cmic department “giulio natta” soft matter lab
TRANSCRIPT
Statistical Optics in colloid science
Roberto Piazza
CMIC DEPARTMENT “Giulio Natta”
SOFT MATTER LAB
International School of Physics “Enrico Fermi”
Course CLXXXIV
MOTIVATION: A PARADOX ● Exploting optical methods has always been crucial for the development of soft matter physics. When I was young, scattering played the leading role. Now that I am growing a little bit older it is the turn of advanced microscopy. Yet these two worlds (real and reciprocal space) are intimately related. To appreciate this, a solid background in statistical optics is crucial
● When I was young, there was no way to enter the soft matter community without a good training in optics, allowing you to build instruments. Now that I am not anymore that young, youngsters have little time to study optics, not to say building your own stuff.
THIS IS A SCHOOL: THE RIGHT PLACE AND TIME TO START - For experimentalists, to really grasp what they are doing - For theoreticians, to understand how experimentalists can cheat them
A VERY BASIC COURSE (If you get bored, the lake is just 50 m away )
1) A brief review of Fourier and statistical optics
2) Light Scattering and Intensity Correlation Spectroscopy
3) Statistical optics in imaging and “mixed” techniques
SUMMARY TIME SCHEDULE TEMPO
1. BASICS OF FOURIER
AND STATISTICAL OPTICS
A general (scalar) diffraction problem:
Incident plane wave
“Screen” (any obstacle
to propagation)
Amplitude and phase modulated diffracted wave
])(exp[),,(),,( x,y,zizyxUzyxU ϕ=
z
?
● FUNDAMENTAL: NO PHYSICAL “EMITTER”!
● TECHNICAL: how do we take the relative phases of the “secondary sources”?
HUYGENS’ PRINCIPLE “Each element of the wave front can be regarded as the centre of a secondary disturbance which
gives rise to a spherical wavelet. The position of the wave front at later time is the
envelope (interference) of all such wavelets”
t0 t1
THE ANGULAR SPECTRUM
● Fourier decomposition along x, y just beyond the screen
⎪⎩
⎪⎨⎧
+=
+−=
∫∫
)](2exp[)0,,()0,,(
)](2exp[)0,,()0,,(
yfxfiffAdfdfyxU
yfxfiyxdxdyUffA
yxyxyx
yxyx
π
π
● Many of the basic aspects of diffraction, and in general of Fourier optics can be easily grasped by selecting a “main” propagation direction (say, z), and expanding the wave front in terms of plane waves propagating with specific components of the wave-vector k along x and y.
Direction cosines are interrelated! 221 βαγ −−=
PHYSICAL MEANING
)exp(),,()(exp[);,,( tizyxPtiitzyxp ωω −=−⋅= rk● SIMPLE MONOCHROMATIC PLANE WAVE
⎪⎩
⎪⎨⎧
++=
++=
)(2zyx
zyx zyx
uuuk
uuur
γβαλπ
x cos-1 α
cos-1 β
cos-1 γ
y z
k
)ˆexp()](exp[),,( zikyxikzyxP γβ+α=
Introducing director cosines (α,β,γ)
Thus, across the plane z= 0, exp[2πι(fxx+fyy)] may be seen as a plane wave traveling with director cosines α =λfx, β =λfy
PROPAGATION OF THE ANGULAR SPECTRUM
PROBLEM: find the relation between )0,,( βαA and ),,( zA βα
SOLUTION: U(x,y,z) satisfy the Helmholtz equation: ( ) 0),,(22 =+∇ zyxUk
zikeAzA γβα=βα )0,,(),,(
( ) 0),,(1),,( 2222
2
=−−+∂∂ zAkzAz
βαβαβα
122 ≤+ βα● For: REAL Simply a change of the relative phases of the components of the angular spectrum (each wave travels different distances between z-planes → phase delay)
PROPAGATING AND EVANESCENT WAVES
γλ ⇒≤+ − )( 222yx ff
122 >+ βα● For: IMAGINARY α and β cannot anymore be interpreted as direction cosines. We have an EVANESCENT wave, with an amplitude decaying as
α
β Propagating waves
Evanescent waves
12 22 −β+απ=γ=µ ikwhere:
● Evanescent waves do not carry energy away from the aperture (like in a waveguide driven below cutoff)
γλ ⇒>+ − )( 222yx ff
)exp( zµ−
PROPAGATION AS A LINEAR SPATIAL FILTER
)0;,();,();,( yxyxyx ffAzffHzffA =
( )⎩⎨⎧ <+−−
=otherwise0
1)()(if)()(1exp),(2222
YxYxyx
ffffikzffH λλλλ
In terms of spatial frequencies:
where:
is the FREE SPACE TRANSFER FUNCTION
● Propagation can therefore be regarded as a linear dispersive spatial filter with a finite bandwidth. ● Transmission is zero outside a circular region C of radius λ-1 in the frequency plane. ● Within C, | H( fx, fy ) |= 1, but with a frequency-dependent phase shift, which is most significant at high f.
)ˆ(0 rU
r̂Original disturbance
)(0 ρA
ρFull angular spectrum
)ˆ(0 rU
r̂
)ˆ(0 rU
r̂Propagating part Evanescent part
= +
EXAMPLE: FREE-SPACE PROPAGATION (DIFFRACTION) FROM A CIRCULAR APERTURE
)(0 ρA
ρ
)(0 ρA
ρρ < 1 ρ > 1
= +
FRAUNHOFER DIFFRACTION – A SIMPLE VIEW (NO HF PRINCIPLE)
● Fraunhofer diffraction just concerns the far–field behavior of the radiation emitted by a planar source, hence the directions of the diffracted waves. Basically, it amounts to determining the propagating part of the angular spectrum ● “Principle of inverse interference” (G. Toraldo di Francia)
If we can find a system of plane waves {Wj} in S2 with suitable amplitudes and phases that, coming from S1 and interfering on Σ, would reproduce exactly A(P) and ϕ(P), then these waves yield exactly the angular spectrum of the far–field Fraunhofer diffraction pattern originated by Σ.
How does it practically work?
• Introduce amplitude density of propagating waves A(α,β). Total contribution of waves propagating in dΩ = dαdβ/(1-α2-β2)1/2:
βαβ−α−
βα γβ+α ddee1
),( i)(i22
zkyxkA
• Amplitude generated by these waves when “back-propagated” to z=0:
∫ βαβ−α−
βα=Σ β+α dde
1),()( )(i22
yxkAv
• Developing the original disturbance a(x,y) in Fourier components and equating the coefficients:
∫+π
λβ−α−
=βα yxyxaA yfxf yx dde),(1),( )(i22
22
THE POINT SOURCE
● Tiny aperture of size a → a(x,y) = A0. Since ϑ=β−α− cos1 22
ϑλ
=ϑ cos)( 20
aAAThis is the radiation pattern by any aperture, when its size goes to 0: we call it the absolute point source (PS).
