Design of invisibility cloaks for reduced observability of objects

ESTEC 13 June 2008 Design of invisibility cloaks for reduced observability of objects F. Bilotti, S. Tricarico, and L. Vegni University ROMA TRE, Italy Department of Applied Electronics Applied Electromagnetics Lab bilotti@uniroma3.it

Outline Cloaking approach Design of a cloak with MNZ materials (microwaves) Design of a cloak with MNZ and ENZ Metamaterials (microwaves) Studying problem: obstacle in the near field of an antenna Metamaterials in the IR and visible regimes Design of a cloak with ENZ Metamaterials in the IR and visible regime Studying problem: reduction of the solar pressure by optical cloakingc 2

Outline Cloaking approach Design of a cloak with MNZ materials (microwaves) Design of a cloak with MNZ and ENZ Metamaterials (microwaves) Studying problem: obstacle in the near field of an antenna Metamaterials in the IR and visible regimes Design of a cloak with ENZ Metamaterials in the IR and visible regime Studying problem: reduction of the solar pressure by optical cloaking 3

Cloaking and Total Scattering Cross-Section Reduction (I-III) An ideal cloak is a device capable of: minimizing the reflected field from the object; minimizing the scattered field from the object; minimizing the absorbed field in the object. The dream is: broadband transparency, no scattering, no shade for any polarization, direction, object! 4

Cloaking and Total Scattering Cross-Section Reduction (II-III) The right figure of merit is the minimization of the Total Scattering Cross-Section (TSCS). The TSCS is given by the sum of the Scattering Cross- Section (SCS) and the Absorption Cross-Section (ACS) (different from Stealth Technology) Both SCS and ACS have to be minimized. 5

Cloaking and Total Scattering Cross-Section Reduction (III-III) Mie theory can describe the scattering phenomena for typical geometries, in order to have an expression for the SCS. Applying the Optical Theorem ACS can be calculated from the SCS. 6

Previously proposed setups (I-IV) Plasmonic covers by Engheta s group no cover with cover Reduced observability of electrically small spherical particles 7

Previously proposed setups (II-IV) Robustness to a slight variation of the basic shape (e.g. bumps and dimples) 8

Previously proposed setups (III-IV) Implementation of the ENG cloak through a real life medium no cover with cover Rotman, Parallel Plate Medium, 1963 Polarization dependent approach! 9

Previously proposed setups (IV-IV) Implementation of the ENG cloak through a real life medium no cover with cover Rotman, Parallel Plate Medium, 1963 Polarization dependent approach! 10

Advantages and drawbacks of the methods (I-III) Plasmonic cover Conformal Mapping Reflection Minimized YES YES Scattering Minimized YES YES Absorption Minimized YES? Phase Uniformity YES YES Macroscopic Object YES YES Broadband YES NO 11

Advantages and drawbacks of the methods (II-III) Plasmonic cover Conformal Mapping Object Independent NO YES Both Polarizations YES NO Uniform YES NO Effective Parameter Characterization YES NO 12

Advantages and drawbacks of the methods (III-III) Losses and mismatch from the ideal parameters affect the performances of the cloak. Ideal parameters lossless case lossy case Reduced parameters Experiment Schurig, Pendry et al., 2006 13

Cloak with ENZ materials at microwaves ŷ ŷ (I-VIII) E i E i a ˆx H i a ˆx H i ẑ ẑ ( a ) ( b) For cylindrical unbounded structures SCS can be expressed in closed form, depending on polarization: ( TM ) 2λ = ( TM σ ) TM σ2-d = εncn cosn ϕ, π n= 0 2 ( TE) 2λ ( TE σ ) TE = σ2-d = εncn cosnϕ π n= 0 2 14

Cloak with ENZ materials at microwaves (II-VIII) Case of structure with finite length can be studied, also for oblique incidence: E i a ẑ ŷ E i θ i ẑ 2 2L 2 2 kl σ3-d σ2-d sin θssinc cosθi + cosθs λ 2 ( ) In the far-field region the SCS of a cylinder with length L is proportional to the SCS of the unbounded one. L θ s θ i ϕ ˆx H i θ s θ i θ i ˆx 15

