## Radiation of a uniformly moving line charge in a zero-index metamaterial and other periodic media |

Optics Express, Vol. 20, Issue 16, pp. 18515-18524 (2012)

http://dx.doi.org/10.1364/OE.20.018515

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### Abstract

Radiation of electromagnetic waves by a uniformly moving charge is the subject of extensive research over the last several decades. Fascinating effects such as Vavilov-Cherenkov radiation, transition radiation and the Smith-Purcell effect were discovered and studied in depth. In this letter we study the radiation of a line charge moving with relativistic constant velocity within an average zero index metamaterial consisting of periodically alternating layers with negative and positive refractive index. We observe a strong radiation enhancement, ∼3 orders of magnitude, for specific combinations of velocities and radiation frequencies. This surprising finding is attributed to a gigantic increase in the density of states at the positive/negative index boundary. Furthermore, we shed light on radiation effects of such a line charge propagating within the more “traditional” structure of periodically alternating layers consisting of positive and different refractive index with focus on frequencies satisfying the quarter wave stack and the half wave stack conditions. We show that the quarter-wave-stack case results in emission propagating vertically to the line charge trajectory, while the half-wave-stack results in negligible radiation. All these findings were obtained using a computationally efficient and conceptually intuitive computation method, based on eigenmode expansion of specific frequency components. For validation purposes this method was compared with the finite-difference-time-domain method.

© 2012 OSA

## 1. Introduction

1. V. L. Ginzburg, “Radiation by uniformly moving sources (Vavilov - Cherenkov effect, transition radiation,and other phenomena),” Usp. Fiz. Nauk **166**, 1033–1042 (1996). [CrossRef]

11. L.Sh. Khachatryan, S.R. Arzumanyan, and W. Wagner, “Self-amplified Cherenkov radiation from a relativistic electron in a waveguide partially filled with a laminated material,” JJ. Phys.: Conf. Ser. **357**, 012004 (2012). [CrossRef]

12. C. Luo, M. Ibanescu, S. G. Johnson, and J. D. Joannopoulos, “Cherenkov radiation in photonic crystals,” Science **299**(5605), 368–371 (2003). [CrossRef] [PubMed]

13. C. Kremers, D. N. Chigrin, and J. Kroha, “Theory of Cherenkov radiation in periodic dielectric media: Emission spectrum,” Phys. Rev. A **79**, 013829 (2009). [CrossRef]

14. C. Kremers and D. N. Chigrin, “Spatial distribution of Cherenkov radiation in periodic dielectric media,” J. Opt. A **11**, 114008 (2009). [CrossRef]

*ε*and

*μ*(a double negative (DNG) material) the Cherenkov radiation cone is reversed compared to the case of a medium with both

*ε*and

*μ*positive (a double positive (DPS) material) [15

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

16. J. Lu, T. Grzegorczyk, Y. Zhang, J. Pacheco Jr., B.-I. Wu, J. Kong, and M. Chen, “Cerenkov radiation in materials with negative permittivity and permeability,” Opt. Express **11**, 723–734 (2003). [CrossRef] [PubMed]

17. S. Xi, H. Chen, T. Jiang, L. Ran, J. Huangfu, B. I. Wu, J. A. Kong, and M. Chen, “Experimental Verification of Reversed Cherenkov Radiation in Left-Handed Metamaterial,” Phys. Rev. Lett. **103**, 194801 (2009). [CrossRef]

18. S. N. Galyamin, A. V. Tyukhtin, A. Kanareykin, and P. Schoessow, “Reversed Cherenkov-Transition Radiation by a Charge Crossing a Left-Handed Medium Boundary,” Phys. Rev. Lett. **103**, 194802 (2009). [CrossRef]

19. S. N. Galyamin and A. V. Tyukhtin, “Electromagnetic field of a moving charge in the presence of a left-handed medium,” Phys. Rev. B **81**, 235134 (2010). [CrossRef]