● Substituting the real aperture on the screen with a source having this amplitude distribution would not change anything. This source is nothing but the Airy disk produced by a circular aperture, and has an central spot of radius ρ ≈0.61λ. THIS “MINIMAL CENTRIC” IS THE MIN. PHYSICAL SOURCE.
● Radiation pattern from a PS (propagating part): ∫ <βα
β+α βαλ
=1|||,|
)(i20 dde yxkaAA
ρπρ
λ=ρ
)2()( 10
JaAA(some calculation)
DIFFRACTION (GENERAL) ● STRATEGY: Find the IMPULSE RESPONSE FUNCTION h(x,y,z) (inverse FT of the free-space transfer function), starting from Weyl expansion of a spherical wave in planar components: [ ] βαγ+β+α
γπ= ∫ dd)(iexp12ii
zyxkkre kr
[ ] βαβ+αγ=⎟⎠
⎞⎜⎝
⎛∂∂π
− ∫ dd)(iexp)iexp(e2 i
2 yxkzkrzk
kr
[ ]rerz
krezyxh
ikrzk
⎟⎠⎞
⎜⎝⎛ −λ=ℑ= γ− i1),,( i1
Derive w.r. z
● HUYGENS’ WAVELETS For r >> λ the IRF is a spherical wave:
rzyxh
kri2/i ecose),,( ϑλ≈ π
PHASE SHIFT (QUADRATURE)
r
z ϑ
OBLIQUITY FACTOR
● RAYLEIGH-SOMMERFELD DIFFRACTION FORMULA Total disturbance in r = convolution of the amplitude beyond the screen with the IRF
yxeUU z
k
ʹ′ʹ′ʹ′−
ʹ′λ
−= ∫ʹ′−
dd),cos()(i)(i
0 uurr
rrrr
Σ P0
z P1
r01
ξ η
x y
APERTURE PLANE (ξ, η)
OBSERVATION PLANE (x, y)
Σ
ϑ
● STANDARD GEOMETRY 222
01 )()( η−+ξ−+= yxzr
∫Σ
Σλ
= drikrPU
izPU 2
01
0110
)exp()()(
● FRESNEL APPROXIMATION
⎪⎩
⎪⎨
⎧⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ η−
+⎟⎠⎞
⎜⎝⎛ ξ−
+≈
zzy
zxzr
22
01 21
211 in phase
factor in denominator
⎪⎩
⎪⎨
⎧
⎥⎦⎤
⎢⎣⎡ +
λ=
η−ξ−ηξηξ= ∫∫)(
2iexp
ie),(
),(),(dd),(
22i
yxzk
zyxh
yxhUyxUkz
Convolution of the amplitude at the aperture with a quadratic IRF
First-order approximation the exact free-space transfer function for
● TRANSFER FUNCTION IN FRESNEL APPROXIMATION
[ ] ( )[ ]22exp),(),( yxikz
yx ffzieyxhffH +−=ℑ= πλ
FRESNEL REGIME → SMALL ANGLES (PARAXIAL APPROX.)
11, −− λ<<λ<< yx ff
● ACCURACY OF THE FRESNEL APPROXIMATION Main contribution to the convolution integral comes from a square region of size Δξ ~Δη ~ 4(λz)1/2 centered in ξ = x, η = y (Notice that receding from the aperture, this region GROWS!)
In fact, Fresnel approx. is quite good even very close
to the aperture
FRAUNHOHER DIFFRACTION ● If we are REALLY far from the aperture, so that: we can further approximate the diffraction integral as:
MAXkz )(2
22 η+ξ>>
∫∫ ⎥⎦⎤
⎢⎣⎡ η+ξ
λπ
−ηξηξλ
=+
)(2iexp),(ddi
ee),(2/)(ii 22
yxz
Uz
yxUzyxkkz
WARNING! A very stringent condition! Circular aperture, r =1 cm → z >> 500 m! (but see what follows…)
● Aside from a multiplicative phase factor (which does not affect INTENSITY), the amplitude in the observation plane is simply the FOURIER TRANSFORM of the aperture amplit. distribution
evaluated at frequencies: zyf
zxf yx λλ
== ;
( )yx ffU ,~
● FRESNEL NUMBER
⎩⎨⎧
>>
≤⇒
λ=
112
F
FF N
NzDN
full Fresnel regime (“near field”) Fraunhofer regime (“far field”)
FRAUNHOFER DIFFRACTION FROM A CIRCULAR APERTURE (radius a )
⎥⎦⎤
⎢⎣⎡
λ=
zakrzakrJ
zArU
zkrkz
/)/(2
iee)( 1
2/ii 2 21
2
/)/(2)( ⎥⎦⎤
⎢⎣⎡
⎟⎠⎞
⎜⎝⎛=
zakrzakrJ
zArIλ
1 - 1 2 3 -2 -3
Normalized Intensity
1
0.5
(2a/λz)r
r = 1.22(λ/a) z
“Airy pattern”
a/2.1 λϑ ≈
FOURIER OPTICS & IMAGING ● FROM FRESNEL TO FRAUHOFER Aperture illuminated by a spherical wave, converging towards P=(0, 0, z0) on the z axis. ● Writing the spherical wave in paraxial approximation, the field at a
point Q = (ξ, η, 0) is: ⎥⎦
⎤⎢⎣
⎡η+ξ−ηξ≈ηξ )(
2exp),(),( 22
00 z
iktUU
P Q
∫∫+
+
=)(2
00
)(2
0
22
0
),(),(ηξ
λπ
ηξηξλ
yxz
iyx
zki
etUddzi
eyxU
On the plane parallel to the aperture and containing P, we have the Fraunhofer diffraction pattern of the amplitude transmittance. Thus, by turning a plane wave into a converging spherical wave, the Fraunhofer regime is “pulled back” close to the aperture.
THIS IS EXACTLY WHAT A LENS DOES
● Using the Fresnel diffraction formula:
FOURIER-TRANFORMING PROPERTIES OF A LENS
f
A) Object placed against a lens with finite extent
LENS PUPIL ⎩⎨⎧
=01
),( yxPInside the lens aperture
Otherwise
( )⎥⎦⎤
⎢⎣
⎡+= ∫∫
+
yvxuf
iyxPyxdxdyUfi
evuUvu
fki
f λπ
λ2exp),(),(),(
)(2
22
● If the object is much smaller than the lens aperture, we can neglect the effect of the lens finite size. Thus, apart from a phase factor (which does not affect intensity), the amplitude distribution in the focal plane is the FT, i.e., the Fraunhofer diffraction pattern of the amplitude transmitted by object.
