Lecture 1: Light Scatteringsol is the wavelength of light in the sol-vent of refractive index n...

$: Lecture 1: Light Scatteringsol is the wavelength of light in the sol-vent of refractive index n sol,with #being the wavelength of light in vacuum. The wave vector k s of the scattered$
Lecture 1: Light Scattering

F. Cichos

http://www.uni-leipzig.de/~mona

1.1 Static Light Scattering1.1.1 Rayleigh Scattering1.1.2 Mie Scattering

1.2 Dynamic Light Scattering1.2.1 Field Autocorrelation Function1.2.2 Intensity Autocorrelation Function

1. Light Scattering

Light - Scattering

Elastic Scattering

Rayleigh Scattering

Mie Scattering

Inelastic Scattering

Brillouin Scattering

Raman Scattering

......

Optical Coherence Tomography - Eye Pupil

Heterodyne Light Scattering - Single Virus

HI Virus

Sindbis Virus

mixture

Mitra et al. ACS Nano 4,1305 (2010).

Dark Field Microscopy

dark field image bright field image

dark field microscopy removes the huge background

Light Scattering

Static Light Scattering

Dynamic Light Scattering

• molecular structure (conformations) • solution structure (molecular interactions)

• dynamics of molecular structure • dynamics of solution structure

http://www.lsinstruments.ch/

204 POLARIZATION AND CRYSTAL OPTICS

Figure 6.2-l Reflection and refraction at the boundary between two dielectric media.

reflected waves are labeled with the subscripts 1, 2, and 3, respectively, as illustrated in Fig. 6.2-l.

As shown in Sec. 2.4A, the wavefronts of these waves are matched at the boundary if the angles of reflection and incidence are equal, 8, = 8,, and the angles of refraction and incidence satisfy Snell’s law,

n, sin 8, = n2 sin 0,. (6.2-1)

To relate the amplitudes and polarizations of the three waves we associate with each wave an x-y coordinate system in a plane normal to the direction of propagation (Fig. 6.2-l). The electric-field envelopes of these waves are described by Jones vectors

We proceed to determine the relations between J2 and J1 and between J3 and J1. These relations are written in the matrix form J2 = tJ1, and J3 = rJ1, where t and r are 2 X 2 Jones matrices describing the transmission and reflection of the wave, respectively.

Elements of the transmission and reflection matrices may be determined by using the boundary conditions required by electromagnetic theory (tangential components of E and H and normal components of D and B are continuous at the boundary). The magnetic field associated with each wave is orthogonal to the electric field and their magnitudes are related by the characteristic impedances, qO/n, for the incident and reflected waves, and q0/n2 for the transmitted wave, where qO = (P,/E,)‘/~. The result is a set of equations that are solved to obtain relations between the components of the electric fields of the three waves.

The algebraic steps involved are reduced substantially if we observe that the two normal modes for this system are linearly polarized waves with polarization along the x and y directions. This may be proved if we show that an incident, a reflected, and a refracted wave with their electric field vectors pointing in the x direction are self-consistent with the boundary conditions, and similarly for three waves linearly polarized in the y direction. This is indeed the case. The x and y polarized waves are therefore separable and independent.

The x-polarized mode is called the transverse electric (TE) polarization or the orthogonal polarization, since the electric fields are orthogonal to the plane of


I yx

0 900 81

II I I

‘PX ‘PX

0 0 900 900 81 81

Figure 6.2-2 Magnitude and phase of the reflection coefficient as a function of the angle of incidence for external reflection of the TE polarized wave (n2/n1 = 1.5).

External Reflection (n, < its). The reflection coefficient yX is always real and negative, corresponding to a phase shift qo, = r. The magnitude 1~~1 = (n2 - n&h1 + n,> at 8, = 0 (normal incidence) and increases to unity at 8, = 90” (grazing incidence). Internal Reflection (n 1 > nz), For small e1 the reflection coefficient is real and positive. Its magnitude is (nl - n2)/(nl + n2) when 8, = 0”, increasing gradually

1

l-4

0 90 6 4 81

Figure 6.2-3 Magnitude and phase of the reflection coefficient for internal reflection of the wave (n1/n2 = 1.5).

