12.

G. V. Eleftheriades, A.K. Iyer, and P.C. Kremer,”Planar negative refractive index media using periodically L-C loaded transmission lines,” IEEE Trans. Microwave Theory Tech. 50, 2702–2712 (2002). [CrossRef]

13.

A. Sanada, C. Caloz, and T. Itoh, “Characteristics of the composite right/left-handed transmission lines,” IEEE Microwave Wireless Components Lett. 14, 68–71, 2004. [CrossRef]

OCIS Codes
(230.1480) Optical devices : Bragg reflectors
(350.3618) Other areas of optics : Left-handed materials
(160.5298) Materials : Photonic crystals

ToC Category:
Photonic Crystals

History
Original Manuscript: March 12, 2009
Revised Manuscript: April 10, 2009
Manuscript Accepted: April 16, 2009
Published: April 27, 2009

Citation
Hui Zhang, Xi Chen, Youquan Li, Yunqi Fu, and Naichang Yuan, "The Bragg gap vanishing phenomena in one-dimensional photonic crystals," Opt. Express 17, 7800-7806 (2009)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-10-7800


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References

  1. V. G. Veselago, "The electrodynamics of substances with simultaneously negative values of permittivity and permeability," Sov. Phys. Usp. 10, 509-514 (1968). [CrossRef]
  2. D. R. Smith, W. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, "A composite medium with simultaneously negative permeability and permittivity," Phys. Rev. Lett. 84, 4184-4187 (2000). [CrossRef] [PubMed]
  3. J. B. Pendry, "Electromagnetic materials enter the negative age," Physics World. 14, 47 (2001).
  4. R. A. Shelby, D. R. Smith, and S. Schultz, "Experimental Verification of a Negative Index of Refraction," Science 292, 77-79 (2001). [CrossRef] [PubMed]
  5. M. W. Feise, I. V. Shadrivov, and Y. S. Kivshar, "Tunable transmission and bistability in left-handed band-gap structures," Appl. Phys. Lett. 85, 1451 (2004). [CrossRef]
  6. S. Enoch, G. Tayeb, P. Sabouroux, N. Guerin, and P. Vincent, "A Metamaterial for Directive Emission," Phys. Rev. Lett. 89, 213902 (2002). [CrossRef] [PubMed]
  7. L. Sungjoon; C. Caloz, T. Itoh, "Metamaterial-based electronically controlled transmission-line structure as a novel leaky-wave antenna with tunable radiation angle and beamwidth," IEEE Trans. Microwave Theory Tech. 53, 161-172 (2005). [CrossRef]
  8. 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]
  9. Y. Weng, Z. G. Wang, and H. Chen, "Band structures of one-dimensional subwavelength photonic crystals containing metamaterials," Phys. Rev. E. 75, 046601 (2007). [CrossRef]
  10. H. T. Jiang, H. Chen, H. Q. Li, Y. W. Zhang, and S. Y. Zhu, "Omnidirectional gap and defect mode of one-dimensional photonic crystals containing negative-index materials," Appl. Phys. Lett. 83, 5386 (2003). [CrossRef]
  11. C. Caloz and T. Itoh, "Transmission line approach of Left-Handed (LH) materials and microstrip implementation of an artificial LH transmission Line," IEEE Trans. Antennas Propag. 52, 1159 (2004). [CrossRef]
  12. G. V. Eleftheriades, A.K. Iyer and P.C. Kremer,"Planar negative refractive index media using periodically L-C loaded transmission lines," IEEE Trans. Microwave Theory Tech. 50, 2702-2712 (2002). [CrossRef]
  13. A. Sanada, C. Caloz, and T. Itoh, "Characteristics of the composite right/left-handed transmission lines," IEEE Microwave Wireless Components Lett. 14, 68-71, 2004. [CrossRef]

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