AMPLITUDE ON THE LENS FOCAL PLANE
f d
B) Object placed in front of the lens
● Propagation over d in Fresnel approximation yields (neglecting pupil):
⎟⎠
⎞⎜⎝
⎛λλλ
=
+⎟⎠
⎞⎜⎝
⎛−
fv
fuU
fvuU
vufd
fk
f ,~i
e),()(1
2i 22
● d = f → EXACT FT between amplitude transmitted by the object and the amplitude in the back focal plane. f f
F.T. pair planes
f
OBJECT
F. T. F.
T.
OBJECT
POINT SOURCE
● Other Fourier-transforming geometries
Two equal lenses separated by their common focal length
An object illuminated by a spherical wave emanating from a point source
The F. T. plane is always the plane
where the source is imaged.
IMAGING PROPERTIES OF A LENS
● PROBLEM: Under what conditions is the field Ui(x, y) in the image plane a “faithful copy” of the field Uo(x, y) leaving the object?
● Wave propagation is linear → ∫∫= ),(),;,(),( ηξηξηξ oi UvuhddvuU
Response in (x, y) due to a unit-amplitude point source in (ξ, η) (POINT SPREAD FUNCTION)
),;,( ηξvuh
● Restating the problem: Under what conditions: ( )ηξδηξ MvMuMvuh ±±= ,),;,(
⎟⎠⎞
⎜⎝⎛=
Mv
MuU
MvuU ii ,1),( M = magnification
),( ηξoU ),( yxUi
z1 z2
A) Neglect phase factors (irrelevant for intensity)
B) Assume that z1 and z2 satisfy: (Lens Law in geom. optics)
THE POINT SPREAD FUNCTION Consider a δ -function (point source) in (ξ, η). If we:
fzz111
21
=+
[ ]∫∫ ⎭⎬⎫
⎩⎨⎧
η−+ξ−λπ
−λ
=ηξ 22
2212 )()(2iexp),(dd1),;,( MvMu
zyxyPx
zzvuh
),( ηξ MvMu ==
● The PSF is the Fraunhofer diffraction pattern of the pupil centered on the image coordinate ((aside from a scaling factor 1/λz1)
● The effect of diffraction is then of convolving the ideal image with the diffraction pattern of the lens pupil (for monochomatic illumination)
PLAYING WITH THE FOURIER TRANSFORM
z1 z2
F fzz111
21
=+
STOCHASTIC PHENOMENA IN OPTICS
SOURCE Spectrum and temporal fluctuations
Size and spatial fluctuations
“SYSTEM” Random static structure Fluctuation dynamics
PROPAGATION Refractive index fluctuations
DETECTOR Photodetection
statistics
2) STATISTICAL OPTICS
TEMPORAL COHERENCE
SPECTRUM OF THE SOURCE A PHYSICAL PROBLEM
SPATIAL COHERENCE
EXTENSION OF THE SOURCE
A GEOMETRICAL PROBLEM
(almost always…)
TEMPORAL COHERENCE (first order)
TEMPORAL FLUCTUATIONS OF THE SIGNAL AMPLITUDE
NON-MONOCHROMATICY
FREQUENCY VIEW TIME VIEW
ω
u(ω)
MICHELSON INTERFEROMETRY
FOURIER-TRASFORM SPECTROSCOPY
ℑ
POWER SPECTRUM CORRELATION FUNCTION ℑ
t
u(t)
COHERENCE TIME τc FINITE BANDWIDTH Δω ωπ
τΔ
=2
c
DESCRIPTION OF QUASI-MONOCHROMATIC FIELDS: ● Complex representation of a monochromatic field:
)cos()( 0 φ−ω= tAtuR ( )[ ]φ−ω−= tAt 0iexp)(u
[ ] [ ])()(2
)( 00 ωωδωωδ φφ −++=ℑ − iiR eeAtu
+ω0 -ω0
A / 2
[ ] )()(u 0ωωδφ −=ℑ iAet
+ω0
A
a) Suppress the negative frequency components b) Double the strength of the positive frequency components
● Non monochromatic signal with FT Associated analytic signal:
)(tuR [ ] )(~)( ωRR utu =ℑ
∫∞
ω−ωωπ
=0
i)(~d1ˆ)( tR eutu
[ ]
[ ]⎪⎩
⎪⎨
⎧
−π=
=
∫∞+
∞− tzzuzt
tutR
R
)(d1)(Im
)()(Re
u
u
(no more info!)
-
NARROWBAND SIGNALS: COMPLEX ENVELOPE
● Real signal having a narrow spectrum centered around ω0 and of width Δω << ω0:
[ ])(cos)()( 0 tttUtuR φω −=
Slowly-varying envelope and phase ⎭
⎬⎫)()(ttU
φ
t
uR( t )
ω + ω0 - ω0
Δω ● Associated analytic signal:
tt eetUt 0i)(i)()( ω−φ=u
tiett 0)()( ω−= Uu
)()()( tietUt φ=UComplex envelope
ω + ω0
TEMPORAL PROPERTIES OF LIGHT SOURCES A) NARROW BAND THERMAL SOURCE (e.g. a spectral lamp, or even, to a good approximation, a
multimode laser) Many microscopic independent emitters, such as a collection of thermally excited atoms. Each atom emits at a single frequency ω0, but collisions induce abrupt phase jumps.
t
● Total signal amplitude (complex envelope) for N identical emitters:
∑∑ =====
N
i
tiN
i iti
NiUetetUt
1
)(
1
)( )()()( φφ UU
A N-step random walk in the complex plane. For large N, r = Re(U) = U cos(φ) and i = Im(U)= U sin(φ) have a joint Gaussian statistics:
i
r
UN
⎟⎟⎠
⎞⎜⎜⎝
⎛ += 2
22
2 2exp
21),(
σπσirirp UN=σwith
● Prob. density for U is a Rayleigh distribution
⎪⎩
⎪⎨
⎧≥
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛−
=
otherwise0
022
2exp2)(
UUUUpU σσ
0
0.2
0.4
0.6
0.8
1.0
0 1 2 3 4 5U / σ
p U(U
)
● STATISTICS OF THE INTENSITY
)0(2
exp21)( 22 >⎟
⎠⎞
⎜⎝⎛−= IIIpI σσ
0
0.2
0.4
0.6
0.8
1.0
0 1 2 3 4 5I / <I>
p( I
))0(exp1)( >⎟⎟
⎠
⎞⎜⎜⎝
⎛−= III
IIpI
22σ=I
TIME VIEW ● MAIN MESSAGE: To a finite signal bandwidth unavoidably correspond TEMPORAL FLUCTUATIONS of the signal
● Time autocorrelation function of the analytic signal:
ttt )()(*)( ττ +=Γ uu
● Normalizing Γ to its zero-time value Itt==Γ 2)()0( u
Ittg )()(*
)0()()(1
τττ
+=
ΓΓ
=uu
“FIELD TIME-CORRELATION FUNCTION
● The envelope of an optical signal with bandwidth Δω does not appreciably change on a time scale τ << τc = 2π / Δω. τc plays than the role of the COHERENCE TIME of the optical signal.
