nemss-2008, middletown single-particle rayleigh scattering of whispering gallery modes: split or not...

NEMSS-2008, Middletown

Single-particle Rayleigh scattering of whispering gallery

modes: split or not to split?

Lev Deych, Joel Rubin

Queens College-CUNY


Acknowledgements

• Thanks go to Thomas Pertsch, Arkadi Chipouline, and Carsten Schimdt of the Friedrich Schiller University of Jena for their hospitality last summer, when part of this work was done

• Partial support for this work came from AFOSR grant FA9550-07-1-0391, and PCS-CUNY grants


WGM in a single sphere

Modes are characterized by angular (l), azimuthal (m), and radial (s) numbers. Poles of the scattering coefficients determine their frequencies and life-times, which are degenerate with respect to m.

( ) ( )

( , )( , )

( , ) ( , )

( ) ( )

; ;

; ( ) scattering coefficients

sc lm lm lm lmlm

N Mlm lm lm lm lm lm

M NM N lmslms M N M N

ls ls

k ka b

na a k

c

x kRx x i

r rE N M


Fundamental modesZ

Y

X

0 0.5 1 1.5 2 2.5 3

0

1

2

3

4

5

6

, 1m l s

Fundamental modes are concentrated in the equatorial plane

ccw; cwm l m l


Fundamental modes and the coordinate system

Y

Z

X

Linear combination of VSH with

A mode is fundamental only with respect to a given coordinate system

Z

Y

X

Single VSH with

or m l m l l m l


Double peak structure of the spectrum in single resonators

Transmission through an optical fiber coupled to a silicon microdisk. M.Borselli,T.J.Johnson,and O.Painter, Opt.Express 13,1515 (2005).

Near field spectrum showing the peak structure caused by coupling to the tip of the near field microscope itself. A. Mazzei,et. al.,Phys. Rev. Lett. 99, (2007).


CW-CCW splitting – origin of the idea

“We have observed that very high-Q Mie resonances in silica microspheres are split into doublets. This splitting is attributed to internal backscattering that couples the two degenerate whispering-gallery modes propagating in opposite directions along the sphere equator”


CW-CCW splitting paradigm

“… backscattering is observed as the splitting of initially degenerate WG mode resonances and the occurrence of characteristic mode doublets.”

“mode splitting has been … explained as the result of the coupling between … degenerate clockwise and counterclockwise modes via back scattering.”

M.L. Gorodetsky, et al. Opt. Soc. Am. B 17, 1051 (2000)

A. Mazzei,et. al.,Phys. Rev. Lett. 99, (2007)

Coupling coefficient


Axial rotational symmetry and CW-CW degeneracy

Why ?

( )R 1 2 2 1

R R R R Abelian group: Only one-dimensional representations: no degeneracy!

Typical answers: 1

Maxwell equations are 2nd order – time reversal is not linked to complex conjugation

Phys. Rev. A, 77, 013804 (2008), Dubetrand, et al

In disks and ellipsoids full rotational symmetry is replaced by an axial rotational symmetry. Degeneracy with respect to m is lifted, but m m

2. Kramers degeneracy D.S. Weiss. Optics Letters, 20, 1835, (1995)

Both answers are wrong


Inversion symmetry and CW-CCW degeneracy

P

im imR e e

for any angle

R P PR

With the inversion, the group is non-Abelian and permits two-dimensional representations.

m m due to inversion symmetry, not rotation

m m


Symmetry, CCW-CW coupling and Rayleigh scattering

Sub-wavelength scatterers = dipole approximation for the scatterer = shape of the scatterer is not important, can be assumed to be spherical

No axial rotation symmetry, but the inversion symmetry is still there = No coupling between cw and ccw modes in the dipole approximation = no lifting of degeneracy

Z

Y

X

For multiple scatterers (surface roughness) the same is true in the single scattering approximation


Mie theory of scattering of WGMModel a scatterer as a sphere and solve the two-sphere scattering problem, using multi-sphere Mie formalism

2( ) ( ) ( ) ( ), , , ,

1 ,

(1,2) (1,2) (1,2) (1,2) (1,2), , 1,2 , , 1,2

,

i i i iscat l m l m i l m l m i

i l m

in l m l m l m l ml m

a b

c d

E N r r M r r

E N r r M r r

( ) (1,2), ,

,

2 !;

2 ( )!( )!

