Design of invisibility cloaks for reduced observability of ... · (different from Stealth Technology) Both SCS and ACS have to be minimized. Cloaking and Total Scattering Cross-Section

Laboratorio di ElettroMagnetismo Applicato

University ROMA TRE, ItalyUniversity ROMA TRE, ItalyDepartment of Applied ElectronicsDepartment of Applied Electronics

Applied Electromagnetics LabApplied Electromagnetics [email protected]@uniroma3.it

ESTECESTEC13 June 200813 June 2008

F. Bilotti, S. Tricarico, and L. VegniF. Bilotti, S. Tricarico, and L. Vegni

Design of invisibility cloaks for Design of invisibility cloaks for reduced observability of objectsreduced observability of objects


2F. Bilotti, S. Tricarico, L. VegniF. Bilotti, S. Tricarico, L. VegniUniversity Roma Tre, ItalyUniversity Roma Tre, Italy

Outline

Cloaking approachDesign 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 antennaMetamaterials in the IR and visible regimesDesign of a cloak with ENZ Metamaterials in the IR and visible regimeStudying problem: reduction of the solar pressure by optical cloakingc



Outline

Cloaking approachDesign 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 antennaMetamaterials in the IR and visible regimesDesign of a cloak with ENZ Metamaterials in the IR and visible regimeStudying problem: reduction of the solar pressure by optical cloaking



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!



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.

Cloaking and Total Scattering Cross-Section Reduction (II-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.

Cloaking and Total Scattering Cross-Section Reduction (III-III)



Previously proposed setups (I-IV)

Plasmonic covers by Engheta’s group

no cover with cover

Reduced observability of electrically small spherical particles



Previously proposed setups (II-IV)

Robustness to a slight variation of the basic shape (e.g.

bumps and dimples)



Previously proposed setups (III-IV)

Implementation of the ENG cloak through a real life medium

Rotman, Parallel Plate Medium, 1963Rotman, Parallel Plate Medium, 1963

no coverno cover with coverwith cover

Polarization dependent approach!Polarization dependent approach!



Previously proposed setups (IV-IV)

Implementation of the ENG cloak through a real life medium

Rotman, Parallel Plate Medium, 1963Rotman, Parallel Plate Medium, 1963

no coverno cover with coverwith cover

Polarization dependent approach!Polarization dependent approach!



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



Plasmonic cover Conformal Mapping

Object Independent NO YES

Both Polarizations YES NO

Uniform YES NO

Effective ParameterCharacterization YES NO

Advantages and drawbacks of the methods (II-III)



Schurig, Pendry et al., 2006

Losses and mismatch from the ideal parameters affect the performances of the cloak.

lossless case lossy caseIdeal parameters

Reduced parameters

Experiment

Advantages and drawbacks of the methods (III-III)



Cloak with ENZ materials at microwaves (I-VIII)

For cylindrical unbounded structures SCS can be expressed in closed form, depending on polarization:

y

z

x

y

z

x

( )a ( )b

iE

iH

iH

iE

a a

( ) ( )2

2-D0

2 ∞

== = ∑TM TM

TM n nn

c cosn ,λσ σ ε ϕπ

( ) ( )2

2-D0

2TE TETE n n

nc cos nλσ σ ε ϕ

π

∞

== = ∑



z

a

y

xϕiθ

iE

sθ

iH

iθ

sθiθ

z

x

iE

iθ

LIn the far-field region the SCS of a cylinder with length L is proportional to the SCS of the unbounded one.

Case of structure with finite length can be studied, also for oblique incidence:

( )2

23-D 2-D

22

⎡ ⎤≈ +⎢ ⎥⎣ ⎦2

s i sL kLsin sinc cos cosσ σ θ θ θλ

Cloak with ENZ materials at microwaves (II-VIII)



b

a

,ε μ

c c,ε μp

p

ε

μ

Cylinder of radius a and electric parameters (εp, µp) surrounded by a cover of radius b.The SCS can be numerically minimized by a proper choice of the cover permeability and permittivityFor the operation at both polarizations, different parameters may be used.

