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### RESONANT MULTIPLE SCATTERING

R.V.MEHTA

Ramanna Fellow

Department of Physics

Bhavnagar University,Bhavnagar 364022

scattering

- Light wave when encounters obstacles either or both of two phenomena can occur
- Obstacle may absorb light
- And/or light may change its direction-this is in general called scattering
- If efficiency of absorption is high the object will appear black
- If efficiency of scattering is high then object will appear white.

Multiple Scattering

- Here light propagation is influenced by more than one obstacles
- Absorption and scattering of individual particles play their roles.
- More likely to occur when number concentration of scatterers are very large.
- Wave character of light is washed out.
- The regime is called radiative transfer.
- Earlier researchers working in optics laboratory were avoiding this regime and was considered ‘dull’.

Astrophysics

- The light scattering in the interstellar medium cannot have our control and here MS plays a decisive role to determine emission and absorption line spectrum of a star.
- Thus development of theory of radiative transfer took place to understand the astrophysical processes.

Electron Physics

- A large interest in MS was generated from quite a different direction.
- It is known that electron scatter from a local variation in potential just like light scatter from local variation in dielectric constant. The resistivity of metals at low temperatures can be determined by MS of electrons from the potentials of impurities. One can explain zero and finite resistivity of perfect and imperfect crystals using wave aspect of electrons.

Electron waves & light waves

- Analogies between the propagation of electron waves in condensed matter and that of electromagnetic waves in strongly scattering media has led to discovery of several new optical phenomena.

New Optical Effects

- Photonic Hall Effect :Lorentz force in semiconductor leads to the well known Hall effect, in which the application of magnetic field to an electron-transporting medium generates a new current( or voltage) perpendicular to the direction of both the original current and field. Similarly the propagation of light in disordered medium is affected by a magnetic field.
- Other effects e.g. Photonic magnetoresistance, anisotropic diffusion , weak coherent backscattering and localization.

What is a Localized State?

- A localized wave is a nonpropagating wave state.
- Light in certain dielectric microstructures exhibits such localized state.
- It is similar to the localized wave functions of electrons in disordered solids.
- Such microstructures exhibits certain new optical effects.

Anderson localization and PBG

- Strictly speaking the Anderson localization refers to nonpropagating waves while PBG denotes a regime which is empty of wave states. But in certain cases the two mechanism overlaps. In disordered PBG materials the crystalline periodicity is perturbed by disorder. A transition region is formed in which propagation is restricted in certain directions.
- In this talk we shall discuss such materials.

Wave Equations

- Electron :[ ( -h2/8π2m*)2 + V(x) ]ψ(x) = Eψ(x)
- Photons :-2E + ω(E)-(2/c2)εfluct(X)E =ε0 ω2/c2E
- Here ε(x) = ε0 + εfluct(x) The later plays an analogous role to the random potential in Schrodinger equn.; it scatters the emwaves.
- Note that ε0 (ω2/c2) which plays the role of energy eigen value is always positive. This precludes bound states of light in deep negative potential wells. (fig.)

Fig.

- The eigen value (ω2/c2)εo(gray line) of the photon must be > the highest of the potential barrier if (ε0+εfluct) is to be real and positive everywhere.

Anderson Localization

- In 1958 Anderson published a paper titled' Absence of diffusion in certain random lattice [Phys.Rev. vol.109,1492,1958]’.
- It describes the vanishing of propagation in the regime of very strong multiple scattering due to interference between electron waves. This serves a model for metal-insulator transition.

Light Localization!

- If electron waves can be localized then why not light waves? This question was first raised and discussed by Sajeev John [Phys. Rev. Lett.Vol.53,2169,1984] .
- Since then MS of light has become an active area of research. Main aim is to localize light and to find suitable conditions like strong scattering regime, and how to realize it experimentally.

Ferrofluid

- We recently found that a mixture of two magnetic colloids viz. a ferrofluid and a dispersion of micron sized magnetic spheres (also known as magneto-rheological fluid) serves an effiecient medium for direct observation of several of the above photonic effects.
- Why?