tiett 0)()( ω−= Uu
CORRELATION AND SPECTRUM ● Power spectrum. uR( t ) cannot usually be F. T, but still has a finite average power, so we can define
TuPRT
TRu
)(~lim)( ωω ∞→=
SPECTRAL DENSITY (or POWER SPECTRUM)
● THE FUNDAMENTAL LINK: WIENER-KINTCHINE THEOREM
Defining a normalized power spectrum of the real signal as:
⎪⎪⎩
⎪⎪⎨
⎧>
= ∫∞
otherwise0
0)(
)(
)(0
ωωω
ω
ω Ru
Ru
Pd
P
P
[ ][ ]⎩
⎨⎧
ℑ=
ℑ=− )()(
)()(1
1
1
ωτ
τω
Pg
gP
COLLISION-BROADENED THERMAL SOURCE
● Each emitter is independent, so we have:
Therefore: )()0()()0()()0()( *
1
** tNttt iiN
i iiNN uuuuUU ===Γ ∑ =
jitji ≠= 0)()0(* uu
The total correlation function coincides therefore with the correlation function for a single emitter: [ ])0()()(
110)()( φφω −−=≡ titii eetgtg
● The phases φ(0) and φ(t) will be correlated only if the atom does not undergo collisions in t. The phase correlation function will then be proportional to the probability p(t’ > t) of colliding at any t’ > t.
( )τω /exp)( 01 ttitg −−=Time-correlation function for a collision-broadened thermal source
● POWER SPECTRUM. Fourier-trasforming g1( t ):
( )220 /1)(11)(
τωωπτω
+−=P (Lorentzian lineshape)
TEMPORAL COHERENCE AND INTERFEROMETRY
D
M1
BS
M2
CP
L1
L2
S
Δl ● Qualitatively, the two beams can interfere only if the difference between the optical paths Δl = l2-l1 is smaller than the coherence length of the source lc.
● Quantitatively, the detected intensity is proportional to the real part of the time correlation function, evaluated at the delay
[ ]( ) clgII /)(Re1 10 Δττ =+=
cl /Δτ =
[ ]CeII τττω /00 )cos(1 −+=
● THERMAL SOURCE
0
0.5
1.0
1.5
2.0
0 1 2 3
τ / τc
I / I 0
ce ττ /−FRINGE
VISIBILITY
SPATIAL COHERENCE: LASER SPECKLES
DIFFUSER e.g.: “magic” Scotch tape
● When a diffuser is illuminated by a laser, a highly “maculated” pattern forms on the screen
(“SPECKLE PATTERN”)
● If we put a lens and enlarge the laser beam, the speckles become smaller.
F ● Conversely, if we move the diffuser towards the lens focus, the speckle pattern becomes much coarser. (a great way to find the real lens focus)
F Why the speckles?
Why their size depends on the beam spot on the diffuser?
SPATIAL COHERENCE: YOUNG’S INTERFEROMETER
● POINT SOURCE (quasi-monochomatic)
d P
α ∼ λ/a
Fringes with a spatial period: dx /λ=Δ
Form in the overlap region of the diffraction patterns generated by the two pinholes of radius a.
D d
z
Rd
≈ϑ
● EXTENDED SOURCE made of independent point emitters
λϑ <<≈zDdD
Fringes form only when:
● If P1 and P2 are very close (so that u1 ≈ u2, v1 ≈ v2 ), the fields
⎩⎨⎧
+=
+=
222
111
)()(
vuUvuU
PP
will be strongly correlated, even if the fields u and v are fully uncorrelated . That is, propagation INDUCES spatial correlations .
P1 P2 U
V
u1 u2 v1
v2
PROPAGATION AND SPATIAL CORRELATIONS
● However, if P2 is moved apart from P1, the phases of the fields coming from U and V change differently:
⎪⎩
⎪⎨
⎧
−≈−ʹ′=Δ
+≈−ʹ′=Δ
zdDrrr
zdDrrr
vvv
uuu
If we want: Dzdrr vu /λλ <<⇒<<Δ−Δ
P1
P2 V
U
ru’ rv rv’
d z D ru
2/ zD=Ωʹ′Δ D
Conversely, the solid angle under which the source is seen is:
'
2
ΔΩ≈λ
cAΔΩ’
z
To the source is then associated a “coherence cone” with solid angle at vertex: which corresponds to an angular aperture: , just like a diffraction pattern!
2)/( Dλ≈ΔΩD/2 λα ≈
EXAMPLES ● Thermal source: D = 1mm, λ =0.5 µm, R = 1m → Ac ≈ 0.25 mm2
● SUN: Ang. diam. 2α ≈ 32’→ ΔΩ’ ≈ 7x10-5 sr → Ac ≈ 0.004 mm2
● STAR (Betelgeuse): 2α ≈ 0.047’’→ ΔΩ’ ≈ 4x10-14 sr → Ac ≈ 6 m2
(TWINKLE TWINKLE LITTLE STAR…)
● Hence, the pinholes must lie within a
2
22
DzAcλ
≈COHERENCE AREA D ΔΩ α
Ac
SMALL PINHOLES SPACING
S1
S2
● S1 and S2 form two displaced sets of fringes ● However, since fringe oscillation is coarse (low frequency), the shift is a small fraction of the period and the sum of the two interferograms still shows fringes
LARGE PINHOLES SPACING
S1
S2
● Fringe oscillation is rapid. ● Therefore, the two sets are in counter-phase and the two interferograms cancel out.