20. D. E. Fernandes, S. I. Maslovski, and M. G. Silveirinha, “Cherenkov emission in a nanowire material,” Phys. Rev. B **85**, 155107 (2012). [CrossRef]

21. V. V. Vorobev and A. V. Tyukhtin, “Nondivergent Cherenkov Radiation in a Wire Metamaterial,” Phys. Rev. Lett. **108**, 184801 (2012). [CrossRef] [PubMed]

22. Z. Mohammadi, C. P. Van Vlack, S. Hughes, J. Bornemann, and R. Gordon, “Vortex electron energy loss spectroscopy for near-field mapping of magnetic plasmons,” Opt. Express **20**, 15024–15034 (2012). [CrossRef] [PubMed]

23. M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A **12**, 1068–1076 (1995). [CrossRef]

25. P. Lalanne and G. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A **13**, 779–784 (1996). [CrossRef]

6. T. Ochiai and K. Ohtaka, “Electron energy loss and Smith-Purcell radiation in two- and three-dimensional photonic crystals,” Opt. Express **13**, 7683–7698 (2005). [CrossRef] [PubMed]

13. C. Kremers, D. N. Chigrin, and J. Kroha, “Theory of Cherenkov radiation in periodic dielectric media: Emission spectrum,” Phys. Rev. A **79**, 013829 (2009). [CrossRef]

14. C. Kremers and D. N. Chigrin, “Spatial distribution of Cherenkov radiation in periodic dielectric media,” J. Opt. A **11**, 114008 (2009). [CrossRef]

26. A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. **181**(3), 687–702 (2010). [CrossRef]

## 2. Formulation of the problem

*y*direction which is moving at a relativistic velocity along the direction of periodicity (

*z*).

*L*is the unit cell size,

*v*is the velocity of the line charge and

*ρ*is charge density per unit length (see schematic in Fig. 1). The moving charge density results in a current density

*J*in the

*z*direction given by: where

*z*is the direction of propagation. The periodic structure consists of two materials defined by their electric permittivity and magnetic permeability

*ε*and

_{i}*μ*(

_{i}*i*= 1, 2) respectively, and by their width

*L*. To calculate the electromagnetic fields that reside in this structure we first calculate the eigenmodes of the structure for the homogeneous problem (in the absence of sources) using the FMM, and afterward extract the coupling of the source to the eigenmodes.

_{i}### 2.1. Homogeneous problem

*z*≤

*L*, to represent Maxwell’s equations in an infinitely periodic structure. Afterwards, the eigenmodes and eigenvalues of the fields are calculated by solving an eigenvalue equation. We now elaborate on these principles. The Floquet-Bloch condition implies that the wavevectors in the

*z*direction are given by

*k*

_{z,m}=

*k*

_{z,0}+

*mK*, where

*K*= 2

*π*/

*L*is the grating vector and

*k*

_{z,0}is the “zero-order” term, which is discussed in more detail in the next paragraph. The FMM allows to express the fields in the periodic structure as Equation (2) can be simplified to:

*Ḏ*=

*W̳X̳C̳*is a column vector with elements corresponding to the Fourier components of the field,

*W̳*is a matrix with elements m,n (m being the Fourier order, and n being the eigenmodes index number [23

23. M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A **12**, 1068–1076 (1995). [CrossRef]

*X̳*is a diagonal matrix with elements exp(

*jk*

_{x,m}

*x*), and

*C̱*is a column vector of coupling constants. Each diffracted order obeys the pseudo-periodic boundary conditions at the two unit cell boundaries

*F*(

*x*,

*z*= 0) =

*F*(

*x*,

*z*=

*L*)exp(

*jk*

_{0}

*L*) [27

27. R. Petit, *Electromagnetic Theory of Gratings* (Topics in Applied Physics) (Springer, 1980). [CrossRef]

*H*and two electric field components

_{y}*E*and

_{x}*E*, and

_{z}*F*describes one of these fields). Using the FMM approach, it is common to solve for the out-of-plane field component (

*H*in our case), by solving numerically an eigenproblem for a truncated number of Fourier orders, resulting in finite sized matrices (2