TE

REFLECTION AND REFRACTION 207

0 BB 900 0 BB 900

01 81

Figure 6.2-4 Magnitude and phase of the reflection coefficient for external reflection of the TM wave (n,/n, = 1.5).

to unity when 8r equals the critical angle 8, = sin- ‘(n,/n,). For 8, > 8,, the magnitude of rX remains unity, corresponding to total internal reflection. This may be shown by using (6.2-8) to write+ cos 8, = -[l - sin28,/sin28,]1/2 = -j[sin28,/sin28, - 1]‘/2, and substituting into (6.2-6). Total internal reflection is accompanied by a phase shift cpX = arg{Y,} given by

tan: = ( sin28 r - sin2B,) 1’2

cos 8, (6.2-9) TE Reflection

Phase Shift

The phase shift cpX increases from 0 at 8, = 8, to r at 8, = 90”, as illustrated in Fig. 6.2-3.

TM Polarization The dependence of the reflection coefficient yY on 8, in (6.2-6) is similarly examined for external and internal reflections:

n ExternaZ Reflection (n, < n,). The reflection coefficient is real. It decreases from a positive value of (n2 - n1)/(n2 + n,) at normal incidence until it vanishes at an angle 8, = e,,

(6.2-10) Brewster Angle

‘The choice of the minus sign for the square root is consistent with the derivation that leads to the Fresnel equations.


0 90 0 0 @B 0, 900 61 81

Figure 6.2-5 Magnitude and phase of the reflection coefficient for internal reflection of the TM wave (n,/nz = 1.5).

known as the Brewster angle. For 8r > en, P,, reverses sign and its magnitude increases gradually approaching unity at 8, = 90”. The property that the TM wave is not reflected at the Brewster angle is used in making polarizers (see Sec. 6.6).

. Internal Reflection (nl > nz). At 8, = O”, rY is negative and has magnitude (nl - n2)/(n1 + n2). As 8, increases the magnitude drops until it vanishes at the Brewster angle 8, = tanP1(n2/nr>. As 8, increases beyond 8,, Y,, becomes positive and increases until it reaches unity at the critical angle BC. For 8, > 8, the wave undergoes total internal reflection accompanied by a phase shift <py = arg{r,} given by

EXERCISE 6.2- 1

Brewster Windows. At what angle is a TM-polarized beam of light transmitted through a glass plate of refractive index iz = 1.5 placed in air (n = 1) without suffering reflection losses at either surface? These plates, known as Brewster windows, are used in lasers (Fig. 6.2-6; see Sec. 14.2D).

Mie Normal Modes98 ABSORI’I‘ION A N D SCA'I'I'ERING BY A SPI-IERE

1

E a N‚„ (

EC!N

E “ Ne13

34

/© \\EaNe14 “) /(\ $‚/ \

TM MODES TE MODES(No Radial H Component) (No Radial E Component)

ELECTRIC TYPE MAGNETIC TYPE ‘E-WAVE H-WAVE

Figure 4.4 Electric field patterns: normal modes (Mic, 1908).

At first glance it may be confusing to see what appear to be free chargesoutside the particle, that is, points where the field lines appear to convergetoward or diverge from. There clearly should be no free charges because eachdiagram represents field lines on the surface of an imaginary sphere in themedium surrounding the particle, which we may take to be free space. Theseapparent charge points are positions on the imaginary sphere at which thetransverse field vanishes, and radial fields cannot be represented on a sphericalsurface. This can be made clearer by considering the radial component of thefield for a particular mode. We have chosen the a1mode, which has particularimportance later in the book; this is the field radiated by an oscillating electricdipole. Therefore, we can refer to the dipole radiation pattern for insight intothe patterns shown in Fig. 4.4. Field lines in the xy plane (0 = 71/2) corre‑

Rayleigh Scattering of Light

particle with

Static Light Scattering - SLSSTATIC LIGHT SCATTERING 109

straight through the index-matching liquid and the polymer solution, forming astrong, unscattered (or forward-scattered) beam. The molecules in the beam pathscatter a tiny fraction of the photons in all directions. The intensity of the scatteredbeam is detected by a photodetector, typically a photomultiplier, placed horizon-tally at an angle ! (scattering angle) from the forward-scattering direction. To pre-vent streak scattering at the air-glass interface, the glass vat has a planar cut at eachside of the path of the direct beam.