Cloak with ENZ materials at microwaves (III-VIII) Cylinder of radius a and electric parameters (ε p, µ p ) surrounded by a cover of radius b. The SCS can be numerically minimized by a proper choice of the cover permeability and permittivity For the operation at both polarizations, different parameters may be used. b a ( a,b) ( a,b ε ε ) ( μ μ) ( + c c p) ( μc μp) ( c ) λ λ μ μ c, n =, a b, n ε 1 μ + μ = = c 0 2n 0 εc, μc ε p a μ b p ε, μ 16

Cloak with ENZ materials at microwaves (IV-VIII) For an unbounded cylinder with the geometrical and electrical parameters: b ε ε = c, n= 0 a ε 1 Min εc c [ σ ] TM a= 10 mm b = 18. a f0 = 3 GHz ε p = 2 μ p = 2 εc, μc ε p a μ an ENZ is needed in order to reduce the SCS in TM polarization: ε = 01., μ = 1 c c b p ε, μ 17

Cloak with ENZ materials at microwaves (V-VIII) SCS minimization rate versus parameter variation in the theoretical model (constant permittivity): 40 30 40 30 σ TM = σ NoCover TM Cover TM σ σ TM [db] 20 10 σ TM [db] 20 10 0-10 -4-2 0 2 4 ε c 0-10 0 1 2 3 4 5 μ c 18

Cloak with ENZ materials at microwaves (VI-VIII) Theoretical SCS minimization rate for Drude-like dispersive permittivity: 40 30 0,4 20 0,2 10 σ TM [db] 0-10 -20 0,0-0,2 ε' r ε'' r -30-40 0 1 2 3 4 5 frequency [GHz] -0,4 2,5 3,0 3,5 frequency [GHz] 19

Cloak with ENZ materials at microwaves (VII-VIII) Theoretical SCS minimization rate versus losses in the metamaterial. 40 σ TM [db] Anyway, since the ENZ regime is considered, 20 ε c =εc jεc losses are expected to be low. 10-3 10-2 10-1 30 ε r '' 20

Cloak with ENZ materials at microwaves (VIII-VIII) Theoretical SCS minimization rate versus geometrical parameters: σ TM [db] 40 30 20 10 εc, μc ε p a μ b p α = b a ε, μ σ TM [db] 40 30 20 10 E i H i y z L = a ϕ ρ x 0 1,0 1,5 2,0 2,5 3,0 α 0 0 1 2 3 4 5 frequency [GHz] 21

Outline Cloaking approach Design of a cloak with MNZ materials (microwaves) Design of a cloak with MNZ and ENZ Metamaterials (microwaves) Studying problem: obstacle in the near field of an antenna Metamaterials in the IR and visible regimes Design of a cloak with ENZ Metamaterials in the IR and visible regime Studying problem: reduction of the solar pressure by optical cloaking 22

Cloak with MNZ materials at microwaves A MNZ is needed in order to reduce the SCS in TE polarization: ε = 1, μ = 01. c c (I-III) σ TE [db] 40 30 20 10 σ TE [db] 40 30 20 10 σ TE = σ NoCover TE Cover TE σ 0-10 0-10 -4-2 0 2 4 0 1 2 3 4 5 μ r ε c 23

Cloak with MNZ materials at microwaves (II-III) Theoretical SCS minimization rate for dispersive permeability: 40 40 30 20 10 30 σ TE [db] 0-10 -20-30 -40 ( μ μ ) 2 2 ( 0 ) ω 2 s 0 μc ( ω ) =μ + ω ω + j ωδ σ TE [db] 20 μ =μ jμ c c c 0 1 2 3 4 5 frequency [GHz] 10-3 10-2 10-1 μ c '' 24