_{y}*N*+ 1 × 2

*N*+ 1) and vectors (2

*N*+ 1 × 1), where 2

*N*+ 1 is the number of Fourier orders used in the calculation. For the TM case, the equation needed to be solved is:

23. M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A **12**, 1068–1076 (1995). [CrossRef]

28. A. Yanai, M. Orenstein, and U. Levy, “Giant resonance absorption in ultra-thin metamaterial periodic structures,” Opt. Express **20**, 3693–3702 (2012). [CrossRef] [PubMed]

*x*′ is the normalized coordinate defined by

*x*′ =

*k*

_{0}

*x*(

*k*

_{0}= 2

*π*/

*λ*

_{0}with

*λ*

_{0}being the vacuum wavelength of the electromagnetic wave), and

*E̳*,

*M̳*are the electric permittivity and magnetic permeability Toeplitz matrices respectively and

*k*

_{z,0}+

*mK*)/

*k*

_{0}. By solving the eigenvalue problem, one obtains both the eigenvector matrix and eigenvalues (

*k*/

_{x}*k*

_{0})

^{2}of the emitted electromagnetic field. What remains to be found is the coupling constants vector

*C̱*, as discussed next.

### 2.2. Inhomogeneous problem

*ω*= 2

*πc*/

*λ*

_{0}. The factor (

*ω*/

*v*) in the exponential argument in Eq. (3) may be interpreted as the spatial frequency of the current source [1

1. V. L. Ginzburg, “Radiation by uniformly moving sources (Vavilov - Cherenkov effect, transition radiation,and other phenomena),” Usp. Fiz. Nauk **166**, 1033–1042 (1996). [CrossRef]

*k*

_{z,c}=

*ω*/

*v*. In an homogeneous medium, the current source will induce an electromagnetic wave with a spatial frequency

*k*

_{z,c}along the

*z*direction, similarly to the spatial frequency of the current source itself. Since

*v*<

*c*/

*n*then

*k*is purely imaginary while for the case of

_{x}*v*>

*c*/

*n*,

*k*is real, resulting in a propagating wave. It is therefore convenient to rewrite Eq. (3) as: It is seen that the only non-zero Fourier spatial component of the source is

_{x}*k*

_{z,c}. We have already mentioned that by using Floquet-Bloch expansion the fields in the periodic structure can be described as asset of plane waves with wavevectors

*k*

_{z,m}=

*k*

_{z,0}+

*mK*. One can therefore interpret the “zeroth order” (

*m*= 0) wavevector of the diffraction problem

*k*

_{z,0}to emanate from the spatial frequency of the source

*k*

_{z,c}. The periodic structure facilitates coupling to higher orders (

*m*≠ 0). This description is in close analogy to the configuration of illuminating a periodic structure by a plane wave, where

*k*

_{z,0}is the incident wavevector component along the direction of periodicity.

*σ*from

*x*= 0, the magnetic field obeys: In order to find the coupling coefficients C, we combine Eqs. (2) and (5), obtaining: Which can be written in matrix form as:

*m*= 0) which is equal to unity. For

*x*→ 0 the matrix

*X̳*is asymptotically equal to the identity matrix

*I̳*. The matrix

*W͇*is obtained by solving the homogeneous problem. It is now possible to solve for the column vector of coupling constants

*C̱*: Having obtained

*C̱*, we have a full description of

### 2.3. Calculation of radiated power

*E*field can be deduced from

_{z}*H*. In matrix form we write [23

_{y}**12**, 1068–1076 (1995). [CrossRef]

*V̳*defined as

*E*, Eq. (8) can be evaluated: The delta function

_{z}*δ*(

*x*) removes the integration over

*x*. By integrating over

*z*, all multiplications between non equal Fourier components of the current source and

*E*are zero from orthogonality, and Eq. (10) can be written in matrix form: where

_{z}*V̱*

_{m=0,n}is a row vector and its elements are the “zeroth order” Fourier components of each eigenmode.

