Cherenkov emission in a nanowire material

Transcription

1 Lisboa 16/11/2012 Cerenkov emission in a nanowire material David E. Fernandes, Stanislav I. Maslovski, Mário G. Silveirina Departamento de Engenaria Electrotécnica e de Computadores Instituto de Telecomunicações - Universidade de Coimbra 2005, it - instituto de telecomunicações. Todos os direitos reservados.

2 Outline Introduction Motivation Cerenkov Radiation Caracterization Model Stopping Power: Boosting te Radiation Emission Density of States Conclusions 2

3 Introduction 3

4 Te Cerenkov Effect Appears wen carged particle moves at a velocity v larger tan te pase velocity of te electromagnetic waves inside te medium v p Velocity Tresold! Emitted radiation beam consists of conical wave fronts wit aperture 2θ: Particle detection in ig energy pysics. 4

5 Motivation 5

6 Motivation for studying periodic structures in te context of te Cerenkov effect Umklapp processes: Periodicity of te wave number ' kx, n kx 2 n / a kx v Appears a possibility of matcing wit k c for low velocities. 0 Possibility of absence of velocity Tresold! May allow for te detection of particles moving at low velocities (not possible wit conventional dielectric structures). 6

7 Nanowire Material: Definition 2D array of conducting wires embedded in a ost material. Te omogenized wire medium is described by an effective dielectric permittivity tensor: xx yy zz yy xx zz k y 1 2 m fv p Granularity is neglected in te effective medium approac Time dispersion in effective permittivity is present. 1 c p a a R ln 2 7

8 Nanowire Material: Wy tis periodic structure? Ultraig density of states Larger number of radiative decay cannels. Density of states proportional to te volume Related to te number of wires! Vacuum: Boosts te interaction between matter and radiation at low velocities! 2 k k k c dn TM V x y z 2 c 2 3 d Wire-Medium: 2 c dn TEM V 0 2 k x 1 d 2 a c S. I. Maslovski, M. G. Silveirina, Pys. Rev. A, 82, (2011). 8

9 Nanowire Material: Wy tis periodic structure? Media wit yperbolic dispersion Enancement of spontaneous emission. Alexander N. Poddubny, Pavel A. Belov, Yuri S. Kivsar, Pys. Rev. A, 84, (2011) Possibility to guide radiation in a specific direction. Cerenkov radiation in wire-medium structures. David E. Fernandes, Stanislav I. Maslovski, Mário G. Silveirina, Pys. Rev. B, 85, (2012) Viktor V. Vorobev,Andrey V. Tyuktin, Pys. Rev. Lett., 108, (2012) 9

10 Cerenkov Radiation Caracterization Model 10

11 Studied Configuration Linear array of carges moving at a constant velocity v confined to te plane y=0. Current density: J x y t en yx v txˆ,, z FT.. x J( x, y, ) en y e z jk x xˆ kx v 11

12 Field Caracterization Te solution of te Maxwell equations caracterizes te interaction between te carges and te structure: H -D t J B t E 0 H z In te frequency domain te general solution of te problem is : H H z zˆ y TEM y TM jkxx TTEM e TTM e e y 0 x, y, TEM y TM y jkxx T y 0 TEM e TTM e e Superposition of 2 modes TEM j p kt c p kt c 4kt c TM j p kt c p kt c 4kt c 1 2 k x

13 Solution of te scattering problem: Field Pattern en z / c k / c k H z x, y, sgn y e e e TM x TEM y TEM x TM y jkxx TEM TM TEM TM It is expected a strong coupling of te EMF inside te nanowire structure (coupling wire-by-wire). TEM waves travel in direction parallel to te wires at a velocity v TEM c Well defined radiation pattern: tan c v 13

14 Log H H z z,max Electromagnetic Field Distribution Snapsot of te magnetic field: Carges are moving for a long time and source of carges is far away. Snapsot of te Magnetic Field at t=0 is enoug to caracterize Cerenkov emission. Log H H z z,max 1 1 v 0.15c v 0.8c R 0.05a R 0.05a a 100nm a 100nm PEC wires PEC wires 14