What is a Ferrofluid

- A ferrofluid is a colloidal dispersion of nanomagnetic particles. Hence for visible light size of the nanoparticles (~10 nm) is always < then the light wavelength. Hence such a disordered medium may be considered to be a homogenous effective medium.
- The fluid exhibits magneto-birefringence.
- Its dielectric constant is found to be a function of applied field.

Magnetorheological Fluid

- When micron sized magnetic spheres are dispersed in a liquid it is known as MR fluid.
- Viscosity changes drastically with field.
- Used in several devices

FF + MRF

- Magnetic holes.
- Under the influence of the field it exhibits field induced structures.
- Relative ε and μ are field dependent.
- For optical effects ε will be polarization dependent.
- Scattering will be mainly governed by the large particles.

Deductions

- When E vector is perpendicular to the field then and then only light disappeared at a critical field HC.
- For any other polarization direction light does not appears but only modulated.
- The combination FF+MRF is unique.

contd

- D(ω)=( ⅓)vE(ω)ltr(ω).

vE and ltr are transport velocity and transport mean free path and are quite different from the phase velocity and extinction mean free path. It has been found that the propagation of light in MS is characterized by the transport velocity and it can be very small near resonance.

Definition of certain parameters

- Frequency dependent complex refractive index: m(ω)≡ (co/vp) +i(co/2ωlext)≡η +ik
- The relation between wave vector and frequency of the coherent beam, k= η(ω)ω/co will give rise to polariton behaviour when the frequency is near resonance of the scatterers.
- Diffusion constant is defined by:

Wave equations

- For electrons and photons: H is Hamiltonian,ε(r) is static dielectric constant. We can write

Comparison

- Vector nature of light and 2nd derivative w.r.t. time are the two differences. It can be shown that the light energy and light potential are respectively given by
- Elight= (ω/c0)2 ; and V(r)light =[1-ε(r)]Elight
- Since for free motion E~p2 and ω2 ~Elight ~p2 the dispersion for electron is parabolic while that for light is linear in frequency.

As ω2 is always positive , bound state in this formalism is lost for light waves since dielectric constatnt is always positive.

Contd.

- From above we can deduce that at small frquency light scattering follow famous Rayleigh law and for dynamical system light and electrons behaves quite differently.

Resonant Multiple Scattering

- The scattering from single particle can be optimized by tuning the wavelength of light into a resonance. For this its size should be matched to the wavelength of the light
- This balancing of length scales together with a suitable shape will tune the particle into a resonator of light.
- Thus , the study of RMS is important topic for search for localization of light.

Single Scattering

- SS is the basis for good understanding of MS. MS will be dominant when (1) particles scatter light with conservation of energy. If there is a loss in SS then MS will be exponentially decreases (2) at resonance MS will be enhanced. At resonance the scatt. cross section reaches to maximum where it is proportional to the square of wavelength. If SS is large then MS is also anticipated to be large. Using tunable lasers a resonant frequency can be tuned.

RG or Born Approximation

- This is simpler to use for SS calculations
- Unfortunately at resonance the Born series diverges.
- The Mie solution is too complicated to use in a general MS theory.
- There are a large number of resonances.
- Only can be used after assuming ISA, Far-field approximation. Even then polarization inclusion is complicated.

Point scatterers

- PS avoids these difficulties. 3D properties of a point scatterer is

characterized by the local potential Vi(r,r;) =

- |Ψ+k> =[1+ G0(E)T i(E )]|k> ……..(1)

Contd.

- T-matrix in terms of Born series is given by
- Ti(E) =Vi + ViG0(E)Vi + ViG0(E) ViG0(E)Vi+..
- Here G0=(E+iε-Ho)-1is a Green function operator and shows that the radiation is emitted by the scatterer.
- In case of light scattering one has to replace the free Hamiltonian and the energy by
- H0→p2Δp and E→(ω/c0)2.

Light Scattering

- The light potential is Vi(ω,r’,r)= -γu(ω/c0)2δ(r-ri)δ(r-r’) where γ is the polarizability. The Green function takes the form given as follows. Here first term is electrostatic part and does not propagate while second one is transverse part and describe traveling wave solutions of Maxwell’s equations.

Contd.