Cloak with MNZ materials at microwaves (III-III) Theoretical SCS minimization rate versus geometrical parameters: y σ TE [db] 40 30 20 εc, μc ε p a μ b p ε, μ σ TE [db] 40 30 20 H i E i z L = a ϕ ρ x 10 0 1,0 1,5 2,0 2,5 3,0 α α = b a 10 0 0 1 2 3 4 5 frequency [GHz] 25

Outline Cloaking approach Design of a cloak with MNZ materials (microwaves) Design of a cloak with MNZ and ENZ Metamaterials (microwaves) Studying problem: obstacle in the near field of an antenna Metamaterials in the IR and visible regimes Design of a cloak with ENZ Metamaterials in the IR and visible regime Studying problem: reduction of the solar pressure by optical cloaking 26

Design of a MNZ-ENZ cloak at microwaves (I-XVII) Full wave simulation of an ideal cloak for TM polarization: Plane wave impinging on a cylinder of length L covered by a dispersive homogeneous cloak 27

Design of a MNZ-ENZ cloak at microwaves (II-XVII) Max SCS values for covered (left) and uncovered (right) structure: Max(σ TM ) [db] -20-30 L = 30 [mm] L = 60 [mm] L = 90 [mm] L = 120 [mm] L = 150 [mm] L = 180 [mm] Max(σ TM ) [db] -10-20 -30 L = 30 [mm] L = 60 [mm] L = 90 [mm] L = 120 [mm] L = 150 [mm] L = 180 [mm] -40 2,5 3,0 3,5 frequency [GHz] -40 2,5 3,0 3,5 frequency [GHz] 28

Design of a MNZ-ENZ cloak at microwaves (III-XVII) Max SCS values versus cylinder length: -10 no cover with cover -20 Max(σ TM ) [db] -30-40 Max(σ TM ) [db] 20 10-50 2,5 3,0 3,5 frequency [GHz] 30 60 90 120 150 180 L [mm] 29

Design of a MNZ-ENZ cloak at microwaves (IV-XVII) Full-wave SCS minimization rate versus geometrical parameters: 20 σ TM [db] 15 10 5 0 εc, μc ε p a μ b p ε, μ -5 L=100 [mm] -10 1,2 1,4 1,6 1,8 2,0 2,2 2,4 2,6 2,8 3,0 α 30

Design of a MNZ-ENZ cloak at microwaves (V-XVII) Simulated SCS for covered (left) and uncovered (right) structure: 31

Design of a MNZ-ENZ cloak at microwaves (VI-XVII) E-field for covered (left) and uncovered (right) structure: 32

Design of a MNZ-ENZ cloak at microwaves (VII-XVII) H-field for covered (left) and uncovered (right) structure: 33

Design of a MNZ-ENZ cloak at microwaves (VIII-XVII) Max SCS values for covered (left) and uncovered (right) structure in TE polarization: Max(σ TE ) [db] -20 L = 30 [mm] L = 60 [mm] L = 90 [mm] L = 120 [mm] L = 150 [mm] L = 180 [mm] -30 Max(σ TE ) [db] -10-20 -30 Ly = 30 [mm] Ly = 60 [mm] Ly = 90 [mm] Ly = 120 [mm] Ly = 150 [mm] Ly = 180 [mm] -40 2,5 3,0 3,5 frequency [GHz] -40 2,5 3,0 3,5 frequency [GHz] 34

Design of a MNZ-ENZ cloak at microwaves (IX-XVII) Max SCS values versus cylinder length in TE polarization : -10 no cover with cover -20 Max(σ TE ) [db] -30 σ TE [db] 20-40 10-50 2,5 3,0 3,5 frequency [GHz] 30 60 90 120 150 180 L [mm] 35

Design of a MNZ-ENZ cloak at microwaves (X-XVII) Simulated SCS for covered (left) and uncovered (right) structure: 36

Design of a MNZ-ENZ cloak at microwaves (XI-XVII) E-field for covered (left) and uncovered (right) structure: 37

Design of a MNZ-ENZ cloak at microwaves (XII-XVII) H-field for covered (left) and uncovered (right) structure: 38