( ) ( ) ( )( )

( )( )

20 01

λ λ μ μμ με εε μ μμ μ

+−−= ≠

− +−= =a,b a,b c pcc n

c cc p

b , n , a b, na

Cloak with ENZ materials at microwaves (III-VIII)



b

a

,ε μ

c c,ε μp

p

ε

μ

0

101 832

2p

p

a mmb . a

f GHzε

μ

⎧ =⎪

=⎪⎪ =⎨⎪ =⎪⎪ =⎩

For an unbounded cylinder with the geometrical and electrical parameters:

an ENZ is needed in order to reduce the SCS in TM polarization: 0 1 1= =c c. ,ε μ

[ ]

01

ε

ε εε

σ

−= =

−

c

c

c

TM

b , na

Min

Cloak with ENZ materials at microwaves (IV-VIII)



SCS minimization rate versus parameter variation in the theoretical model (constant permittivity):

-4 -2 0 2 4

-10

0

10

20

30

40

σ TM [d

B]

εc

0 1 2 3 4 5-10

0

10

20

30

40

σ TM [d

B]

μc

NoCoverTM

TM CoverTM

=σσσ

Cloak with ENZ materials at microwaves (V-VIII)



0 1 2 3 4 5-40

-30

-20

-10

0

10

20

30

40

σ TM [d

B]

frequency [GHz]

Theoretical SCS minimization rate for Drude-like dispersive permittivity:

2,5 3,0 3,5

-0,4

-0,2

0,0

0,2

0,4

ε'r ε''r

frequency [GHz]

Cloak with ENZ materials at microwaves (VI-VIII)



Theoretical SCSminimization rate versus losses in the metamaterial.

Anyway, since the ENZ regime is considered, losses are expected to be low. 10-3 10-2 10-1

20

30

40

σ TM [d

B]

εr''

′ ′′ε = ε − εc c cj

Cloak with ENZ materials at microwaves (VII-VIII)



b

a

,ε μ

c c,ε μp

p

ε

μ

1,0 1,5 2,0 2,5 3,0

0

10

20

30

40

σ TM [d

B]

α0 1 2 3 4 5

0

10

20

30

40

σ TM [d

B]

frequency [GHz]

Theoretical SCS minimization rate versus geometrical parameters:

α =ba

y

xz

ρ

ϕiE

iHa

L = ∞

Cloak with ENZ materials at microwaves (VIII-VIII)



Outline




-4 -2 0 2 4

-10

0

10

20

30

40

σ TE [d

B]

μr

0 1 2 3 4 5-10

0

10

20

30

40

σ TE [d

B]

εc

NoCoverTE

TE CoverTE

=σσσ

A MNZ is needed in order to reduce the SCS in TE polarization: 0 11= =c c, .ε μ

Cloak with MNZ materials at microwaves (I-III)



10-3 10-2 10-1

20

30

40

μc''

σ TE [d

B]

0 1 2 3 4 5

-40

-30

-20

-10

0

10

20

30

40

frequency [GHz]

σ TE [d

B]

Theoretical SCS minimization rate for dispersive permeability:

′ ′′μ = μ − μc c cj

( ) ( )( )

20

2 20

∞∞

μ −μ ωμ ω = μ +

ω −ω + ωδs

c j

Cloak with MNZ materials at microwaves (II-III)



1,0 1,5 2,0 2,5 3,0

0

10

20

30

40

σ TE [d

B]

α0 1 2 3 4 5

0

10

20

30

40

frequency [GHz]

σ TE [d

B]

Theoretical SCS minimization rate versus geometrical parameters:

b

a

,ε μ

c c,ε μp

p

ε

μ

α =ba

y

xz

ρ

ϕ

iE

iHa

L = ∞

Cloak with MNZ materials at microwaves (III-III)




Outline



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



2,5 3,0 3,5

-40

-30

-20 L = 30 [mm] L = 60 [mm] L = 90 [mm] L = 120 [mm] L = 150 [mm] L = 180 [mm]

Max

(σTM

) [dB

]

frequency [GHz]2,5 3,0 3,5

-40

-30

-20


Max

(σTM

) [dB

]

frequency [GHz]

Max SCS values for covered (left) and uncovered (right) structure:

Design of a MNZ-ENZ cloak at microwaves (II-XVII)