Figure 2.31 is a top view of the sample geometry. The incident beam has a wavevector ki. The wave vector is parallel to the propagation direction of the beam andhas a magnitude of 2"!(#!nsol), where #!nsol is the wavelength of light in the sol-vent of refractive index nsol, with # being the wavelength of light in vacuum. Thewave vector ks of the scattered beam has nearly the same magnitude as that of ki. Inthe static light scattering (often abbreviated as SLS) in which the molecules areassumed to be motionless, the two magnitudes are exactly equal. In reality, motionsof the molecules make ks different from ki, but the change is so small (typically lessthan 0.01 ppm) that we can regard "ki " $ "ks ". The change in the wave vector uponscattering is called the scattering vector. The scattering vector k is defined as

(2.43)

The inset of Figure 2.31 allows the magnitude of "k " $ k to be convenientlycalculated as

(2.44)k $4" nsol

# sin

!

2 scattering wave vector

k # ks % ki

photodetector

forwardscattered

beam

incidentbeam

scatteredbeam

θ

index-matchingliquid

polymersolution

Figure 2.30. Schematic of the geometry around a sample cell in a light-scattering measure-ment system. A photodetector detects the light scattered by a polymer solution in the beampath into a direction at angle ! from the forward direction. The vat is filled with an index-matching liquid.

sample

For the forward-scattered beam, k ! 0. With an increasing ", k increases. Figure2.32 shows how k changes with " for water (nsol ! 1.331) at 25°C and He-Ne laser(# ! 632.8 nm) as a light source and for toluene (nsol ! 1.499) at 25°C and Ar$

laser (# ! 488.0 nm; there is another strong beam at 514.5 nm). For the first sys-tem, k spans from 3.46 % 106 m &1 at " ! 15° to 2.56 % 107 m &1 at " ! 150°.

Two pinholes or two vertical slits are placed along the path of the scattered beamto restrict the photons reaching the detector to those scattered by the molecules in asmall part of the solution called the scattering volume. The scattering volume is anintersection of the laser beam with the solid angle subtended by the two pinholes(Fig. 2.33).

Polymer molecules, especially those with a high molecular weight, scatter thelight strongly. In the following subsections, we will first learn the scattering bysmall particles and then find why it is strong for the polymer molecules. We willalso learn what characteristics of the polymer molecules can be obtained from thescattering pattern.

2.4.2 Scattering by a Small Particle

Small particles (solvent molecules and monomers constituting the polymer) sus-pended in vacuum can scatter the light. They are called scatterers. An electromag-netic wave, also called radiation, enters the isotropic particle to cause polarizationin the direction of the electric field of the incident wave (Fig. 2.34). The polariza-tion is a displacement of the spatial average of the positively charged nuclei with

110 THERMODYNAMICS OF DILUTE POLYMER SOLUTIONS

Figure 2.31. Top view of the geometry around the sample cell. The wave vector ki of the in-cident beam changes to ks when scattered. Two pinholes or two slits specify the scatteringangle. The inset defines the scattering wave vector k.

pinholes

θscatteredbeam

ki

/2θ

k

incidentbeam

unscatteredbeam

index-matchingliquid

polymersolution

testtube

photodetector

vat

ki

ks

ks

ki

sample

scattering intensity is averaged over time

Scattering by Two Volume Elements

1

2

scattering vector

Rayleigh scattering from two point particles1

2

incoherent average

Rayleigh scattering from two point particles1

2

incoherent average 2 4 6 8 10 12

1

2

3

4

Scattering Vector

length scale

no destructive interference of scattered waves

possible destructive interference of scattered waves

500400300200100

0

q-1

[nm

]

3.02.01.0 φ [rad]

Multiple Particle Scattering

i

j

700

600

500

400

300

200

100

0120010008006004002000

speckle

Multiple Scattering from Inhomogeneous Particle

R

Form Factor

Homogeneous Sphere

Rayleigh-Debye-Ganz Scattering

Assumptions

no reflections at the boundary

no additional phase shift in the particle

Form Factor Polystyrene Spheres 5 µm

0.1

2

468

1

2

468

10

2

46

Inte

nsity

[a.u

.]