Design of a MNZ-ENZ cloak at microwaves (XIII-XVII) E-field for covered structure: 39

Design of a MNZ-ENZ cloak at microwaves (XIV-XVII) H-field for covered structure: 40

Design of a MNZ-ENZ cloak at microwaves (XV-XVII) Both polarization setup: 40 ( kr) 20 ε eff ε d 0 2 N 2 2 1 2 3 4 f @GHzD -20-40 μ eff 41

Design of a MNZ-ENZ cloak at microwaves (XVI-XVII) TM polarization results (E-field) of a dielectric cylinder (ε r = 2) without the cover (left) and with cover (right). 42

Design of a MNZ-ENZ cloak at microwaves (XVII-XVII) TE polarization results (H-field) of a dielectric cylinder (ε r = 2) without the cover (left) and with cover (right). 43

Outline Cloaking approach Design of a cloak with MNZ materials (microwaves) Design of a cloak with MNZ and ENZ Metamaterials (microwaves) Studying problem: obstacle in the near field of an antenna Metamaterials in the IR and visible regimes Design of a cloak with ENZ Metamaterials in the IR and visible regime Studying problem: reduction of the solar pressure by optical cloaking 44

Studying problem: obstacle in proximity of an antenna (I-VIII) Electrically small covered cylinder in vacuum ŷ ŷ E i E i H i εc 1 a= b, n= 0 εc εp ẑ ( μ 1) ( μ + μ c c p) ( μc μp) ( μc + 1) a = 2n b, n a ˆx ( a ) c( b) 0 H i ẑ a μ 1 a= b, n= 0 μc μp ( ε 1) ( ε + ε c c p) ( εc εp) ( εc + 1) a = 2n b, n ˆx 0 45

Studying problem: obstacle in proximity of an antenna (II-VIII) Electrically small covered metallic cylinder in vacuum ŷ ŷ E i E i a ˆx H i a ˆx H i ẑ ẑ ( a ) ( b) ( ) ( 1 εc ) ( 1 ε ) 2 2 2 c0 jπk0a μ + α μ μ = < ε < c a b, n + 1 c 2n 0 0 1 c 46

Studying problem: obstacle in proximity of an antenna (III-VIII) Half-wavelength dipole working at 3 GHz. d = λ λ 2 a = 005λ. L = 05λ. 47

Studying problem: obstacle in proximity of an antenna (IV-VIII) Electrically small covered metallic cylinder in vacuum with emishpeares 48

Studying problem: obstacle in proximity of an antenna (V-VIII) 49

Studying problem: obstacle in proximity of an antenna (VI-VIII) Dipole alone Dipole+Obstacle Dipole+Cloaking 50

Studying problem: obstacle in proximity of an antenna (VII-VIII) Possible setup: parallel plate plasma column surrounding the metallic cylinder. ŷ ŷ E i E i a ˆx H i a ˆx H i ẑ ẑ ( a ) ( b) 51

Studying problem: obstacle in proximity of an antenna (VIII-VIII) 52

Outline Cloaking approach Design of a cloak with MNZ materials (microwaves) Design of a cloak with MNZ and ENZ Metamaterials (microwaves) Studying problem: obstacle in the near field of an antenna Metamaterials in the IR and visible regimes Design of a cloak with ENZ Metamaterials in the IR and visible regime Studying problem: reduction of the solar pressure by optical cloaking 53

State-of-the-art of Metamaterials at THz and optical frequencies (I-III) ENZ and ENG materials at optical frequencies are hard to find in nature. A layered structure (Ag-SiO 2 ), under certain conditions, can be effectively used to have such materials in the visible. 54

State-of-the-art of Metamaterials at THz and optical frequencies (II-III) a) Yen, et al. ~ 1THz (2-SRR) 2004 Katsarakis, et al (SRR 5 layers) - 2005 b) Zhang et al ~50THz (SRR+mirror) - 2005 c) Linden, et al. 100THz (1-SRR) - 2004 d) Enkrich, et al. 200THz (u-shaped)- 2005 55