30 60 90 120 150 180

10

20

Max

(σTM

) [dB

]

L [mm]2,5 3,0 3,5

-50

-40

-30

-20

-10

Max

(σTM

) [dB

]

no cover with cover

frequency [GHz]

Max SCS values versus cylinder length:

Design of a MNZ-ENZ cloak at microwaves (III-XVII)



1,2 1,4 1,6 1,8 2,0 2,2 2,4 2,6 2,8 3,0-10

-5

0

5

10

15

20

σ TM [d

B]

α

L=100 [mm]

Full-wave SCS minimization rate versus geometrical parameters:

b

a

,ε μ

c c,ε μp

p

ε

μ

Design of a MNZ-ENZ cloak at microwaves (IV-XVII)



Simulated SCS for covered (left) and uncovered (right) structure:

Design of a MNZ-ENZ cloak at microwaves (V-XVII)



E-field for covered (left) and uncovered (right) structure:

Design of a MNZ-ENZ cloak at microwaves (VI-XVII)



H-field for covered (left) and uncovered (right) structure:

Design of a MNZ-ENZ cloak at microwaves (VII-XVII)



2,5 3,0 3,5

-40

-30


Max

(σTE

) [dB

]

frequency [GHz]2,5 3,0 3,5

-40

-30

-20

-10 Ly = 30 [mm] Ly = 60 [mm] Ly = 90 [mm] Ly = 120 [mm] Ly = 150 [mm] Ly = 180 [mm]

Max

(σTE

) [dB

]

frequency [GHz]

Max SCS values for covered (left) and uncovered (right) structure in TE polarization:

Design of a MNZ-ENZ cloak at microwaves (VIII-XVII)



30 60 90 120 150 180

10

20

L [mm]

σ TE [d

B]

Max SCS values versus cylinder length in TE polarization :

2,5 3,0 3,5-50

-40

-30

-20

-10

Max

(σTE

) [dB

]

no cover with cover

frequency [GHz]

Design of a MNZ-ENZ cloak at microwaves (IX-XVII)



Simulated SCS for covered (left) and uncovered (right) structure:

Design of a MNZ-ENZ cloak at microwaves (X-XVII)



E-field for covered (left) and uncovered (right) structure:

Design of a MNZ-ENZ cloak at microwaves (XI-XVII)



H-field for covered (left) and uncovered (right) structure:

Design of a MNZ-ENZ cloak at microwaves (XII-XVII)



E-field for covered structure:

Design of a MNZ-ENZ cloak at microwaves (XIII-XVII)



H-field for covered structure:

Design of a MNZ-ENZ cloak at microwaves (XIV-XVII)



Both polarization setup:

1 2 3 4f @GHzD

-40

-20

20

40 ( )

2

20

2⎛ ⎞⎜ ⎟⎝ ⎠≈ −eff d

N

k rε ε

effμ

Design of a MNZ-ENZ cloak at microwaves (XV-XVII)



Design of a MNZ-ENZ cloak at microwaves (XVI-XVII)

TM polarization results (E-field) of a dielectric cylinder (εr = 2) without the cover (left) and with cover (right).



Design of a MNZ-ENZ cloak at microwaves (XVII-XVII)

TE polarization results (H-field) of a dielectric cylinder (εr = 2) without the cover (left) and with cover (right).



Outline




Electrically small covered cylinder in vacuum

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

y

z

x

y

z

x

( )a ( )b

iE

iH

iH

iE

a a

( )( )

( )( )

2

10

10

1

εε ε

μ μμμμ μ

⎧ −= =⎪

−⎪⎪⎨

+−⎪= ≠⎪ +−⎪⎩

c

c p

c pcncc p

a b, n

a b, n( )( )

( )( )

2

10

10

1

μμ μ

ε εεεε ε

⎧ −= =⎪

−⎪⎪⎨

+−⎪= ≠⎪ +−⎪⎩

c

c p

c pcncc p

a b, n

a b, n



Electrically small covered metallic cylinder in vacuumy

z

x

y

z

x

( )a ( )b

iE

iH

iH

iE

a a

( ) ( )( )

2 2 2 20 01

1 0 0 11

επ μ α μ μ ε

ε−⎡ ⎤∝ + − − ⇒ = ≠ ⇒ < <⎣ ⎦ +

cnc cc

c j k a a b, n

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



0 05λ=a .