302520151050 angle [°]

Multiple Scattering from Inhomogeneous Particle

Guinier approximation

expanding in sin(qr)/qr

with pair distance function g(r)

R

radius of gyration

Gaussian Chain - Form Factor of Polymers

Debye function

1.0

0.8

0.6

0.4

0.2

P(q

)1086420

qRg

Form Factors for Polymers

Let us calculate P(k) for a spherical molecule of radius Rs stuffed uniformlywith monomers that scatter light with the same intensity. Now we use Eq. 2.76 todirectly integrate with respect to r and r! in the sphere:

(2.91)

where the integral is carried out over the volume Vsp ! (4"!3)Rs3 of the sphere.

After some calculations (Problem 2.19), we find that

(2.92)

For a rodlike molecule with length L, it can be shown that (Problem 2.20)

(2.93)

Figure 2.44 summarizes P(k) ! P(k) for three polymer conformations of a sim-ple geometry. Figure 2.45 compares PGaussian(k), Psphere(k), and Prod(k) plotted as afunction of kRg. The three factors are identical for kRg « 1 as required. At higherkRg, the three curves are different.

We now calculate the form factor Pstar(k) for an nA-arm star polymer with auniform arm length N1. When calculating the average of exp[ik · (r # r!)], it isnecessary to distinguish two cases for r and r!: (1) being on the same arm and (2)being on different arms. The former takes place with a probability of 1!nA. Then,

(2.94)

where the subscripts 1 and 2 correspond to the two cases, and ⟨⟨exp[ik·(r # r$)]⟩⟩stands for the average of ⟨exp[ik · (r # r$)]⟩ with respect to the two monomers over

Pstar(k) !1nA

⟨⟨exp[ik%(r # r$)]⟩⟩1 & "1 #1nA

#⟨⟨exp[ik %(r # r$)]⟩⟩2

Prod(k) ! x#1$2x

0

sin zz

dz # " sin xx #2 with x ! kL!2

Psphere(k) ! [3x#3(sin x # x cos x)]2 with x ! kRs

Psphere(k) !1

Vsp2 $

Vsp

dr$Vsp

dr$ exp[ik%(r # r$)] ! % 1Vsp

2 $Vsp

dr exp(ik %r)%2

126 THERMODYNAMICS OF DILUTE POLYMER SOLUTIONS

shape

spherical

rodlike

Gaussian

Rs

L

[3x−3(sin x − xcos x)]2

2x−2[1 − x−2(1 − exp(−x2))]

x−1∫ z−1sin z dz − (x−1sin x)20

2x

Rg2 x P(k)

(3/5)Rs2 kRs

L2/12 kL/2

kRgb2N/6

Rg

Figure 2.44. Polymers with a simple geometry and their form factors.

Scattering Form Factor

STATIC LIGHT SCATTERING 127

the length of the arm(s). Using, Eq. 2.78, we have

(2.95)

where Rg12 ! N1b2!6 ! Rg

2!(3 " 2!nA) is the mean square radius of gyration of thearm, with Rg being the radius of gyration of the whole star polymer (see Eq. 1.84).In the second average,

(2.96)

where r0 is the position of the core of the star polymer, and the average in the lastequation is calculated for a single arm as

(2.97)

Thus,

(2.98)

The difference in Pstar(k) between a 2-arm star (! linear chain) and a 6-arm staris not as striking as the difference between a Gaussian chain and a rodlike mole-cule. At low kRg, all the curves overlap (not shown), as required. At kRg1 » 1, thesecond term becomes negligible, and the scattering comes mostly from two nearbymonomers on the same arm. The difference in Pstar(k) is, however, clearly seen inthe plot of (kRg)2Pstar(k) as a function of kRg. Figure 2.46 compares the form factor

Pstar(k) !1nA

fD(kRg1) # "1 "1nA

#"(kRg1)"2[1"exp("(kRg1)2)]#2

!1N1$N1

0d n exp( " 1

6 k2nb2) ! (kRg1)"2[1 " exp("(kRg1)2)]

⟨⟨exp[ik$(r " r0)]⟩⟩ !1N1$N1

0d n ⟨exp[ik$(r " r0)]⟩

! ⟨⟨exp[ik $(r " r0)]⟩⟩ ⟨⟨exp[ik $(r0 " r%)]⟩⟩ ! % ⟨⟨exp[ik$(r " r0)]⟩⟩ %2⟨⟨exp[ik $(r " r%)]⟩⟩2 ! ⟨⟨exp[ik $(r " r0) # ik$(r0 " r%)]⟩⟩

⟨⟨exp[ik$(r " r%)]⟩⟩1 ! fD(kRg1)

Figure 2.45. Form factor P(k) for a spherical molecule, a rodlike molecule, and a Gaussianchain, plotted as a function of kRg.