⎟⎠⎞
⎜⎝⎛ −+⎟
⎠⎞
⎜⎝⎛ −=
crtPK
crtPKtQ 2
221
11 ,,),( uuu
COMPLEX DEGREE OF COHERENCE ● The field at a point Q on the screen is the superposition of the diffraction pattern from the two pinholes. For a narrowband source, this is simply related to fields on the pinholes at earlier times:
● INTENSITY: ⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ −
Γ++=crrKKQIQIQI 12
1221)2()1( Re2)()()(
),(),(*)( 2112 ττ +=Γ tPtP uu
MUTUAL COHERENCE FUNCTION
Γ12(τ) is like Γ(τ), but evaluated at two different point in space, so it is a space-time correlation function that, in terms of the positions of P1 and P2, can be written: ),(),(*);,( 2121 ττ +=Γ tt rururr
● To separate out effects purely due to spatial coherence, let us consider a quasi-monochomatic source, such that all delays are Then, we can write with:
cτ<<ωττ ieJ −=Γ 1212 )(
),(),(*),(),(* 212112 tttPtPJ rUrUuu ==
( ) ( )21
12
PIPIJ
=µ DEGREE OF SPATIAL COHERENCE
Normalized:
MUTUAL INTENSITY
P1
P2
Q1
Q2
1ϑ
2ϑ
Σ
SOURCE
( )∫∫ Σ
−−
ΣΣΣ=
21
21)(2121221
)cos()cos(,1),( 21
rrePPJddQQJ rrik ϑϑ
λ
PROPAGATION OF THE MUTUAL INTENSITY
Using Fresnel diffraction for both fields, one finds:
This looks rather terrific, but…
THE VAN CITTERT-ZERNIKE THEOREM ● Suppose that the source S is fully spatially incoherent, so that:
)()(),( 211021 PPPCIPPJ −= δ
( )∫Σ−− Σ= d
rrePICQQJ rrik
21
21)(1221
)cos()cos(),( 21 ϑϑλ
● In paraxial approximation:
( ) ( )∫Σ−
⎥⎦⎤
⎢⎣⎡ Δ+Δ= yxz
iIddz
CeQQJi
ηξλπ
ηξηξλ
ψ 2exp,)(
),( 0221
Σ
η
ξ
y
x P
Q1 Q2
● Therefore, apart from a scaling and phase factor: The mutual intensity is the Fourier transform of the intensity distribution across the source
(van Cittert-Zernike theorem)
● The VCZ theorem may “resemble” the expression for Fraunhofer diffraction, but this is just a formal analogy. Indeed:
a) The physical quantities are totally different (mutual intensity as FT
of the source intensity, rather than field on a screen as FT of the field at the source)
b) The range of validity is very different! The VCZ theorem is solely based on the paraxial approximation, and holds true also in the Fresnel regime!
∫∫ΔΔ
==ΔΔ−
ηξηξ
λλ
ηξηξµ
ψ
ddI
zyzxIe
ddIC
QQJyxi
),(
)/,/(~
),(
),(),( 21
● From the VCZ theorem, the spatial degree of coherence is then :
● From the VCZ theorem:
2
2
22
2
),(
)()(),(~
)(),(
)()()/,/(~
∫∫
∫∫
=ΔΔΔΔ
=
dxdyyxI
fdfdffIz
dxdyyxI
ydxdzyzxIA
yxyx
c λλλ
COHERENCE AREA
( )∫∞+
∞−ΔΔΔΔ= )()(,ˆ 2 ydxdyxAc µ● We shall formally define:
but from the properties of the FT (Parseval’s theorem):
∫∫ =⇒= dxdyyxIfdfdffIyxIffI yxyxyx22),()()(),(~),(),(~
2
22
2
2
2 )(
),(
),()(
II
Az
dxdyyxI
dxdyyxIzA
Sc
λλ ==
∫∫ Where AS is the
area of the source
EXAMPLE: UNIFORMLY BRIGHT SOURCE
AS
● If we have an incoherent source (or illuminated aperture) of area AS with uniform intensity I0:
20
22 III == ( )S
c AzA2λ
=
which will therefore correspond to the typical speckle size at distance z, consistently with what we qualitatively found previously.
But…Does a fully incoherent source REALLY exist? A fully incoherent wave field would have an infinitesimally fine spatial structure. Therefore its angular spectrum would consist only of evanescent waves, and cannot propagate! Then, there must be correlation at least on the scale of λ. Writing: is than a limiting approximation, valid for an optical system that cannot resolve structures finer than λ
)(),( 2121 PPPPJ −∝δ
THE SPECKLE FIELD ● The field at each point P on the screen is a random sum of contributions from each point of the source, so has a Gaussian statistics for the field components (similarly to a chaotic source).
● Around P there is a correlated region of average size AC and random shape.
● The INTENSITY statistics of the speckles is exponential, that is “most speckles are dark”.
0
0.2
0.4
0.6
0.8
1.0
0 1 2 3 4 5I / <I>
p( I
)
● The “granular” speckle pattern depends only on the geometry of the source.
NOT ALWAYS TRUE, AS WE WILL SEE!
● We define the (normalized) intensity correlation function (or “degree of second order temporal coherence”) as:
222)()(*
)()()(*)(*
)(
)()()(
t
t
t
t
tttttt
tItItI
guu
uuuu ττττ
++=
+=
PROPERTIES
[ ]
[ ]
[ ]1)(0)0()(0)
1)0(1)0()
*)()()()()
122
12
2
2
1122
≤≤≤≤
=≥=
=−=−
ττ
ττττ
gggiii
gI
Igii
ggggi
For a monochromatic source, obviously: 1)(2 ≡τg
INTENSITY CORRELATION (Second order temporal coherence)
CHAOTIC SOURCE
● However, due to the independence of the emitters, a given term averages to zero, unless the containing only products of a field times its complex conjugate relative to the same emitter.