λ=d

0 5λ=L .

2λ

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

Half-wavelength dipole working at 3 GHz.



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

Electrically small covered metallic cylinder in vacuum with emishpeares



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



Dipole alone Dipole+Obstacle Dipole+Cloaking

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



y

z

x

y

z

x

( )a ( )b

iE

iH

iH

iE

a a

Possible setup: parallel plate plasma column surrounding the metallic cylinder.

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



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




Outline



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-SiO2), under certain conditions, can be effectively used to have such materials in the visible.



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

State-of-the-art of Metamaterials at THz and optical frequencies (II-III)



At optical frequencies the kinetic inductance of the electron should be taken into account.

A saturation in the scaling is expected.

State-of-the-art of Metamaterials at THz and optical frequencies (III-III)



Outline




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.



d1

d2

z

yε1µ1

Design of a cloak with ENZ Metamaterials at THz and/or optical frequencies (II-XV)

1 2xy

z 1 2

2

1

1

1 1 11dd

ε + ηεε =

+η

⎛ ⎞η= +⎜ ⎟ε + η ε ε⎝ ⎠

η =



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.

d1 d2

d1

d2 object εobj, μobj

εobj, μobj ηε η ε ε

⎛ ⎞= +⎜ ⎟+ ⎝ ⎠1 2

1 1 11r

ε ηεεη

+=

+1 2

1z

η = 2

1

dd

ηε η ε ε

⎛ ⎞= +⎜ ⎟+ ⎝ ⎠1 2

1 1 11z

ε ηεεη

+=

+1 2

1r

η = 2

1

dd

2

1

2

1

Design of a cloak with ENZ Metamaterials at THz and/or optical frequencies (III-XV)



With a plasmonic material (Ag) and silica (SiO2), 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.

Design of a cloak with ENZ Metamaterials at THz and/or optical frequencies (IV-XV)



500 550 600 650 700

-20

-15

-10

-5

0

Rea

l par

t of t

he re

lativ

e pe

rmitt

ivity

Frequency [THZ]

SiO2

Silver Layered medium

Choosing a proper value for ηit is possible to obtain the desired permittivity.

In this case is not possibile to get the same values (εr = εz)

Design of a cloak with ENZ Metamaterials at THz and/or optical frequencies (V-XV)



b

a

,ε μ

c c,ε μp

p

ε

μ[ ]

[ ]0

p

p

a 50 nmb 1 8 a

f 600 THz2

1

ε

μ

⎧ =⎪

=⎪⎪ =⎨⎪ =⎪⎪ =⎩

.


an ENZ is needed in order to reduce the SCS in TM polarization: 0 32 1ε μ= =c c. ,

[ ]

01

ε

ε εε

σ

−= =

−

c

c

c

TM

b , na

Min

Design of a cloak with ENZ Metamaterials at THz and/or optical frequencies (VI-XV)

[ ]1

0 21100 22

η

ε

=⎧⎪ =⎨⎪ ≈⎩ c

.d nm

.



Max SCS values versus cylinder length in TM polarization

500 550 600 650 700-140

-135

-130

-125

-120

Max

val

ue o

f RC

S

Frequency [THz]

L=250 [nm] L=500 [nm] L=720 [nm] L=960 [nm]

500 550 600 650 700

-140

-135

-130

-125 with cover, L=500 [nm] no cover, L=500 [nm]

Max

val

ue o

f RC

S

Frequency [THz]

Design of a cloak with ENZ Metamaterials at THz and/or optical frequencies (VII-XV)



Simulated RCS for uncovered (left) and covered (right) structure:

Design of a cloak with ENZ Metamaterials at THz and/or optical frequencies (VIII-XV)



E-field for uncovered (left) and covered (right) structure:

Design of a cloak with ENZ Metamaterials at THz and/or optical frequencies (IX-XV)



H-field for uncovered (left) and covered (right) structure:

Design of a cloak with ENZ Metamaterials at THz and/or optical frequencies (X-XV)



b

a

,ε μ

c c,ε μp

p

ε

μ[ ]