0

0.2

0.4

0.6

0.8

1

kRg

rod

Gausssphere

P(k

)

0 1 2 3 4 5

Mie Scattering

R

What is the particle refractive index is to strong?• expand the incident plane wave into spherical waves • expand the scattered wave into spherical waves • match the boundary conditions at the particle interface (Fresnel) • solve wave equation

no reflections at the boundary

no additional phase shift in the particle

Rayleigh-Debye-Ganz Scattering

D.W.H. July 2009

4

For each scattering angle (I,T), the Equations (6) and (7) represent the intensities

(W/cm2) of scattered radiation vertically and horizontally polarized with respect to the scattering

plane, respectively, which is defined by the incident ray (of intensity Io) and the scattered ray,

noting the polarization state of the incident ray as shown in Figure 2,

22

12 2sin

4oI I i

rIO IS

, (6)

22

22 2cos

4oI I i

rTO IS

. (7)

For perfectly spherical particles, polarized incident radiation produces similarly polarized

scattered radiation; hence the scattering problem may be redefined in terms of the polarization

states with respect to the scattering plane. Accordingly, equations (6) and (7) may be recast in

terms of the differential scattering cross sections (cm2/sr), namely

'2

1VVoVV r

II V (8)

'2

1HHoHH r

II V . (9)

In these two equations, the subscripts refer to the state of polarization of the incident and

scattered light, respectively, with orientation defined by the scattering plane. Specifically, the

subscripts VV refer to both vertically polarized incident light and vertically polarized scattered

light with respect to the scattering plane (i.e. I = 90o). Similarly, the subscripts HH refer to both

horizontally polarized incident light and horizontally polarized scattered light with respect to the

scattering plane (i.e. I = 0o). For unpolarized incident light, the scattering is given by the

following

'2

1scatoscat r

II V , (10)

D.W.H. July 2009

8

22

2'

4iHH S

OV . (20)

As before, the above two equations are averaged to define the differential scattering cross section

for unpolarized incident light, which gives the relation

)(8 212

2' iiscat �

SOV (21)

In this formulation, the intensity functions are calculated from the infinite series given by

� � � � � �2

11

2 1 cos cos1 n n n n

n

ni a bn n

S T W Tf

�ª º �¬ ¼�¦ , (22)

� � � � � �2

21


n

ni a bn n

W T S Tf

�ª º �¬ ¼�¦ . (23)

In the equations (22) and (23), the angular dependent functions Sn and Wn are expressed in terms

of the Legendre polynomials by

� � � �(1) coscos

sinn

n

P TS T

T , (24)

� � � �(1) coscos n

n

dPd

TW T

T , (25)

where the parameters an and bn are defined as

� � � � � � � ��

' '

' 'n n

n

n nn

n n

m m ma

m m mD D D D

[ D D D [ D< < � < <

< � <

, (26)

� � � � � � � ��

' '

' 'n n

n

n nn

n n

m m mb

m m mD D D D

[ D D D [ D< < �< <

< �<

. (27)

The size parameter D is defined using Equations (1) and (2) as

2 o

o

amSDO

. (28)

D.W.H. July 2009

8

22

2'

4iHH S

OV . (20)



)(8 212

2' iiscat �

SOV (21)


� � � � � �2

11


n

ni a bn n

S T W Tf

�ª º �¬ ¼�¦ , (22)

� � � � � �2

21


n

ni a bn n

W T S Tf

�ª º �¬ ¼�¦ . (23)



� � � �(1) coscos

sinn

n

P TS T

T , (24)

� � � �(1) coscos n

n

dPd

TW T

T , (25)


� � � � � � � ��

' '

' 'n n

n

n nn

n n

m m ma

m m mD D D D

[ D D D [ D< < � < <

< � <

, (26)

� � � � � � � ��

' '

' 'n n

n

n nn

n n

m m mb

m m mD D D D

[ D D D [ D< < �< <

< �<

. (27)


2 o

o

amSDO

. (28)