∑ =++=+
N
jijiji tttttItI
1,)()()(*)(*)()( uuuu τττ
● We should evaluate the double sum:
+++=+ ∑iiiii tttttItI )()()(*)(*)()( uuuu τττ
[ ]∑≠++++++
jijjiijjii tttttttt )()(*)()(*)()(*)()(* uuuuuuuu ττττ
● Since all emitters are identical:
[ ]22 )()(*)()(*)1(
)()()(*)(*)()(
τ
τττ
++−+
+++=+
ttttNN
ttttNtItI
iiii
iiii
uuuu
uuuu
Where we split the averages because of independence
● For large N, the first term can be neglected, and N(N - 1) ≈ N2 :
[ ]222 )()(*)()(*)()( ττ ++=+ ttttNtItI iiii uuuu
222 )()(*)( ttNtI ii uu=
212 )(1)( ττ gg += SIEGERT RELATION
● For a collision-broadened thermal source:
● Therefore, for a chaotic source, g2(τ ) does not yield any additional information, and can be directly obtained from |g1(τ )|
( )cg τττ /||2exp1)(2 −+=
● More generally, defining a second order complex degree of coherence:
tt
t
PIPItPItPI)()(),(),(
)(21
21)2(12
ττγ
+=
2211221
)2(12 );,(1);,( τγτγ rrrr +=then, for a chaotic source:
2. LIGHT SCATTERING AND INTENSITY CORRELATION
SPECTROSCOPY
LIGHT SCATTERING BASICS Scattered radiation - wave-vector ks - frequency ωs - detected along V/H axis by setting analyzer A
Scattering plane defined by (ki, ks)
Incident radiation - wave-vector ki - frequency ωi - polarized along V/H axis by setting polarizer P
SCATTERING WAVE-VECTOR q For quasi-elastic scattering (ωi ≈ ωs) 2
sin42
sin2||0
ϑλπ
=ϑ
==nkq q
0
2||||λπ
===nkis kk
MOLECULAR SCATTERING (Rayleigh) ● Single molecule (or particle with size a <<λ) Scattered field Es can depend only on: - incident field E0 - particle polarizability α (prop. to particle volume V) - particle and surrounding medium refractive indices np, ns - distance from detector (in far-field must decrease as R-1)
ip
sip
s Inn
fRVIE
nn
fRVE ⎟
⎠
⎞⎜⎝
⎛λ
∝⇒⎟⎠
⎞⎜⎝
⎛λ
∝ 242
2
2
~ a6 Dimensional analysis
● Full electromagnetic treatment - Incident plane wave Ei=E0eik.rni - Induced dipole P =aEi Radiation zone
R>>λ
Diff. scattering cross-section
4
2
20
2
dd
λα
ε
⋅π=
Ωfiσ nn
NO FLUCTUATIONS NO SCATTERING ● Perfectly uniform medium (no refractive index fluctuations)
For each volume element v… …There is always a v’ that scatters with the same amplitude and Δφ=π/2
● Ideal gas: Poisson fluctuations Here (and only here) Is prop. to N
Attractive interactions ↓
Stronger fluctuations ↓
more scattering
Repulsive interactions ↓
weaker fluctuations ↓
less scattering
SCATTERING FROM A FLUCTUATING MEDIUM
● Generic medium with a fluctuating dielectric tensor : )()( ,t,t rδεIrε +ε=
tifif
ssif tt
RIkI ω
∞+
∞−
δεδεεπ
−=ω ∫ i*20
230
2e),()0,(d
32),( qqqSpectral density:
220
230
2)(
32)( qq if
ssif R
IkI δεεπ
−=Total scatt. intensity: Probe of mean-square fluctuations at q
[ ] t
ji
tifif
sif
ijttII ω−⋅∞+
∞−∑∫ αα=ω i
,
)0()(i*0 ee)()0(d),( rrqq
( )[ ]∑ −⋅α=ji
ijifsif II
,
20 iexp||)( rrqq
● DISCRETE MOLECULAR POSITIONS ⎟⎠⎞
⎜⎝⎛ −δ=ρ ∑1
)()( i,t rrr
),()(),( 20 ωα⋅=ω qnnq SII fi
sif
)()()( 20 qnnq SII fi
sif α⋅=scalar α
PARTICLE SCATTERING
SCATTERING AMPLITUDE S(θ,φ) (Vertical incid. polariz/Vertical detection)
),(ie),(
ie),(),(
)(i
0
)(i
tzEkr
SEkr
StrE i
zrktkr
s
−ω−
ϕϑ=ϕϑ=
)(i0e),( tkz
i EtzE ω−=θ
φ
● S is in general a complex quantity: ),(e),(),( ϕϑαϕϑ=ϕϑ iFS
Scattered intensity: 022
2 ),(),;( Irk
FrI ϕϑ=ϕϑ
● Scattering cross section σs(by equating total scattered power to
the incident power falling a surface area): ∫ Ωϕϑ= d),(1σ 22s Fk
● In general, the particle may scatter and absorb, with an absorption
cross-sect ion σa → EXTINCTION cross-section: σext = σs + σa
THE OPTICAL THEOREM ● Whatever the illuminating source, the scattering pattern from the particles contains a faithful reproduction of the incident field that spatially overlaps with the transmitted radiation, but with a different phase ● It is the interference between this “copy” of the incident field and the transmitted beam that yields both the power reduction of the transmitted field and its phase delay in traversing the medium.
INCIDENT PLANE WAVE
z=0
In the forward direction, all contributions add in phase
SUM FIELDS, NOT INTENSITIES! ∑= i iSS )0()0(
[ ])0(Re42 Skext
πρσ −=
[ ]
[ ]⎪⎩
⎪⎨
⎧
−=
+=
)0(Re2'
)0(Im21
2
2
Sk
n
Sk
n
πρ
πρ REFRACTION
ABSORPTION
Complex refr. index 'i~ nnn +=
Extinction cross section
LORENZ - MIE SCATTERING ● Scattering from a spherical particle (exact)
[ ]∑∞
=
+++
=1
)(cos)(cos)1(12
nnnnnVV ba
nnnE ϑτϑπ
● Expansion coefficients very complicated! (combination of Riccati-Bessel functions) However…
⎪⎩
⎪⎨
⎧
=
=
ϑϑ
ϑτ
ϑϑ
ϑπ
d)(cosd)(cos
sin)(cos)(cos
)1(
)1(
nn
nn
P
P● Angular functions
By increasing n, the angular pattern becomes “richer”
● Mode amplitude depends only on x =ka, m=np/n, and decreases fast with x. Hence, for small particles, we need only the lowest order terms.
)()1(45
)(21
94
)2()1)(2(
52i
21
32i
7521
76
2
2
25
22
223
2
2
1
xOxmib
xOxmmx
mmmx
mma
+−−=
+⎟⎠
⎞⎜⎝
⎛+−
++
−−−
+−
−=
Dipole First real term
● By increasing a/λ, more and more mode superimpose. the contributions from different modes tend to cancel each other, except in the forward direction, where all of them present a maximum with the same sign, and, to a lesser extent, in the backward direction, where signs from successive modes alternate.
● In the limit of a very large particle, we would expect the extinction cross section to coincide with the are of the beam geometrically intercepted by the particle, hence that Q = σext/πa2 →1. Instead one finds Q = σext/πa2 →1. This “paradox” comes from the fact that half of the scattering is contained in a strongly peaked diffraction cone (ovlp. with transmitted!)
Special scattering regimes ● Rayleigh-Gans-Debye (RGD a) x(m-1) = ka(np/n -1)<< 1 → The phase shift of the field in passing
+ through the particle is negligible b) m ≈ 1 → No reflections or refractions
Each volume element of the particle “sees” the external field
Simply integrate over volume elements field (a “collection” of Rayleigh scatterers)
23i
22
42
0
de1)(
)()(4
)(
∫ ⋅=
−=
V
p
rV
kP
kPnnrkVIqI
rq
π
FORM FACTOR
Homogeneous sphere
A curious story with the Optical Theorem (read!)