*z*direction

*x*= 0, the time averaged Poynting vector is multiplied by two: In Eq. (12), “T” denotes the Hermitian complex conjugate transpose. It is seen that

*S*=

_{z}L*P*.

## 3. Simulation results

### 3.1. Comparison with FDTD

*y*direction and extending over the unit cell in the

*z*direction) with angular frequency

*ω*and

*J*(

_{z}*x*,

*z*,

*t*) =

*ρδ*(

*x*)exp(

*jk*

_{z,c}

*z*) exp(

*jωt*). This current source represents the steady state solution of our problem, i.e. it doesn’t assume a moving line charge explicitly. As such, the current source excitation fits the purpose of validating the results obtained by FMM (being a frequency domain method the FMM provides a steady state solution) with an independent method. In Fig. 2 we show the radiated power per unit frequency as calculated by both FDTD and FMM. The consistency between the two methods is clearly evident.

### 3.2. Vertical emission from a periodic structure satisfying the QWS condition

*x*direction) by a line charge moving along the horizontal (

*z*) direction within a 1D periodic structure. We focus our attention on a specific frequency (out of the continuum of frequencies emitted by the line charge) for which the QWS condition is satisfied. In a homogeneous medium, vertical emission is obtained when

*k*

_{z,c}approaches zero. Unfortunately, such a case occurs only if the line charge is moving with an infinitely high velocity, and therefore vertical emission cannot be obtained in a homogenous medium. As known from the theory of 1D photonic crystals [30], Bloch wavevectors at one of the Brillouin Zone (BZ) edges (e.g.

*k*=

_{B}*π*/

*L*, where

*k*is the Bloch wavevector) with frequencies that obey the QWS condition, are Bragg reflected to the opposite Brillouin zone edge (

_{B}*k*= −

_{B}*π*/

*L*). We now ask ourselves what would happen if we assume an excitation by a specific

*k*

_{z,c}inducing a phase difference between the two boundaries of the QWS unit cell that is equivalent to

*k*= ±

_{z}*π*/

*L*, i.e. excitation at the edge of the 1st BZ. For such a case, all diffracted orders are located at the edges of this BZ. Obviously, these diffracted orders cannot contribute to power flow in the

*z*direction, because all orders obey the QWS condition. Moreover, plane waves that monotonically decay in either the positive or negative z direction, cannot contribute to power flow in the

*x*direction because their energy density vanishes. However, the superposition with equal phase and amplitude of two such plane waves with their

*k*at the opposite BZ edges, will result in a standing wave with zero phase difference across the unit cell, located at the center of the BZ (

_{z}*k*= 0). Such a wave does not decay monotonically in the

_{z}*z*direction and thus can propagate along the

*x*direction.

*v*= 2

*Lc*/

*λ*,

*ε*

_{1}= 1.2,

*ε*

_{2}= 1 and

*μ*

_{1}=

*μ*

_{2}= 1. In Fig. 3(a) the Poynting vector in the

*x*direction (

*S*) is shown, providing a clear evident for the radiation of light in the vertical direction away from the line charge. In Figs. 3(b) and 3(c) we plot the real and imaginary parts of

_{x}*H*(the colorbar is normalized by

_{y}*ρ*). Indeed, one can learn from these two figures that away from the source (located at

*x*= 0) the magnetic field has a negligible phase difference across the unit cell, as expected from the above mentioned discussion. In Fig. 3(d), we plot the magnitude of the Fourier components of the magnetic field amplitudes, (calculated at

*x*= ±100

*λ*to diminish the effect of the non-propagating modes). As can be seen, there are two major non vanishing Fourier components (

*N*= 0,1), with equal amplitude. These two Fourier components correspond to plane waves with

*k*= ±

_{z}*π*/

*L*, in agreement with the above mentioned discussion. The phases of these two Fourier components (not shown) are identical. All Fourier orders are distributed symmetrically around

*N*= 0,1.

