Interferentie (continued…)
Coherentie
in tijd
in plaats
De Michelson interferometer in geval van
beperkte coherentie
Optical coherence tomography
De Fabry-Perot interferometer (etalon)
1
Leerdoelen
In dit college behandelen we:
• Coherentielengte en coherentietijd
• Beperkte zichtbaarheid van interferentie
• Optical coherence tomography
• Multi-beam interferentie: etalons
• Anti-reflectie coatings
• Hecht: 4.10; 9.6
2
Coherentie, incoherentie, partiële coherentie
Als kijken naar één persoon redelijk goed voorspelt wat iemand anders doet,
dan zijn deze mensen gecorreleerd.
Coherent
Partieel coherent
3
Incoherent
Correlatie functies
We kunnen de correlatie van een optisch veld wiskundig beschrijven
met de covariantie functie
waarbij de haken een tijdsgemiddelde voorstellen.
Als het veld op r1 en het veld op r2 ongecorreleerd zijn, dan zal dit
tijdsgemiddelde nul zijn.
Als de velden wél (partieel of volledig) gecorreleerd zijn, dan is de functie
 ongelijk aan nul.
Coherentie theorie, is het onderwerp van Hfd. 12. Wij zullen het hier
verder niet behandelen
4
Young’s double slit experiment
Interference between light waves!
• Light passing through a small aperture produces a coherent light wave
• This coherent light illuminates two slits
• Because there is a fixed phase relation between the emitted waves,
interference will be visible
Spatially coherent source
5
Two Slits and Spatial Coherence
If the spatial coherence length is less
than the slit separation, then the
relative phase of the light transmitted
through each slit will vary randomly,
washing out the fine-scale fringes, and
a one-slit pattern will be observed.
Max
Fraunhofer diffraction patterns
Good spatial
coherence
Poor spatial
coherence
6
The Spatial Coherence Length
A plane wave is also considered perfectly spatially coherent.
The spatial coherence length is the transverse distance over which
the wave-fronts remain flat:
Wave-fronts
Spatial
Coherence
Length, xc
xc
x
Since there are two transverse dimensions, we should also define the
spatial coherence area, Ac = xc yc.
Spatial coherence can be limited if there are
multiple waves of the same frequency but with
different directions, that is, vectors.
7
The spatial coherence depends on the
emitter size and its distance away.
Light source
q
d
A distance D from a source of
diameter d:
Dk ≈ k sinq
≈ (2p/l) d/D
D
So the spatial coherence area Ac is:
æ 2π ö D 2 l 2 l 2
Ac » ç ÷ » 2 =
W
d
è Dk ø
2
where W = d2/D2 is the
solid angle subtended
by the source.
Starlight is spatially very coherent because stars are very far away.
So even an uncorrelated source like a star, made up of atoms that 8
emit independently, can produce sharp fringes here on Earth.
How quickly will a broadband light wave
deviate from a perfect sine wave in time?
Suppose the light wave has two frequencies:
Etot (z,t) = E0 expi(k1z - w1t) + E0 expi(k2 z - w2t)
The two frequencies will become
significantly out of phase with
each other in a time, tc:
E
w1t c - w 2t c = 2π
Þ t c = 2π / (w1 - w 2 )
t
So the phase will drift on a
time scale of: ~ 2p/Dw = 1/Dn
where:
Dw = w1 - w2 = 2πDn
9
Shorter light pulses
have broader spectra
Duration of a light pulse:
1
Dt »
a
Width of the spectrum:
Dw » a
So:
F(w)
f(t)
t
w
t
w
t
w
1
Dw »
Dt
The shorter the pulse,
the broader the spectrum!
This also has implications for the coherence time of a light source:
a broader spectrum leads to a reduced coherence time.
10
The Temporal Coherence Time
A harmonic plane wave is considered perfectly temporally
coherent.
The temporal coherence time is how long the beam remains
sinusoidal in time at a single wavelength (and amplitude):
Wave-fronts
Temporal
Coherence
Time, tc
tc
x
Temporal coherence asks what if we
add many waves with different w’s
for a given beam direction.
A monochromatic plane wave has an infinite coherence time.
11
The coherence time is the reciprocal of
the spectral width (bandwidth).
The largest frequency difference in the light wave will yield the
shortest phase-drift time, which we call the coherence time:
t c = 1/ Dv
where Dn is the light bandwidth (the width of the spectrum).
Sunlight and light bulbs are temporally incoherent—and have
very small coherences times (a few femtoseconds)—because
their bandwidths are very large (the entire visible spectrum).
Lasers can have much longer coherence times—as long as
about a second; that's >1014 cycles of the electric field!
12
Spatial
and
Temporal
Coherence
Beams can be coherent,
partially coherent or
incoherent in both
space and time.
Only a plane wave is
perfectly temporally and
spatially coherent.
Wave-fronts
Spatial and
Temporal
Coherence
xc
tc
Temporal
Coherence;
Spatial
Incoherence
xc
tc
Spatial
Coherence;
Temporal
Incoherence
xc
tc
Spatial and
Temporal x
Incoherence
xc
t, z
tc
13
Spatial and
Temporal
Coherence:
Another
Picture
Spatial and
Temporal
Coherence
Temporal
Coherence;
Spatial
Incoherence
Spatial
Coherence;
Temporal
Incoherence
Spatial and
Temporal
Incoherence E,x
t, z
14
Many Different Ray Angles Interfering in
Space
We considered
focusing rays in pairs
with symmetrically
propagating directions
and added up all the
fields at the focus,
each yielding fringes
with a spacing of
l/2sin(q), where q is
the ray angle relative
to the axis.
In addition, we required all such fringes to be perfectly in phase at the
focus.
15
Spatial Coherence: Interference in Space
of Many Beams from a Perfect Lens
We added up all the spatial fringes from crossing beams at the focus
of an infinitely large perfect lens and found that we could focus a
beam to a diameter of
~l/2 (= 2p/Dk):
E
Irradiance
x
The interference is coherent and constructive at x = 0, yielding a
maximally intense small beam there.
But it is coherent and destructive elsewhere.
Interference of Beams in Space from a
Lousy Lens
Now, what if the lens is very badly made, and all the spatial fringes
are randomly out of phase with each other?
The fringes now shift by random amounts:
E
Irradiance
x
Analogous to the temporal case, the fluctuations are on a length scale
similar to the perfect in-phase case, but are much less intense and
there are many more spikes.
Dk = Dk/2p
Also analogously, the coherence length is: Lc ~ 2p/Dk = 1/Dk, where Dk
is the range of off-axis k’s (for this beam and also the perfect focus).
Does a broad spectrum always produce a
short pulse?
No! Only if all the waves arrive in phase, do they add up to a short
pulse. Otherwise, there will only be rapid fluctuations on the
timescale of the coherence time.
Flat
spectral
phase
Frequency
Intensity vs. time
Locked
phases
Short
pulse
Time
Complex
spectral
phase
Frequency
Intensity vs. time
Random
phases
Light
bulb
Time
How to Make Spatially and Temporally
Coherent Light from a Light Bulb
A light bulb is neither spatially nor temporally coherent.
temporally
19
Michelson interferometer with a
low-coherence source
Interference will only be visible if
the arm lengths are within a
distance ½ c τc of each other.
Light source
Low-Coherence Source
Detector
Detector
Coherent Source
l/2
Mirror Displacement
½ c τc
Mirror Displacement
20
Optical Coherence Tomography
Michelson interferometer with a low coherence source and a biological
sample as one of the end mirrors.
Different reflecting layers in a sample can be distinguished
B-Scan if they are
spaced by more than ½ c τc.
A-line
21
Optical coherence tomography of a
human fovea
6 x 6 mm
6 x 1.1 mm
3.1 x 0.6 mm
B. Cense et al. Opt. Express 12, 2435-2447 (2004)
22
Interference of more than two waves: the FabryPerot interferometer (Fabry-Perot etalon)
• A Fabry-Perot interferometer consists of two parallel reflecting surfaces.
• An etalon is a Fabry-Perot interferometer consisting of a piece of glass
with parallel reflecting surfaces.
• The reflected and transmitted waves are an infinite series of multiply
reflected and spatially overlapping waves.
r, t = reflection, transmission coefficients from air to glass
r′,t′ = reflection, transmission coefficients from glass to air.
L
Transmitted
wave: E0t
Incident wave: E0
Reflected
wave: E0r
nair = 1
n
nair = 1
d = round-trip phase
difference in medium = 2kL
(normal incidence)
tt  E0
2  id