● Anomalous Diffraction (AD) “Tenuous” particles (common situation in microscopy of biological objects) Take into account phase changes, but: m ≈ 1 → No reflections or refractions a >> l → almost ray-tracing
For a large particle, in general: SCATTERING = DIFFRACTION + REFRACTIONS/REFLECTIONS But for low optical contrast, this regime is reached very slowly
)1(2
)cos1(2sin212 2
−=
⎥⎦
⎤⎢⎣
⎡ −+−=
mkapp
pppQ
Normalized extinction (exact):
Dynamic Light Scattering (DLS)
Quasi-Elastic Light Scattering (QELS)
Laser Light Scattering (LLS)
Photon Correlation Spectroscopy (PCS)
Self-Beating Spectroscopy (SBS)
Bosons are particles that differ from fermions because they behave differently
Corriere della Sera, 3/7/2012
Intensity Correlation Spectroscopy (ICS)
How physicists (re)discover the radio!
ICS vs. INTERFEROMETRY (OR STANDARD SPECTROSCOPY)
Note: a quadratic detector does not measure the instantaneous intensity, but rather its time-average over many optical cicles!
)()()( * tEtEtI = (actually an irradiance)
The scattering volume: another “chaotic” source:
● Total scattered field: ∑ =
⋅=M
jv
sjAE
1)(iti ee rqω Again a sum of
random phasors!
V
vj
● Split V into M>>1 small volumes, which have anyway a large size compared to sample spatial correlation
- Es has a gaussian distribution - The intensity has an exponential distribution (speckle field) BEWARE!
a) The scattered field is gaussian only provided it is fully fluctuating (does not work for non-ergodic samples)
b) If the number of particles N in each sub-volume is small, the Siegert relation must be corrected for number fluctuations:
22
12)()0(
)(1)(N
tNNtgg
δδτ ++=
g1(t) → Physics of Brownian motion
● Langevin equation: )(1dd t
mtRvv
+−= γ
● Velocity autocorrelation and particle displacement: ( )⎪
⎩
⎪⎨
⎧
−−=
=
−
−
t
tB
DDttRmTkt
γ
γ
γe166)(
e3)()0(
2
vv
ttD d)()0(31
0∫∞
= vv● Self-diffusion coefficient:
● Intermediate scattering function (time- FT of dynamic structure factor):
[ ]2||i
21-i ee),(
R(0)R(t)R(0)R(t)q −−⋅ ==q
s tqF
(this is not trivial, and requires the process to be gaussian!)
● Correlation functions: ⎩⎨⎧
−=
=Γ−
−Γ−
t
tt
gg
22
i1
e1)(ee)(
τ
τ ω 2Dq=Γ
Decay rate
TIME FOR NUMBERS DLS measurement on: - Small surfactant micelles (radius a ≈2 nm → D ≈ 1.1x10-10 m2/s ) - Using a blu laser (λ = 488 nm), at max q = 3.4x107 nm-1
Δω = Dq2 ≈ 130 kHz LUDICROUSLY SMALL! No conventional spectroscopic measurement can resolve such a small broadening, because it is much smaller than the source bandwidth (even for a well-stabilized single-mode laser, it is very hard to reduce the bandwidth to less than 1MHz)!
●THE SOLUTION: Forget usual spectroscopic or interferometric methods, which are related to the field correlation function, and look for intensity correlation, the ultimate spectroscopy (no lower brodening limit!)
THE MAGIC OF ICS: TIME VIEW
( ) )(),()(e)(),( 00)(i tEtqBtEqbtqE
i
tis
i ==∑ ⋅rq
Fluctuations in the incident field E0(t) and Brownian fluctuation embodied in B(q,t) are statistically independent. Hence, the full correlation function factorizes as: ),(),(),( 111 tqgtqgtqg sourcesample=
0),(0),( 11BcB tt
source tqgtqgτττ <<<<>>
→⇒→ BAD NEWS! But:
● FIELD CORRELATION SPECTROSCOPY (INTERFEROMETRY)
● Scattered field:
0
0.2
0.4
0.6
0.8
1.0
0.001 0.01 0.1 1τ
g 1(τ)
CHAOTIC SOURCE & SAMPLE
samplec
sourcec ττ 1.0=
)(1 τg
)()(1 τSg
)()( 1)(
1 ττ gg SC.F. “dies out” before g1(τ ) starts decaying
● INTENSITY CORRELATION SPECTROSCOPY
DECAYS TO 1! )(2 tg source ),(),( 22 tqgtqg sample
t Bc ττ <<<<→
1
2
3
4
0.001 0.01 0.1 1!
g 2(!)
samplec
sourcec ττ 01.0=
This is exactly what we are looking for! After a short time, we can forget about the source and we see only sample correlations!
Ideal monochromatic sources would be a DISASTER for DLS!
FREQUENCY VIEW (or, how physicists re-discovered the radio)
Crystal radio (no battery!)
Antenna (HF input)
Headset (Audio output)
Tunable resonant filter
Rectifier
QUADRATIC DETECTOR
Cat-whisker Impurity
Galena crystal (lead sulfide)
AMPLITUDE MODULATION (not Marconi, but Fessenden!) [ ] ttmfAtv cωcos)(1)( +=
tmAtmtAtv mmcm ωωωω cos)(2cos2cos1[)( 222 +±++=
2ωc ωm
QUADRATIC DETECTION
cm
mttfωω
ω
<<
= cos)(
[ ]ttmAtAtv mcmcc )cos()cos(2
cos)( ωωωωω −+++=
ωc
Low-pass filter
tmAtv mfilt ωcos)( 22 =
DLS: )()()( tvtftv c= (carrier-suppressed AM)
DLS AND SPATIAL COHERENCE ● Illumination: What matters is only the power the source emits in a single coherence area → A single transversal mode laser (a practical need)
● Detection: No more than one speckle (more just lowers the contrast)
D
● First (and rather dull) way: two pinholes
● Later (more clever): imaging by a stopped-down lens on a controllable slit GIVE SPECKLES THE SIZE YOU LIKE!
(and select the scattering volume) D
● Today: monomode fiber TOTALLY DIFFERENT (ask Jaro Rička)
D
Again about radios: Heterodyne detection and Doppler Velocimetry
Signal
Local Oscillator at ωc
● Superheterodyne detection (E. Amstrong) Mix the incoming AM signal with a LOCAL OSCILLATOR at the carrier frequency ωc
{ } tAAttAAtvtvtV mLmmLLO ωωω cosR.F.atsignalscos)cos1()()()( 2 +=+==
Again the modulating audio signal, but amplified by vLO!