State-of-the-art of Metamaterials at THz and optical frequencies (III-III) At optical frequencies the kinetic inductance of the electron should be taken into account. A saturation in the scaling is expected. 56

Outline Cloaking approach Design of a cloak with MNZ materials (microwaves) Design of a cloak with MNZ and ENZ Metamaterials (microwaves) Studying problem: obstacle in the near field of an antenna Metamaterials in the IR and visible regimes Design of a cloak with ENZ Metamaterials in the IR and visible regime Studying problem: reduction of the solar pressure by optical cloaking 57

Design of a cloak with ENZ Metamaterials at THz and/or optical frequencies (I-XV) Metals exhibit a real negative permittivity in the visible. Layered structures of plasmonic and non-plasmonic materials can exhibit ENZ behavior in the visible range. If the thicknesses of the slabs are electrically small, the resulting composite material is described through constitutive parameters depending only on the ratio between the thicknesses of the labs and the constitutive parameters of the two different materials. 58

Design of a cloak with ENZ Metamaterials at THz and/or optical frequencies (II-XV) y ε = xy ε +ηε 1+η 1 2 ε 1 µ 1 d 1 d2 z 1 1 1 η = + ε z 1+η ε1 ε2 d2 η= d 1 59

Design of a cloak with ENZ Metamaterials at THz and/or optical frequencies (III-XV) object ε obj, μ obj 1 1 1 η d d 1 1 = + 2 2 ε r 1+ η ε 1 ε 2 d 2 d 2 1 ε1+ ηε2 ε z = 1+ η 1 ε obj, μ obj 1 1 1 η = + ε z 1+ η ε 1 ε 2 ε1+ ηε 2 ε r = 1+ η d η = 2 d1 d η = 2 d1 Applying the same idea to cylindrical geometry it is possible to obtain in both configurations a close-to-zero real permittivity along the axis of the cylinder. 60

Design of a cloak with ENZ Metamaterials at THz and/or optical frequencies (IV-XV) With a plasmonic material (Ag) and silica (SiO 2 ), it is possible to obtain an ENZ material along the axis of the cylinder in the visible. With high plasma frequency it is necessary to use the non radial component of the permittivity tensor. 61

Design of a cloak with ENZ Metamaterials at THz and/or optical frequencies (V-XV) Real part of the relative permittivity 0-5 -10-15 -20 500 550 600 650 700 Frequency [THZ] SiO 2 Silver Layered medium Choosing a proper value for η it is possible to obtain the desired permittivity. In this case is not possibile to get the same values (ε r = ε z ) 62

Design of a cloak with ENZ Metamaterials at THz and/or optical frequencies (VI-XV) For an unbounded cylinder with the geometrical and electrical parameters: b ε ε = c, n= 0 a ε 1 Min εc c [ σ ] TM [ ] a = 50 nm b= 18a. f0 = 600 THz εp = 2 μp = 1 [ ] εc, μc ε p a μ an ENZ is needed in order to reduce the SCS in TM polarization: ε = 032., μ = 1 c η = 021. d1 = 10 ε 022. c c [ nm] b p ε, μ 63

Design of a cloak with ENZ Metamaterials at THz and/or optical frequencies (VII-XV) Max SCS values versus cylinder length in TM polarization -120-125 with cover, L=500 [nm] no cover, L=500 [nm] Max value of RCS -125-130 L=250 [nm] -135 L=500 [nm] L=720 [nm] L=960 [nm] -140 500 550 600 650 700 Frequency [THz] Max value of RCS -130-135 -140 500 550 600 650 700 Frequency [THz] 64

Design of a cloak with ENZ Metamaterials at THz and/or optical frequencies (VIII-XV) Simulated RCS for uncovered (left) and covered (right) structure: 65

Design of a cloak with ENZ Metamaterials at THz and/or optical frequencies (IX-XV) E-field for uncovered (left) and covered (right) structure: 66