- The first term represents electrostatic contribution and is diverging and given by 1/3u-1I and second term gives Maxwell’s solution of propagating part. This is diverging term but can be cutoff by parameter Λ with an additional factor 2/3. This leads to

matrix elements in momentum space can be found from the above eqn. The 1/3 part is absorbed into modified polarizability.Intrducing a length scale 4πΓ=1/Λ we get the final results

In the above ω02 ≡4πΓc02/αsph. Contrary to electron scattering in this dielectric simple model there will be resonance. It is shown that the scattering cross section is given by σscat (ω)= |t(ω)|2/6π

- Compared to Mie solution by sphere above has only one resonance hence simplify the case.

Delay time and Dwell time

- Dynamics of propagation of scattering waves are characterized by delay and dwell times. Former represents time duration of scattering process and later represents the time that wave spent inside the potential region.
- The former is the characteristic of resonant scattering.

Dwell time

- Lagendijk and Tigglen has shown that using simple geometrical optics one can derive dwell time.

Contd.

- In the figure a plane wave incident on a bigger particle having refractive index m=η+ik. In the particle this wave can be represented as
- ψ(s,t) =exp [ikms -iωt)
- Where s is an arbitrary coordinate along path of the wave. The amplitude decays exponentially and
- the emerging wave amplitude will be
- | ψout|2 =exp[-2kkL]
- Here L is the path length inside the particle.

Contd.

- Since the speed of light in the particle is
- co/η , the residence time becomes coL/η.
- |ψout|2 represents albedo (scatt/ext.cross sections)a which is the deviation from optical theorem due to dissipation of energy inside the particle.| This leads to dwell time

τd (ω) = κ→0 [1-a(ω)/2ωk]m

It has been shown that at resonance for m=2.73 and x=4.59 the dwell time is as large as 850a/co

This 5 time larger than thick samples used for MS experiments.

Magnetic Spheres

- In the above discussion it was assumed that the scatterers are non-magnetic.
- Pinheiro et al have shown that magnetic contribution may lead to resonances even In small particle limit (PRL,84,1435,2000).Effects are found on the Ioffe-Regel localization parameter, energy transport velocity and diffusion constant. Since magnetic permeability of most of the material is ~1 at optical frequencies it is suggested to work at microwave frequencies to varify these deductions.

Our work

- In all these theoretical as well as experimental work either the medium was nonmagnetic or the scatterers were nonmagnetic.
- We have prepared a medium –which may perform as ‘magnetorheological fluid’ in which large –micron sized magnetic spheres are dispersed in a ferrofluid. The later is a stable fluid in which nanomagnetic particles are colloidally dispersed in a liquid carrier. Ferrofluid performs as magnetizable medium.

Contd.

- Thus in this dispersion both the medium as well as scatterers are magnetic.
- It is known that dielectric constant of a ferrofluid can be modified by application of external magnetic field. This leads to variation of relative dielectric constant of micron sized particles w.r.t. the field. In short the parameter εscatt. in equations of Mie amplitudes an and bn. Thus field tuning of these parameters may lead to resonance under sutable conditions. (OL,33,1987,2008).

Experimental findings

- Zero total scattering

Theoretical Considerations

- Pinheiro et al have also shown that even in single scattering also Localization is favoured.
- We have considered scattering by micron sized particle whose dielectric constant can be varied by varying magnetic field. Using Langevin formalism expression for the dielectric constant is derived and is given by
- mf = (m∞ - m0)[ L(ξ))] + m0
- Where L(ξ) ={ cothξ – ξ-1}

Then using Mie expressions for scattering coefficients amplitude functions s1 and s2 were calculated

References

- 1.A.Lagendijk and B.A.Van Tiggelen,Physics Report270(1996)143-215
- 2. R.V.Mehta et al, PRL 96,127402(2006)
- 3.E.V.Mehta et al,PRB,,74, 195127(2006)
- 4.J.N.Desai, Current Sci. 93,452 (2007)
- 5.R.V.Mehta et al Current Sci.,93,1071(2007)

Acknowledgement

- Prof. R.V.Upadhyay, Prof. S.P.Bhatnagar, Dr. R.J.Patel, Dr. B.Chdasama, and several coworkers and students.

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