tt r e E0
tt  (r 2 eid ) 2 E0
tt  (r 2 eid )3 E0
23
Transmitted wave through an FP-interferometer
E0t = tt ¢E0 + tt ¢ r 2e -id E0 + tt ¢ (r 2e -id ) 2 E0 + tt ¢ (r 2e -id )3 E0 +...
= tt¢ E0 éë1+ (r 2e -id ) + (r 2e -id )2 +...ùû
(
E0t = tt¢E0 / 1- r 2e -id
(
)
1
1- x
lim 1+ x + x 2 + x 3 +......+ x n =
n®¥
)
2
Sec. 4.10:
Set r = -r′ and
tt′ = 1 – r2 = 1 – R
2
E0t
tt¢
(1- R) 2
The transmission is: T º
=
=
2 -id
E0
1- r e
(1- Re -id )(1- Re +id )
é
ù é
ù é
ù
(1- R)2
(1- R)2
(1- R)2
=ê
ú=ê
ú=ê
ú
2
2
2
2
2
1+
R
2Rcos(
d
)
1+
R
2R[12sin
(
d
/
2)]
12R
+
R
+
4Rsin
(
d
/
2)]
ë
û ë
û ë
û
Divide above and below the line by: (1  R) 2
1
T=
1+ F sin 2 d / 2
(
)
with :
F=
4R
(1- R) 2
24
Etalon Transmittance vs. Thickness,
Wavelength, or Angle
1
T=
1+ F sin 2 d / 2
(
Transmittance
1
0.5
0
-2p
-p
Transmission maxima
occur when d / 2 = mp:
R = 18%
F=1
R = 5%
F = 0.2
R = 87%
F = 200
0
p
)
2pL/l = mp ; m×l/2=L
2p
3p
4p
d = nk0 L = 2πnL / l0
or:
l = 2L / m
The transmittance varies significantly with thickness, angle, and
wavelength.
As the reflectance of each surface (R) approaches 1 (F increases),
the widths of the high-transmission regions become very narrow. 25
Etalon ‘Free Spectral Range’ (FSR)
The Free Spectral Range is the frequency or wavelength range between
transmission maxima.
Transmittance
1
lFSR =
Free Spectral
Range
0.5
fFSR=c/2L
0
-2p
4p L
4p L