Design of a cloak with ENZ Metamaterials at THz and/or optical frequencies (X-XV) H-field for uncovered (left) and covered (right) structure: 67

Design of a cloak with ENZ Metamaterials at THz and/or optical frequencies (XI-XV) For an unbounded cylinder with the geometrical and electrical parameters: b ε ε = c, n= 0 a ε 1 Min εc c [ σ ] TM [ ] a = 50 nm b= 18a. f0 = 600 THz εp = 2 μp = 2 [ ] εc, μc ε p a μ an ENZ is needed in order to reduce the RCS in TM polarization: ε = 01., μ = 1 c η = 023. d1 = 10 ε 01. c c [ nm] b p ε, μ 68

Design of a cloak with ENZ Metamaterials at THz and/or optical frequencies (XII-XV) Max SCS values versus cylinder length -120-120 without cover, L=800 [nm] with cover, L=800 [nm] Max value of RCS -130-140 L=200 [nm] L=400 [nm] L=600 [nm] L=800 [nm] Max value of RCS -130-140 -150 500 550 600 650 700 Frequency [THz] -150 500 550 600 650 700 Frequency [THz] 69

Design of a cloak with ENZ Metamaterials at THz and/or optical frequencies (XIII-XV) Simulated SCS for uncovered (left) and covered (right) structure: 70

Design of a cloak with ENZ Metamaterials at THz and/or optical frequencies (XIV-XV) E-field for uncovered (left) and covered (right) structure: 71

Design of a cloak with ENZ Metamaterials at THz and/or optical frequencies (XV-XV) H-field for uncovered (left) and covered (right) structure: 72

Outline Cloaking approach Design of a cloak with MNZ materials (microwaves) Design of a cloak with MNZ and ENZ Metamaterials (microwaves) Studying problem: obstacle in the near field of an antenna Metamaterials in the IR and visible regimes Design of a cloak with ENZ Metamaterials in the IR and visible regime Studying problem: reduction of the solar pressure by optical cloaking 73

Radiation Pressure over a covered nanoparticle (I-XXII) For covered spherical non metallic particles the total normalized RCS can be expressed as: σ 2 Q = t t = ( 2 + 1) { + } σ 2 l Re al bl a= g 2l+ 1 ( ka) l= 1 In the Rayleigh limit the coefficients b l go rapidly to zero, while the others disappear when ( ) ( l 1) εc ε εc lεp + + b εc εp + εc + lε ( ) ( l 1) bl al ε pr = 2 εcr = 3 μpr = 1 μcr = 1 b= 2a ε a c ε p b l ka ε 74

Radiation Pressure over a covered nanoparticle 8 1 2 3 4 5 6 7 8 9 8 1 2 3 4 5 6 7 8 9 (II-XXII) a ε c b ε p ε ka 6 4 2 1 a l 0 ka 6 4 2 1 bl 0 al εr = 3 μr = 1 0 εr = 3 μr = 1 εr = 3 μr = 1 0 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 l l bl εr = 3 μr = 1 1 1 0.8 - l - 1 0.8 - l - 1 0.6 0.4 2 3 4 0.6 0.4 2 3 4 0.2 5 0.2 5 1 2 3 4 5 ka 1 2 3 4 5 ka 75

Radiation Pressure over a covered nanoparticle (III-XXII) a ε c ε p b ε σ 2 = t t = + + σ 2 l l g l= 1 ( ka) ( 2 1) Re{ } Q l a b 4 ( ) εr = 321. + j 01. μr = 1 -a/λ - 3 2 Q s Q t Q s Q s a < 01. λ Rayleigh limit a > 2λ σ σ s g 1 2.5 5 7.5 10 12.5 15 ka 76

Radiation Pressure over a covered nanoparticle (IV-XXII) Optical forces on illuminated particles can be deduced from the Lorentz law or Maxwell stress tensor. Gradient forces move particles towards the spots of light intensity. Scattering forces are explained by the transfer of momentum between the electromagnetic waves scattered by the particle and the particle itself. In 2D lossless problems with S-polarization light these forces can be derived from a scalar potential, that is with light propagating in the cross-section plane of the particles Are optical forces derived from a scalar potential? 23 July 2007 Vol. 15, No. 15 OPTICS EXPRESS, Daniel Maystre and Patrick Vincent 77