● Similar trick in DLS, by mixing the scattered light with a tiny fraction of the incident beam (|ELO|2 >> |Es|2) We get the real part of g1(t), not its modulus square!
[ ])(Re1)( 12 tgCtg h +=
)exp()cos(1)()exp()iexp(),( 22
21 tDqtCtgtDqttqg h −⋅+=⇒−⋅= VqVq● Particles in a moving fluid:
WE DETECT ABSOLUTE MOTION (impossible with homodyne)!
ϕ
z
d SPECKLE SIZE generated by d: ξ
λλξλ
== zz
zd
So the speckle size coincides with the CORRELATION LENGTH! (and we obtain the PHYSICS of the source)
NEAR FIELD
EFFECTIVE SOURCE SIZE: ξλzd =
“STRUCTURED” SOURCES: NEAR FIELD SPECKLES
ξλ
ϕ ≈ξ D
(M. Giglio, 2000)
● A source with correlations extending for ξ
COHERENCE AREA DIFFRACTION PATTERN
FAR FIELD
(just geometry…)
ϕλ
ϑ <<≈D
● PUZZLE: This seems to violate the VCZ theorem (saying that the speckle size should vanish for z → 0). Where is the catch?
IN THE ASSUMPTION OF A FULLY INCOHERENT SOURCE!
● Generalized VCZ theorem for sources with finite spatial coherence (R. Cerbino, 2007)
“Deep Fresnel region” (DFR) ⎟⎟
⎠
⎞⎜⎜⎝
⎛=<<=<< !
2
λλξ DzDzz Fc
“Usual” Fresnel region
- In the DFR (z<<zc) :
- z>>zc → Usual VCZ
[ ]),(.F.T),(),( 2211*
yx qqIyxEyxEJ ==Σ
Far-field scattering pattern from the source
Mutual Intensity does not change
in the DFR
(Cerbino R., Vailati A., 2009)
FOURIER PLANE
CCD
IMAGE PLANE
BEAM STOP
● Sensitive to misalignment (hard to suppress incident beam) ● Suffers from stray-light
SMALL-ANGLE LIGHT SCATTERING (“TRADITIONAL”)
SPECKLE FIELD
CCD
MAGNIFIED SPECKES
OBJECTIVE
NEAR-FIELD SCATTERING
● Optically simple ● FREE from stray light (heterodyning) ● Unfortunately, does not work for fully non-ergodic systems
IMAGE DIFFERENCE → SPECKLE FIELD SOURCE CORRELATIONS
SCATTERED INTENSITY STRUCTURE (OR FORM) FACTOR
3. A FEW WORDS
ABOUT IMAGING
At this point…
Photon Correlation Imaging (PCI) (L. Cipelletti, 2009)
Imaging through a stopped-down lens
“Speckled” image of the beam
FT plane
z1 z2
f
All regions correspond to the same q (nothing to do with FT of image plane)
Front f.p.
Back f.p.
Image plane
1st F.T.
2nd F.T.
AN APPARENTLY STUPID QUESTION: Why do we see (large) non absorbing particles under the microscope?
A phase object
A phase object
(with border)
● If a 3D sphere can be shown to be equivalent to a 2D disk with a suitable refractive index profile, no chance! ● In the anomalous diffraction approximation, rigorously true (so, we just see cell borders)
The microscope: a paradise of Fourier optics
“SOURCE” PLANES
“OBJECT” PLANES
F. T. PAIRS
RESOLVING POWER ● INCOHERENT IMAGING: RAYLEIGH CRITERION We shall say that two incoherent point sources, imaged by a lens with pupil w, are “barely resolved” if the center of the Airy pattern of one point coincides with the first minimum of the second.
≈0.73
1
Minimum resolvable separation: wziλ
δ 61.0≈
● EXAMPLE: astronomical telescope Sources are always mutually incoherent and, since they are at ∞, the are imaged on the focal plane. Than, is the angular separation of the stars and
f/δϑ =
wλ
ϑ 61.0min ≈
ϑ
zi = f
2w
∞
∞ δ
● COHERENT IMAGING The result depends on relative phase of the two point sources. Indeed, we can write (in normalized image coordinates):
[ ] [ ]211
)61.0()61.0(2
)61.0()61.0(2)(
++
+−−
=xxJe
xxJxI i
ππ
ππ ϕ
b) Sources in quadrature (ϕ = π/2): identical to incoherent
a) Sources in phase (ϕ = 0): NO DIP at all!
c) Sources in counter-phase (ϕ = π): 100% DIP! (resolution doubles)
RESOLVING POWER: MICROSCOPE ● Microscope optics is definitely NOT paraxial. The Rayleigh criterion in the non-paraxial case becomes:
ONAnλ
θλ
δ 22.1)sin(
22.1 =≈θ = angle subtended by the exit pupil viewed from the image plane NAO = numerical aperture of the objective.
Specimen
Condenser
Objective
Back focal plane
● However, the degree of spatial coherence is controlled by the numerical aperture of the condenser NAC (see next) ● Full coherent illumination for NAC → 0.
● Maximum resolution is obtained when the NA of condenser and objective are comparable
Try to look at a dilute suspension (3%) of 100 nm PS particles
CERBINO & TRAPPE EXPERIMENT (a replica by Eleonora Secchi)
● Standard microscope in white light ● High N.A. objective (no stop-down tricks), but ● Moderately stopped-down condenser
DDM
BUT NOW TAKE THE DIFFERENCE BETWEEN TWO IMAGES AT SLIGHTLY DIFFERENT TIMES (OR WITH THE TIME AVG.)
Scattering (3D) and image (2D) q-vectors
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎠
⎞⎜⎝
⎛+=
2||2
||2
21
kq
2nd term O(θ)→ for small θ: 2||
2 qq ≈
POLYCHROMATIC In general different q may correspond to the same q||, but negligible for:
||
1q
<<Δλ
KÖHLER ILLUMINATION
APERTURE STOP Controls the size of the illuminated area
FIELD STOP Controls the spatial
coherence of the source
A “coherence mosaic” source:
● Scattering from tiny uncorrelated volumes (little problems with turbidity)
Actually, a 3D mosaic!
● Allows for (moderate) z-scan
wt wl
λπ 2
lt
ww ≈
E.G: Approaching the metastable critical point of a depleted colloid
GHOST PARTICLE IMAGE VELOCIMETRY (GPIV) Tracing flow with d=50 nm PS tracers
Flux
Coarse-graining in regions of interest
FLOW RECONSTRUCTION
0
200
400
600
0 200 400 600 800
Δz= 5um Δz= 20um
velocity [um/s]
z [u
m]
z scan
A 3-D TECHNIQUE