### 3.3. Horizontal emission from a periodic structure satisfying the HWS

*v*=

*Lc*/

*λ*. In Fig. 4 the field amplitudes of

*H*and

_{y}*E*are plotted. It is seen that in each half plane (

_{z}*x*> 0 and

*x*< 0), the magnetic field distribution corresponds to a “plane wave” propagating in the

*z*direction. The

*E*component decays exponentially along the

_{z}*x*direction and therefore

*S*must be zero away from the source, as no power flux in the

_{x}*x*direction can exist with a vanishing

*E*component. Indeed, substituting our FMM results in Eq. (11) results in a null for the first nine significant digits behind the decimal point. The nearly zero radiated power is also understood from the following observation: the real and imaginary parts of the current distribution given by Eq. (4) are shifted by

_{z}*π*/2 compared to the real and imaginary parts of

*E*shown in Fig. 4. Therefore, the work done by the

_{z}*E*field on the charges vanishes.

_{z}## 4. Radiation in zero real average refractive index DPS – DNG periodic structures

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

*k*

_{x,n}, the Poynting vectors in the DPS and DNG media, are anti parallel. If the real part of the refractive index in the two media is opposite (as obtained for example for the following combination of parameters:

*ε*

_{1}=

*ε*+

_{R}*jε*,

_{I}*ε*

_{2}= −

*ε*+

_{R}*jε*,

_{I}*μ*

_{1}=

*μ*+

_{R}*jμ*,

_{I}*μ*

_{2}= −

*μ*+

_{R}*jμ*where the subscripts R and I denote the real and imaginary parts of the material parameters respectively), and the duty cycle is set to

_{I}*L*

_{1}=

*L*

_{2}) the overall energy flow will cancel out after averaging over a unit cell, if the field intensities in both media are equal. This case of opposite real part of the refractive index is a special case of a structure having average zero real part of the refractive index [28

28. A. Yanai, M. Orenstein, and U. Levy, “Giant resonance absorption in ultra-thin metamaterial periodic structures,” Opt. Express **20**, 3693–3702 (2012). [CrossRef] [PubMed]

31. S. Nefedov and S. A. Tretyakov, “Photonic band gap structure containing metamaterial with negative permittivity and permeability,” Phys. Rev. E **66**, 036611 (2002). [CrossRef]

33. S. Kocaman, M. S. Aras, P. Hsieh, J. F. McMillan, C. G. Biris, N. C. Panoiu, M. B. Yu, D. L. Kwong, A. Stein, and C. W. Wong, “Zero phase delay in negative-refractive-index photonic crystal superlattices,” Nat. Photonics **5**, 499–505 (2011). [CrossRef]

*v*/

*c*and

*λ*/

*L*, calculated for the following parameters:

*ε*

_{1}= 1 + 1 × 10

^{−3}

*j*,

*ε*

_{2}= −1 + 1 ×10

^{−3}

*j*,

*μ*

_{1}= 1 + 1 ×10

^{−3}

*j*,

*μ*

_{2}= −1 + 1 ×10

^{−3}

*j*and

*L*=

*λ*= 2

*L*

_{1}= 2

*L*

_{2}. Very narrow peaks are observed. Moreover, the calculated work reaches very high values: possibly three orders of magnitude higher than in “ordinary” DPS structures (compare with the values in Fig. 2). Figure 5(b) is based on similar data used in Fig. 5(a), however the

*v*/

*c*axis is now rescaled to (

*λ*/

*L*)*(

*v*/

*c*). With this choice of scaling, the calculated work follows vertical lines located at

*λ*/

*L*)*(

*v*/

*c*) = 1/

*p*(p is a natural number),

*k*

_{z,c}

*L*= 2

*πc*/(

*λv*) = 2

*πp*. Comparing to Eq. (4), it follows that when (

*λ*/

*L*)*(

*v*/

*c*) is equal to one of these discrete values, there is zero phase difference between the two boundaries of the unit cell. It was shown that for such a case, the local density of states (LDOS) is greatly enhanced [34