Radiation Pressure over a covered nanoparticle ŷ (V-XXII) ŷ The total force for an unbounded dielectric lossless cylinder can be expressed as: E i H i ( a ) 2 2 ( b) ε ε ε ε 0 0 2 in in xy = UdΩ + 2 Re E Ω + Ω n t,xy d 4ε En d Ω Ω 0 Ω ( ) ( ) ( ) f E n ( ) ( 2 2 μ ) 0 z ε z ẑ a ( ) 1 ε ε ε ε f = Ω+ Im z H E d Re E E d Ω 0 0 in z ˆ 2ω 4ε z z n z 2 Ω Ω 1 ε 1 0 U = 4 ε H E ˆx Are optical forces derived from a scalar potential? 23 July 2007 Vol. 15, No. 15 OPTICS EXPRESS, Daniel Maystre and Patrick Vincent H i E i ẑ a ˆx 78

Radiation Pressure over a covered nanoparticle (VI-XXII) For S-polarization or P-polarization either H z or E z vanishes. In particular for S- polarization the total force reduce to: 1 2 Ω fxy = UdΩ= ( ε0 ε) Ez dω 4 ˆn Ω Ω Ω fz = 0 ωμ0 f = Im P H, P= ( ε ε0 ) E 2 z Total optical force derives from a scalar potential, and the elementary optical forces inside Ω are directed towards the smallest values of the potential, that is the spots of electric power density. Are optical forces derived from a scalar potential? 23 July 2007 Vol. 15, No. 15 OPTICS EXPRESS, Daniel Maystre and Patrick Vincent 79

Radiation Pressure over a covered nanoparticle (VII-XXII) Magnitude of the Radial Force Approximated total force on a dielectric lossless electrically small SiO 2 cylinder 10-20 10-21 10-22 a=12.5 [nm] 10-23 0,0 0,5 1,0 1,5 2,0 2,5 3,0 ε r f ε 0 ε 0 1 2 = UdΩ= ( ε0 ε) Ez dω 4 xy Ω Ω z = ˆz z = n n n= E lim E z ( ρϕ) ˆ n z 0 l ( 0 μερ) E, E j J k e = E = lim E = 0 2 je 0 0l0 1 ak0h1 0 2 z () ( ak ) jnϕ 80

Radiation Pressure over a covered nanoparticle (VIII-XXII) Approximated total force on a dielectric lossless SiO 2 cylinder Magnitude of the Radial Force 10-18 10-19 10-20 a=50 [nm] 10-21 0,0 0,5 1,0 1,5 2,0 2,5 3,0 ε r f ε 0 ε 0 1 2 = UdΩ= ( ε0 ε) Ez dω 4 xy Ω Ω z = ˆz z = n n n= E lim E z ( ρϕ) ˆ n z 0 l ( 0 μερ) E, E j J k e = E = lim E = 0 2 je 0 0l0 1 ak0h1 0 2 z () ( ak ) jnϕ 81

Radiation Pressure over a covered nanoparticle (IX-XXII) Approximated total force on a dielectric SiO 2 cylinder 10-18 Magnitude of the Radial Force 10-19 10-20 a=50 [nm] 10-21 0,0 0,5 1,0 1,5 2,0 2,5 3,0 ε r F=2.11 10 18 [N/m] 82

Radiation Pressure over a covered nanoparticle (X-XXII) Force density field inside an electrically small silica cylinder: a = 12.5 nm Ω Ω ˆn 83

Radiation Pressure over a covered nanoparticle (XI-XXII) Approximated total force on a dielectric lossless covered SiO 2 cylinder f ωμ0 xy = Im ( ε ε ) E H dω + Im ( εc ε ) E H dω 2 0 1 0 2 Ω1 Ω2 Magnitude of the Radial Force 10-20 10-21 10-22 a=12.5 [nm] 0,0 0,5 1,0 1,5 2,0 ε = 213. ε = 05. c b ε ε = c 18. a ε 1 c εc, μc ε p a μ b p ε, μ ε r 84