D.W.H. July 2009

8

22

2'

4iHH S

OV . (20)



)(8 212

2' iiscat �

SOV (21)


� � � � � �2

11


n

ni a bn n

S T W Tf

�ª º �¬ ¼�¦ , (22)

� � � � � �2

21


n

ni a bn n

W T S Tf

�ª º �¬ ¼�¦ . (23)



� � � �(1) coscos

sinn

n

P TS T

T , (24)

� � � �(1) coscos n

n

dPd

TW T

T , (25)


� � � � � � � ��

' '

' 'n n

n

n nn

n n

m m ma

m m mD D D D

[ D D D [ D< < � < <

< � <

, (26)

� � � � � � � ��

' '

' 'n n

n

n nn

n n

m m mb

m m mD D D D

[ D D D [ D< < �< <

< �<

. (27)


2 o

o

amSDO

. (28)

D.W.H. July 2009

9

The Ricatti-Bessel functions < and [ are defined in terms of the half-integer-order Bessel

function of the first kind (Jn+1/2(z)), where

� � � �12

1 22n nzz J zS

�§ ·< ¨ ¸© ¹

. (29)

Equation (30) describes the parameter [n

� � � � � � � �12

1 22n n n nzz H z z i zS[ �

§ · < � &¨ ¸© ¹

, (30)

where Hn+1/2(z) is the half-integer-order Hankel function of the second kind, where the parameter

Xn is defined in terms of the half-integer-order Bessel function of the second kind, Yn+1/2(z),

namely

� � � �12

1 22n nzz Y zS

�§ ·& �¨ ¸© ¹

. (31)

Finally, the total extinction and scattering cross sections are expressed as

^ `¦f

�� 0

2

Re)12(2 n

nnext banSOV (32)

¦f

�� 0

222

)()12(2 n

nnscat banSOV , (33)

noting that the absorption cross section is readily calculated from the above two.

D.W.H. July 2009

9

The Ricatti-Bessel functions < and [ are defined in terms of the half-integer-order Bessel

function of the first kind (Jn+1/2(z)), where

� � � �12

1 22n nzz J zS

�§ ·< ¨ ¸© ¹

. (29)

Equation (30) describes the parameter [n

� � � � � � � �12

1 22n n n nzz H z z i zS[ �

§ · < � &¨ ¸© ¹

, (30)

where Hn+1/2(z) is the half-integer-order Hankel function of the second kind, where the parameter

Xn is defined in terms of the half-integer-order Bessel function of the second kind, Yn+1/2(z),

namely

� � � �12

1 22n nzz Y zS

�§ ·& �¨ ¸© ¹

. (31)

Finally, the total extinction and scattering cross sections are expressed as

^ `¦f

�� 0

2

Re)12(2 n

nnext banSOV (32)

¦f

�� 0

222

)()12(2 n

nnscat banSOV , (33)

noting that the absorption cross section is readily calculated from the above two.

http://zakharov.zzl.org/lstar.php

Mie Scattering - 5 µm diameter

0.0001

0.001

0.01

0.1

1

10

Inte

nsity

[a.u

.]

150100500 angle [°]

0

45

90

135

180

225

270

315

forward scattering

log intensity


Mie Scattering - 1 µm diameter

10-4

10-3

10-2

10-1

100

Inte

nsity

[a.u

.]

150100500 angle [°]

0

45

90

135

180

225

270

315

forward scattering

log intensity


Mie Scattering - 0.02 µm diameter

10-5

10-4

10-3

10-2

10-1

Inte

nsity

[a.u

.]

150100500 angle [°]

0

45

90

135

180

225

270

315

dipole scatteringhttp://zakharov.zzl.org/lstar.php

Multiple Particle Scattering - Structure Factor

i

j different particles

identical particles

structure factor

Pair Distribution Function

isotropic distribution

structure factorFourier trafo of

pair distribution function

Example: Hard sphere S(q), Percus Yevick

Average over isotropic distribution of viz.

Static Light Scattering - SLS

internal structure solution structure• powerful standard technique • not limited to spherical objects

Lecture 1: Light Scatteringsol is the wavelength of light in the sol-vent of refractive index n...

Documents

Transcript of Lecture 1: Light Scatteringsol is the wavelength of light in the sol-vent of refractive index n...