[ ]0

p

p

a 50 nmb 1 8 a

f 600 THz2

2

ε

μ

⎧ =⎪

=⎪⎪ =⎨⎪ =⎪⎪ =⎩

.


an ENZ is needed in order to reduce the RCS in TM polarization: 0 1 1= =c c. ,ε μ

[ ]

01

ε

ε εε

σ

−= =

−

c

c

c

TM

b , na

Min

Design of a cloak with ENZ Metamaterials at THz and/or optical frequencies (XI-XV)

[ ]1

0 23100 1

η

ε

=⎧⎪ =⎨⎪ ≈⎩ c

.d nm

.



500 550 600 650 700-150

-140

-130

-120

Max

val

ue o

f RC

S

Frequency [THz]

L=200 [nm] L=400 [nm] L=600 [nm] L=800 [nm]

500 550 600 650 700-150

-140

-130

-120 without cover, L=800 [nm] with cover, L=800 [nm]

Max

val

ue o

f RC

S

Frequency [THz]

Max SCS values versus cylinder length

Design of a cloak with ENZ Metamaterials at THz and/or optical frequencies (XII-XV)



Simulated SCS for uncovered (left) and covered (right) structure:

Design of a cloak with ENZ Metamaterials at THz and/or optical frequencies (XIII-XV)




Design of a cloak with ENZ Metamaterials at THz and/or optical frequencies (XIV-XV)




Design of a cloak with ENZ Metamaterials at THz and/or optical frequencies (XV-XV)



Outline




b

a

ε

cε

pε

Radiation Pressureover a covered nanoparticle (I-XXII)

For covered spherical non metallic particles the total normalized RCS can be expressed as:

( )( ) 2

1

2 2 1σσ

∞

== = + +∑t

t l lg l

Q l a bka

Re

( ) ( )( ) ( )

2 11

1c c p

lc p c

l la b

l l

ε ε ε ε

ε ε ε ε+

⎡ ⎤− + +⎣ ⎦=− ⎡ + + ⎤⎣ ⎦

l

l

l

ba

2 3

1 1pr cr

pr cr

ε ε

μ μ

= =

= =

ka

2b a=

In the Rayleigh limit the coefficients bl go rapidly to zero, while the others disappear when



b

a

ε

cε

pε

1 2 3 4 5 6 7 8 90

2

4

6

81 2 3 4 5 6 7 8 9

0

1

1 2 3 4 5 6 7 8 90

2

4

6

81 2 3 4 5 6 7 8 9

0

1

l

la↑

↓

ka lb↑

↓

3 1r rε μ= =

l

ka

3 1r rε μ= =

1 2 3 4 5

0.2

0.4

0.6

0.8

1

5

4

3

2

1

- -l

ka

la 3 1r rε μ= =

1 2 3 4 5

0.2

0.4

0.6

0.8

1

5

4

3

2

1

- -l

ka

lb 3 1r rε μ= =

Radiation Pressureover a covered nanoparticle (II-XXII)



Radiation Pressureover a covered nanoparticle (III-XXII)

b

a

ε

cε

pε

2.5 5 7.5 10 12.5 15

1

2

3

4

- -a / λ

ka

( )3 2 1 0 1 1r r. j .ε μ= + =

sQ

sQ

sQ

tQ

( )( ) 2

1

2 2 1σσ

∞

== = + +∑t

t l lg l

Q l a bka

Re

0 12

λλ σ σ

< ⇒> ⇒ →s g

a . Rayleigh limita



“Are optical forces derived from a scalar potential?” 23 July 2007 Vol. 15, No. 15 OPTICS EXPRESS, Daniel Maystre and Patrick Vincent

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

Radiation Pressureover a covered nanoparticle (IV-XXII)



y

z

x

y

z

x

( )a ( )b

iE

iH

iH

iE

a a

The total force for an unbounded dielectric lossless cylinder can be expressed as:

( ) ( )

( ) ( ) ( )

( )