Radiation Pressure over a covered nanoparticle (XII-XXII) Magnitude of the Radial Force Approximated total force on a dielectric lossless covered SiO 2 cylinder f ωμ0 xy = Im ( ε ε0) E H dω 1 + Im ( εc ε0) E H dω2 10-18 10-19 10-20 2 a=50 [nm], SiO 2 Ω1 Ω2 ε = 213. ε 025. c [ σ ] Min = 18. εc,b TM εc, μc ε p a μ b p ε, μ 0,5 1,0 1,5 2,0 ε c 85

Radiation Pressure over a covered nanoparticle (XIII-XXII) Force density inside an small SiO 2 covered cylinder. ε c =0.5 ε c =3 SiO 2 SiO 2 86

Radiation Pressure over a covered nanoparticle (XIV-XXII) From the expression of the force density a similar expression can be derived for metallic particles ( ) 1 f = d E+ d E 4 { } Force of Optical Radiation Pressure on a Spheroidal Metallic Nanoparticle October 2007 Vol. 33, No. 10 OPTICS EXPRESS, N. I. Grigorchuk P. M. Tomchuck In the Rayleigh approximation for a dielectric spherical particle 3 a ( εc ε)( 2εc + εp) ( 2εc + ε)( εc εp) a ( 2ε + εc)( 2εc + εp) 2 ( εc ε)( εc εp) 3 P 3ε b E d= V P b 3 3 int 87

Radiation Pressure over a covered nanoparticle (XV-XXII) Radius ratios for a covered sphere ε c 7.5 5 ε p ε c = 2 εc 2.5 0-2.5-5 -7.5 1 a b 0 a= 3 εc = ε ( εc ε)( 2εc + εp) ( εc εp)( 2εc + ε) b ε p - 7.5-5 -2.5 ε 0 2.5 5 7.5 p ε p = ε 88

Radiation Pressure over a covered nanoparticle Magnitude of the dipole moment (XVI-XXII) ε p b/a=1 ε p b/a=1.1 ε p b/a=1.2 ε c ε c ε c ε p b/a=1.3 ε p b/a=1.4 ε p b/a=1.5 ε p ε c ε c ε c 89

Radiation Pressure over a covered nanoparticle (XVII-XXII) A silver sphere with a layered cover -130 Max value of RCS -135-140 no cover with cover η = 02. a = 50 b = 18. a d1 = 10 εc 05. [ nm] [ nm] -145 500 550 600 650 700 Frequency [THz] 90

Radiation Pressure over a covered nanoparticle (XVIII-XXII) A silver sphere with a layered cover Max value of RCS -130-135 -140-145 500 550 600 650 700 Frequency [THz] no cover η=0.2 η=0.16 η=0.17 η=0.18 η=0.21 Max value of RCS -130-135 -140 no cover with cover -145 550 560 570 580 590 600 610 620 630 640 650 Frequency [THz] 91

Radiation Pressure over a covered nanoparticle (XIX-XXII) A silver sphere with a layered cover -130 Max value of RCS -130-135 Max value of RCS -135-140 no cover with cover -140 500 550 600 650 700 Frequency [THz] -145 550 560 570 580 590 600 610 620 630 640 650 Frequency [THz] 92

Radiation Pressure over a covered nanoparticle (XX-XXII) E-field for uncovered (left) and covered (right) structure: 93

Radiation Pressure over a covered nanoparticle (XXI-XXII) H-field for uncovered (left) and covered (right) structure: 94

Radiation Pressure over a covered nanoparticle (XXII-XXII) A silica sphere with a layered cover -180 Max value of RCS -185-190 -195-200 600 650 700 Frequency [THz] a=12.5 [nm] η = 015. b = 16. a d1 = 10 εc 07. [ nm] 95