= 2p
l
l  lFSR

4p L
l

4p L[1  lFSR / l ]
l
lFSR
= 2p

-p
0
p
2p
d = kL = 2π L / l
3p
4p
4p L
4p L

= 2p
l
l[1  lFSR / l ]
1 1  lFSR / l
1
l2


=
 lFSR =
l
l
2L
2L
26
The line width lLW is an etalon
transmittance peak's full-width-halfmax (FWHM).
Pick a peak (it’s easiest to use the
one centered at d = 0), and find the
value of d that yields T = ½. Then
set this value of d equal to dLW/2.
Assume that the reflectivity is close to
one and that d << 1.
The line
width is:
lLW =
l2  1 R 
πL 

R 
Transmittance
Etalon Line Width
1
T=
1  F sin 2 d / 2 
4R
F=
(1  R) 2
lLW
l
d = 2π L / l
The line width is the accuracy
with which an etalon can
measure wavelength.
27
The Interferometer or Etalon Finesse
The Finesse, F , is the ratio of the free spectral range and the
line width:
l2
R
L
F » 2
= π
æ
ö
1- R
l 1- R
ç
÷
πL è R ø
Taking
R »1
F » π / (1- R)
The Finesse is the number of wavelengths the interferometer can
resolve.
28
Anti-Reflection Coatings
Notice that the center of
the round glass plate
looks like it’s missing.
It’s not! There’s an
anti-reflection coating
there (on both the front
and back of the glass).
Such coatings have
been common on
photography lenses and
are now common on
eyeglasses.
Multilayer
Coatings
Air
An AR coating usually consists of
many layers with thickness l/4,
usually with alternating high and
low refractive indices.
n0
nL
nH
nL
The phase shifts at each interface
depend on the refractive index
change.
Using many layers allows precise
control over the reflection
coefficient R.
nH
nL
nH
Glass substrate
ns
Bewijs van de stelling op slide 24
31
Tot slot
Wat hebben we gezien:
• Coherentielengte en coherentietijd
• Beperkte zichtbaarheid van interferentie
• Optical coherence tomography (oogheelkunde)
• Multi-beam interferentie: etalons, toepassing in de spectroscopie
• Anti-reflectie coatings
32