L

iinc l m L m lm lL L

l m

i L

L m L m

E NExcites a fundamental

ccw WGM

1

2

radius of the main sphere

radius of the scatterer

distance between the centers

R

R

dZ

Y

Scattered field

Internal field


Scattering coefficients

(1) (1) (1) (2) (2) (2) (1), , 1 2 , 1 2; 1

l ll m l ml m l L m l m lm l m l l m lm

l l

a a A a a A

r r r r

Application of the Maxwell boundary conditions gives, for the scattering coefficients (neglecting cross-polarization coupling)

( )cos

2 11 1 ( 1) ( 1)

2 1

( , , , , ) ji

j

l lm pl m l l

lm

p j

i jp l l

i mmj ii

mmi l

lA i i l l l l p p

h

l l

r m l m l p kr P ekr

r rX

0; 0 l m l mji ji lm i j lm mmA A kd

r r

In the chosen coordinate system translation coefficients are diagonal in m

Translation coefficients describe coupling between spheres


Dipole approximation

1(2) (2) (1)1, 1 1 1 2

(1) (1) (1) (2) 1, , 1 1 2

1

1 ;

0 for 1

l l mm l m m

l

ml m l L m m lm

mlm

a a A

a a A

A m

r r

r r

In the dipole approximation(2) 0 for 1lma l

1

1

(1) 1

1

(1) (2)(1) (1) 1

, , 1(1) (2) ', 1 ' '

'

(1) (1), ,

, 11 1

1

Lm mL m lm

l L m l L l l m ml m l m l m

l

l L m l L

A Am

a A A

m

Now equation for the scattering coefficients can be solved exactly


Convergence of the sum over l

(1) 1

1

2 12' 1 1

' ' 3, 1

( )

ll m m

l m l m

R RlA A

kd d d

Translation coefficients grow with l, therefore there is an issue of convergence

of the sum over l in the equation for scattering coefficients. For 1l

one obtains proving convergence

11 2 2 1; For 0, , 1 1 slow convergence

Rd R R R R

d

d

1R2R

To improve numerical convergence we introduce

33 2 22 112 1

33 3 2 21 21

1 1 1

( ) 4( )

l

l

dR R dRs l

kd d kRkd d R

and present (1) 1 (1) 1 (1) 1

1 1 1

2 2 2 11 1 2 1

3

3 1 / 2 3 1 / 21 1

2 2( )

lL llm m Lm m lm m

l m lm L m Lm l m lml l L

i m i m RA A s A A A A l

kd d


Single mode approximation and resonances

(1) 1 (1) 1

1 1

1 1 231 1 1 / 2

2l Llm m Lm m

l m lm L m Lml

iA A A A m s

1

1

(1) 1

1

(1) (2)(1) (1) 1

, ,(1)(2) 2 (2),1 1

(1) (1), ,

, 13

1 1 1 / 22

1

Lm mL m lm

l L m l Ll L Lm ml m

L m Lm

l L m l L

A Am

ia A A m s

m

(1)

(1) 1

1

(1) 1 1

1 1

1

11 2 (2) (2)1 1

1 1 1(1) 1 2 (2) (2) (2)1 1 1

[ ] 0, 1,

3[ ] 1 1 / 2 1 0, 1,

2

3[ ] 1 1 / 2 1 1 0, 1,

2

L

L Lm mL m Lm

L lLm m lm ml L m Lm m lm

m l L

im s A A m l L

im s A A A A m l L

A resonance at the original single sphere frequency, unmodified

Two new frequencies for 1, 0m m

Weak resonances from terms with l L


Approximate expressions for the shifted frequencies:

2 2 5

2 51 1

( ) ( )2

2 1 3 2 1lm lm

lm l l l l l

kd kdf fx x px i p x

l R d l R d

Scattering induced resonances

12 2 2

3 32 22 2

2

1 1( ) ( )

1 21 1

2 2 1

( )lm l lm l lm lkd x u h kd v

n m np R sR

n n

f h kd

Effective polarizability of the scatterer is renormalized by higher l terms. This explains experimental fundings of Mazzei et al A. Mazzei,et. al.,Phys. Rev. Lett. 99, (2007).


Rayleigh scattering of WGM

A. Mazzei,et. al.,Phys. Rev. Lett. 99, (2007).

4Compare to: ;l l lx x x

To treat WGM’s scattering within a framework developed for plane waves leads to wrong results.

Famous Rayleigh law for scattering cross section is replaced with law for WGMs. This change in scattering law is traced to changes in asymptotic behavior of Hankel function from

45

1/ 1/

2 5;ll lx x x


Numerical results

Exact numerical computation for TM39 mode. Terms with angular momentum up to 50 were included.

The third peak is too weak to be seen here.

Relative height of the peaks depends on the size of the scatterer and distance d.

The result is the same when a cw mode is excited: degeneracy is not lifted

1mSingle-sphere resonance


Conclusion• An exact ab initio theory of Rayleigh (dipole) scattering

of WGM of a sphere based on multisphere Mie theory is derived

• The picture of scattering based on coupling between cw and ccw modes is proven wrong.

• It is shown that one of peaks in the optical response corresponds to the single sphere resonance, while the other comes from excitation by the scatterer of WGM with azimuthal numbers

• Quadratic dependence of peak’s width versus shift is explained by renormalization of the effective polarizability due to interaction with high order modes

1m

nemss-2008, middletown single-particle rayleigh scattering of whispering gallery modes: split or not...

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