15 Electromagnetic Field Distribution j v 0.5c R 0.05a 0.33 fs 1 1 tan c/ v As ɛ(0) increases te angle of te of te radiation pattern decreases. Tere is no longer radiation aead of te position of te carges. 15

16 Electromagnetic Field Distribution Log H H z z,max Interference pattern propagation constants depend on k x. 1 Beam is still very directive: ~ tan c/ v 2 1 m p j z p z R 0.05a a 100nm 16

17 Electromagnetic Field Distribution No direct interaction between carges and Wire-Medium No bremsstralung radiation!! Log H H z z,max Log H H z z,max 1 v 0.1c y0 0.5a R 0.05a 1 v 0.7c y0 0.5a R 0.05a 17

18 Stopping Power 18

19 Stopping Power Integral Our approac: Emitted radiation causes energy loss. Carges remain at constant velocity due to an external force. We may calculate te stopping power troug te integral: P 0 3 E loc J d r Our configuration: x-component of te Instantaneous power provided to eac carge is: P N eve vt, y, t 0 z loc, x 0 local field tat acts on te carges Pd PWM 19

20 Comparison of te values of te stopping power Ratio between te stopping powers of te wire-medium and of te dielectric material (water) P P : WM d d 0 1 j R 0.05a T 20º C ps 5.6 a) Silver Nanowires and ost is vacuum b) PEC Nanowires and ost is vacuum c) Silver Nanowires and ost is water c 1 v 0.48c 20

21 Comparison of te values of te stopping power Ratio between te stopping powers of te semi-infinite wire-medium and of te semi-infinite dielectric material (water) P P : WM d d 0 1 j R 0.05a T 20º C ps 5.6 a) Silver Nanowires and ost is vacuum b) PEC Nanowires and ost is vacuum c) Silver Nanowires and ost is water Half of Stopping Power because we only ave one emispere of wire-medium!! c 1 v 0.48c 21

22 Density of States 22

23 Effect of density of states in te value of stopping power: As seen before te number of potonic states is independent of te frequency and varies wit 1/a 2. P P 0 max a) a 100 nm b) a 300 nm c) a 600nm As te lattice period decreases te stopping power increases! 23

24 Conclusions 24 St. Petersburg, May 30, 2012

25 Conclusions Te nanowire material allows for Cerenkov emission wit no velocity tresold. Te stopping power caracteristic of te nanowire material can be more tan 200 times larger tan in oter more conventional structures suc as dielectrics. Te enancement of te amount of emitted Cerenkov radiation is due to te extremely large density of potonic states. Our study may ave interesting applications in te context of particle detection based on te Cerenkov penomena. David E. Fernandes, Stanislav I. Maslovski, Mário G. Silveirina, Pys. Rev. B, 85, (2012) 25

26 Tank you for your time! 26

27 Tank you for your time! 27

28 Tank you for your time! 28

29 Maximum carge velocity : At te edge of te first Brillouin zone te amplitude of te magnetic field sould be less tan 10% of it s maximum value. a) v 0.3c b) v 0.7c c) v 0.9c d) v 0.98c z p z R 0.05a a 100nm 1 If silver nanowires are embedded in a vacuum medium our model is valid to velocities up to v 0.98c!! 29

30 Maximum integration region in stopping power calculations: kx Maximum frequency of integration maximum value of k x k max a kx / v max v a Relative error of considering an finite integration region: I 1 Integration up to I2 Integrating up to max a) a 100 nm b) a 250 nm c) a 700nm For a=100 nm te error is muc less tan 10% in most of te cases. 30

31 Field Caracterization Imposing te classical boundary conditions (continuity of H z and E x ): H z y H y z y0 0 Te scattering problem is underdetermined! We need 2 additional boundary conditions to solve te problem: te microscopic electric current and te additional potential must be continuous at y=0: I y 0 y0 jk E I a j k 2 2 t t ˆ y t y Ht kt y c M. G. Silveirina, IEEE Transactions on Antennas and Propagation, vol.54, no.6, pp (2006). y a 0 k t E t kt Et jc kt y c 31

32 32