2 2200

0

0 0

2 200

2 4

12 4 2

1 14

ε εε εε

ε ε ε εω ε

εμ ε

ε

∗

Ω ∂Ω ∂Ω

∗ ∗

∂Ω ∂Ω

−− ⎡ ⎤= −∇ Ω+ ∂Ω + ∂Ω⎣ ⎦

− −⎡ ⎤ ⎡ ⎤= ∇ ×∇ ∂Ω + ∂Ω⎢ ⎥ ⎣ ⎦⎣ ⎦

⎛ ⎞= − −⎜ ⎟⎝ ⎠

∫ ∫ ∫

∫ ∫

f E n

f z E

in inxy n t ,xy n

inz z z n z

z z

Ud Re E d E d

Îm H E d Re E d

U H E“Are optical forces derived from a scalar potential?”

23 July 2007 Vol. 15, No. 15 OPTICS EXPRESS, Daniel Maystre and Patrick Vincent

Radiation Pressureover a covered nanoparticle (V-XXII)



Ω

∂Ωn

For S-polarization or P-polarization either Hz or Ez vanishes. In particular for S-polarization the total force reduce to:

( )

( )

20

00

14

0

2

ε ε

ωμε ε

Ω Ω

∗

= −∇ Ω = − ∇ Ω

=

⎡ ⎤= × = −⎣ ⎦

∫ ∫f

f P H P E

xy z

z

z

Ud E d

f

Im ,

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

Radiation Pressureover a covered nanoparticle (VI-XXII)



Approximated total force on a dielectric lossless electrically small SiO2 cylinder

( )

( ) ( )

( ) ( )

20

0 0

00 0 10

0 012

0

14

2

ϕ

ε

ε

ε ε

ρ ϕ με ρ

Ω Ω∞

=−∞

→

→

= −∇ Ω = − ∇ Ω

= =

= = −

∇ =

∫ ∫

∑

f

E

0

l

l

xy z

n jnz z n n

n

z

z

Ud E d

ˆ Ê , E j J k e

j Elim E E

ak H ak

lim E

z z

0,0 0,5 1,0 1,5 2,0 2,5 3,010-23

10-22

10-21

10-20

Mag

nitu

de o

f the

Rad

ial F

orce

εr

a=12.5 [nm]

Radiation Pressureover a covered nanoparticle (VII-XXII)



Approximated total force on a dielectric lossless SiO2cylinder

( )

( ) ( )

( ) ( )

20

0 0

00 0 10

0 012

0

14

2

ϕ

ε

ε

ε ε

ρ ϕ με ρ

Ω Ω∞

=−∞

→

→

= −∇ Ω = − ∇ Ω

= =

= = −

∇ =

∫ ∫

∑

f

E

0

l

l

xy z

n jnz z n n

n

z

z

Ud E d

ˆ Ê , E j J k e

j Elim E E

ak H ak

lim E

z z

0,0 0,5 1,0 1,5 2,0 2,5 3,010-21

10-20

10-19

10-18

Mag

nitu

de o

f the

Rad

ial F

orce

εr

a=50 [nm]

Radiation Pressureover a covered nanoparticle (VIII-XXII)



Approximated total force on a dielectric SiO2 cylinder

0,0 0,5 1,0 1,5 2,0 2,5 3,010-21

10-20

10-19

10-18

Mag

nitu

de o

f the

Rad

ial F

orce

εr

a=50 [nm]

Radiation Pressureover a covered nanoparticle (IX-XXII)

F=2.11 · 10−18 [N/m]



Ω

∂Ωn

Force density field inside an electrically small silica cylinder: a = 12.5 nm

Radiation Pressureover a covered nanoparticle (X-XXII)



b

a

,ε μ

c c,ε μp

p

ε

μ

Approximated total force on a dielectric lossless covered SiO2 cylinder

( ) ( )1 2

00 1 0 22

ωμε ε ε ε∗ ∗

Ω Ω

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥= − × Ω + − × Ω⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦∫ ∫f E H E Hxy cIm d Im d

2 130 5

1 81

εε

ε εε

==

−= ≈

−

c

c

c

..

b .a

0,0 0,5 1,0 1,5 2,010-22

10-21

10-20

Mag

nitu

de o

f the

Rad

ial F

orce

εr

a=12.5 [nm]

Radiation Pressureover a covered nanoparticle (XI-XXII)



b

a

,ε μ

c c,ε μp

p

ε

μ

Approximated total force on a dielectric lossless covered SiO2 cylinder

( ) ( )1 2

00 1 0 22

ωμε ε ε ε∗ ∗

Ω Ω

⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥= − × Ω + − × Ω⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦∫ ∫f E H E Hxy cIm d Im d

[ ]

2 130 25

1 8ε

εε

σ

=≈

=c,b

c

TM

..