34. I. Shadrivov, A. Sukhorukov, and Y. Kivshar, “Complete Band Gaps in One-Dimensional Left-Handed Periodic Structures,” Phys. Rev. Lett. **95**, 193903 (2005). [CrossRef] [PubMed]

35. F. J. García de Abajo and M. Kociak, “Probing the photonic local density of states with electron energy loss spectroscopy,” Phys. Rev. Lett. **100**(10), 106804 (2008). [CrossRef] [PubMed]

36. F. J. García de Abajo, “Optical excitations in electron microscopy,” Rev. Mod. Phys. **82**, 209–275 (2010). [CrossRef]

*z*axis is not balanced by a counter propagating field, and thus there is no build up of the field. However, in the DPS-DNS case the propagation along the

*z*axis in both media is opposite, giving rise to a resonance effect and high LDOS.

*JE*

^{*}is large due to the localized

*E*field component, we did not observe coupling to radiative modes. This can be understood by the fact that the Poynting vector, averaged over the unit cell is identically zero due to the opposite power flow in both media.

_{z}*γ*= 1 ×10

_{ε}^{−3}

*ω*and

_{pε}*γ*= 1 ×10

_{μ}^{−2}

*ω*. For simplicity, the DPS medium is assumed to be air, i.e.

_{p}_{μ}*ε*

_{1}= 1,

*μ*

_{1}= 1. For these assumed parameters, at

*λ*/

*L*= 2, the DNG constitutive parameters take the following values:

*ε*

_{2}= −0.99 +5.6 ×10

^{−3}

*j*and

*μ*

_{2}= −1.01 +8.5 ×10

^{−3}

*j*. Comparing to the previous results, we find that the work is now lower by one order of magnitude (peak value is now 55 [

*ρ*

^{2}/(

*cε*

_{0})] in the vicinity of

*v*/

*c*= 0.5 and

*λ*/

*L*= 2, where the FP condition is nearly satisfied). Additionally, the high work trajectory previously located at (

*λ*/

*L*) *(

*v*/

*c*) = 1 is no longer observed, because of the detuning from refractive index matching for wavelengths other than

*λ*/

*L*= 2.

## 5. Conclusions

## Acknowledgments

## References and links

1. | V. L. Ginzburg, “Radiation by uniformly moving sources (Vavilov - Cherenkov effect, transition radiation,and other phenomena),” Usp. Fiz. Nauk |

2. | P. M. van den Berg, “Smith-Purcell radiation from a line charge moving parallel to a reflection grating,” J. Opt. Soc. Am. |

3. | P.M. van den Berg, “Smith-Purcell radiation from a point charge moving parallel to a reflection grating,” J. Opt. Soc. Am. |

4. | S. J. Smith and E. M. Purcell, “Visible light from localized surface charges moving across a grating,” Phys. Rev. |

5. | S. L. Chuang and J. A. Kong, “Enhancement of Smith-Purcell radiation from a grating with surface-plasmon excitation,” J. Opt. Soc. Am. A |

6. | T. Ochiai and K. Ohtaka, “Electron energy loss and Smith-Purcell radiation in two- and three-dimensional photonic crystals,” Opt. Express |

7. | L. Schächter, |

8. | I. A. Fainberg and N. A. Khizhniak, “Energy Loss of a Charged Particle Passing Through a Laminar Dielectric,” Sov. Phys. JETP |

9. | B. M. Bolotovskii, “Theory of Cerenkov radiation (III),” Sov. Phys. Usp. |

10. | K. F. Casey and C. Yeh, “Transition Radiation in a Periodically Stratified Plasma,” Phys. Rev. A |

11. | L.Sh. Khachatryan, S.R. Arzumanyan, and W. Wagner, “Self-amplified Cherenkov radiation from a relativistic electron in a waveguide partially filled with a laminated material,” JJ. Phys.: Conf. Ser. |

12. | C. Luo, M. Ibanescu, S. G. Johnson, and J. D. Joannopoulos, “Cherenkov radiation in photonic crystals,” Science |

13. | C. Kremers, D. N. Chigrin, and J. Kroha, “Theory of Cherenkov radiation in periodic dielectric media: Emission spectrum,” Phys. Rev. A |

14. | C. Kremers and D. N. Chigrin, “Spatial distribution of Cherenkov radiation in periodic dielectric media,” J. Opt. A |

15. | V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of |

16. | J. Lu, T. Grzegorczyk, Y. Zhang, J. Pacheco Jr., B.-I. Wu, J. Kong, and M. Chen, “Cerenkov radiation in materials with negative permittivity and permeability,” Opt. Express |

17. | S. Xi, H. Chen, T. Jiang, L. Ran, J. Huangfu, B. I. Wu, J. A. Kong, and M. Chen, “Experimental Verification of Reversed Cherenkov Radiation in Left-Handed Metamaterial,” Phys. Rev. Lett. |

18. | S. N. Galyamin, A. V. Tyukhtin, A. Kanareykin, and P. Schoessow, “Reversed Cherenkov-Transition Radiation by a Charge Crossing a Left-Handed Medium Boundary,” Phys. Rev. Lett. |

19. | S. N. Galyamin and A. V. Tyukhtin, “Electromagnetic field of a moving charge in the presence of a left-handed medium,” Phys. Rev. B |

20. | D. E. Fernandes, S. I. Maslovski, and M. G. Silveirinha, “Cherenkov emission in a nanowire material,” Phys. Rev. B |

21. | V. V. Vorobev and A. V. Tyukhtin, “Nondivergent Cherenkov Radiation in a Wire Metamaterial,” Phys. Rev. Lett. |

22. | Z. Mohammadi, C. P. Van Vlack, S. Hughes, J. Bornemann, and R. Gordon, “Vortex electron energy loss spectroscopy for near-field mapping of magnetic plasmons,” Opt. Express |

23. | M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A |

24. | L. Li, “Use of Fourier series in the analysis of discontinuous periodic structures,” J. Opt. Soc. Am. A |

25. | P. Lalanne and G. Morris, “Highly improved convergence of the coupled-wave method for TM polarization,” J. Opt. Soc. Am. A |

26. | A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: A flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. |

27. | R. Petit, |

28. | A. Yanai, M. Orenstein, and U. Levy, “Giant resonance absorption in ultra-thin metamaterial periodic structures,” Opt. Express |

29. | J. D. Jackson, |

30. | J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, |

31. | S. Nefedov and S. A. Tretyakov, “Photonic band gap structure containing metamaterial with negative permittivity and permeability,” Phys. Rev. E |

32. | L. Wu, S. He, and L. F. Shen, “Band structure for a one-dimensional photonic crystal containing left-handed materials,” Phys. Rev. B |

33. | S. Kocaman, M. S. Aras, P. Hsieh, J. F. McMillan, C. G. Biris, N. C. Panoiu, M. B. Yu, D. L. Kwong, A. Stein, and C. W. Wong, “Zero phase delay in negative-refractive-index photonic crystal superlattices,” Nat. Photonics |

34. | I. Shadrivov, A. Sukhorukov, and Y. Kivshar, “Complete Band Gaps in One-Dimensional Left-Handed Periodic Structures,” Phys. Rev. Lett. |

35. | F. J. García de Abajo and M. Kociak, “Probing the photonic local density of states with electron energy loss spectroscopy,” Phys. Rev. Lett. |

36. | F. J. García de Abajo, “Optical excitations in electron microscopy,” Rev. Mod. Phys. |

**OCIS Codes**

(050.0050) Diffraction and gratings : Diffraction and gratings

(240.6690) Optics at surfaces : Surface waves

(300.2140) Spectroscopy : Emission

**ToC Category:**

Metamaterials

**History**

Original Manuscript: June 22, 2012

Revised Manuscript: July 15, 2012

Manuscript Accepted: July 16, 2012

Published: July 27, 2012

**Citation**

Avner Yanai and Uriel Levy, "Radiation of a uniformly moving line charge in a zero-index metamaterial and other periodic media," Opt. Express **20**, 18515-18524 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-16-18515

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