Min .

0,5 1,0 1,5 2,0

10-20

10-19

10-18

Mag

nitu

de o

f the

Rad

ial F

orce

εc

a=50 [nm], SiO2

Radiation Pressureover a covered nanoparticle (XII-XXII)



Force density inside an small SiO2 covered cylinder.

εc=3εc=0.5

SiO2 SiO2

Radiation Pressureover a covered nanoparticle (XIII-XXII)



From the expression of the force density a similar expression can be derived for metallic particles

( ) 14

∗ ∗= ∇ + ×∇×f d E d E

( )( ) ( )( )

( )( ) ( )( )

3

3

3

3

2 23

2 2 2

ε ε ε ε ε ε ε εε

ε ε ε ε ε ε ε ε

− + − + −⇒ =

+ + − − −

P E d Pc c p c c p

int

c c p c c p

ab V

ab

In the Rayleigh approximation for a dielectric spherical particle

“Force of Optical Radiation Pressure on a Spheroidal Metallic Nanoparticle”October 2007 Vol. 33, No. 10 OPTICS EXPRESS, N. I. Grigorchuk P. M. Tomchuck

Radiation Pressureover a covered nanoparticle (XIV-XXII)



Radius ratios for a covered spherecε

pεcε ε=

2p

cε

ε = −

pε ε=-7.5 -5 -2.5 0 2.5 5 7.5

-7.5

-5

-2.5

0

2.5

5

7.5

0

1

∉

pε

cε ab

↑

↓( )( )( )( )

32

2

ε ε ε ε

ε ε ε ε

− +=

− +

c c p

c p ca b

Radiation Pressureover a covered nanoparticle (XV-XXII)



Magnitude of the dipole moment

εp b/a=1 b/a=1.1 b/a=1.2εp εp

εc εc εc

εp

b/a=1.3 b/a=1.4 b/a=1.5εp εp εp

εc εc εc

Radiation Pressureover a covered nanoparticle (XVI-XXII)



A silver sphere with a layered cover

500 550 600 650 700-145

-140

-135

-130

no cover with coverM

ax v

alue

of R

CS

Frequency [THz]

Radiation Pressureover a covered nanoparticle (XVII-XXII)

[ ]

[ ]1

0 250

1 8

100 5

η

ε

=⎧⎪ =⎪⎪⎪ =⎨⎪⎪ =⎪

≈⎪⎩ c

.a nmb .ad nm

.




500 550 600 650 700

-145

-140

-135

-130

Max

val

ue o

f RC

S

Frequency [THz]

no cover η=0.2 η=0.16 η=0.17 η=0.18 η=0.21

550 560 570 580 590 600 610 620 630 640 650-145

-140

-135

-130

Max

val

ue o

f RC

S

Frequency [THz]

no cover with cover

Radiation Pressureover a covered nanoparticle (XVIII-XXII)




500 550 600 650 700-140

-135

-130

Max

val

ue o

f RC

S

Frequency [THz]550 560 570 580 590 600 610 620 630 640 650

-145

-140

-135

-130

Max

val

ue o

f RC

S

Frequency [THz]

no cover with cover

Radiation Pressureover a covered nanoparticle (XIX-XXII)




Radiation Pressureover a covered nanoparticle (XX-XXII)




Radiation Pressureover a covered nanoparticle (XXI-XXII)



A silica sphere with a layered cover

Radiation Pressureover a covered nanoparticle (XXII-XXII)

600 650 700-200

-195

-190

-185

-180

Max

val

ue o

f RC

S

Frequency [THz]

a=12.5 [nm] [ ]1

0 15

1 6

100 7

η

ε

=⎧⎪⎪ =⎪⎨⎪ =⎪

≈⎪⎩ c

.b .ad nm

.

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