Download Introduction to Nonlinear Optics

Introduction to Nonlinear Optics
Robert W. Boyd
University of Ottawa
University of Rochester
University of Glasgow
P resented at the International School of Physics “Enrico Fermi,”
Course 190 - Frontiers in Modern Optics, 30 June - 5 July, 2014.
Outline of Presentation
Part I. Tutorial Introduction to Nonlinear Optics
Part II. Recent Research in Quantum Nonlinear Optics
Why Study Nonlinear Optics?
It is good fundamental physics.
It leads to important applications.
It is a lot of fun.
Demonstrate these features with examples in remainder
of talk.
1. What is Nonlinear Optics?
Nonlinear Optics and Light-by-Light Scattering
The elementary process of light-by-light scattering has never
been observed in vacuum, but is readily observed using the
nonlinear response of material systems.
Nonlinear material is fluorescein-doped boric acid glass (FBAG)
n2
14
n2(silica) [But very slow response!]
M. A. Kramer, W. R. Tompkin, and R. W. Boyd, Phys. Rev. A, 34, 2026, 1986.
W. R. Tompkin, M. S. Malcuit, and R. W. Boyd, Applied Optics 29, 3921, 1990.
Opomjofbs!Pqujdt
UIJSE!FEJUJPO
Spcfsu!X/!Cpze
Some Fundamental Nonlinear Optical Processes: I
Second-Harmonic Generation
(2)
2
2
Dolgaleva, Lepeshkin,
and Boyd
Some Fundamental Nonlinear Optical Processes: II
Sum-Frequency Generation
1
(2)
3
=
+
1
2
2
3
2
1
Some Fundamental Nonlinear Optical Processes: III
Difference-Frequency Generation
amplified!
1
(2)
3
=
2
1
2
2
1
3
Optical Parametric Oscillation
Parametric Downconversion: A Source of Entangled Photons
The signal and idler photons are entangled in:
(a) polarization
(b) time and energy
(c) position and transverse momentum
(d) angular position and orbital angular momentum
Entanglement is important for:
(a) Fundamental tests of QM (e.g., nonlocality)
(a) Quantum technologies (e.g., secure communications)
Some Fundamental Nonlinear Optical Processes: III
Third-Harmonic Generation
(3)
3
3
Some Fundamental Nonlinear Optical Processes: IV
Intensity-Dependent Index of Refraction
self-phase modulation
cross-phase modulation
E( )
E( )
(3)
i
NL
(strong wave)
E( ' )
(probe wave)
where
E( ) e
(3)
E( ' ) e
i NL
Role of Material Symmetry in Nonlinear Optics
(2) vanishes identically for a material
possessing a center of inversion symmetry (a centrosymmetric medium).
non-centrosymmetric medium
U(x)
parabola
parabola
actual
potential
x
centrosymmetric medium
U(x)
parabola parabola
actual
potential
x
2. Coupled Wave Equations and Harmonic
Generation
Treatment of Second-Harmonic Generation – I
z
E1, ω
χ
(2)
E2, 2 ω
L
Let
Ẽ1(z, t) = E1(z)e i!t + c.c. = A1ei(k1z !t) + c.c.
Ẽ2(z, t) = E2(z)e i2!t + c.c. = A2(z)ei(k2z 2!t) + c.c.
where k1 = n1!/c and k2 = n22!/c.
(1)
(2)
We have assumed that the pump wave E1 at frequency ! is undepleted by the nonlinear interaction. We take A2 to be a function of z
to allow the second harmonic wave to grow with z. We also set
P̃2(t) = P2 e i2!t where P2 = ✏0 (2)E12 = ✏0 (2)A21 ei2k1z (3)
The generation of the wave at 2! is governed by the wave equation
r2Ẽ2
n2 @ 2Ẽ2
1 @ 2P̃2
=
.
2
2
2
2
c @t
✏0c @t
(4)
Treatment of Second-Harmonic Generation – II
Note that the first 2term in the wave equation3 is given by
6 @ 2 Ã
7
@ Ã2
6
7
2
2
2
6
6
r Ẽ2 = 64 2 + 2ik2
k2 A27775 ei(k2z 2!t)
(5)
@z
@z
2
3
6
7
@ Ã2
6
7
2
6
6
⇡ 642ik2
k2 A27775 ei(k2z 2!t)
(6)
@z
The second form is the slowly varying amplitude approximation.
Note also that
2P̃
@ 2Ã2
@
2
2
i(k
z
2!t)
2P ei(2k1z 2!t) (7)
2
=
(2!)
A
e
=
(2!)
2
2
@t2
@t2
By combining the above equations, we obtain
dA2
4! 2 (2) 2 i k z
2ik2
= 2
A1 e
where k = 2k1 k2. (8)
dz
c
The quantity k is known as the phase (or wavevector) mismatch
factor, and it is crucially important in determining the efficiency of
nonlinear optical interactions
Treatment of Second-Harmonic Generation – III
For the case
k = 0, Eq. (8) becomes
dA2
4! 2 (2) 2
2ik2
= 2
A1
(9)
dz
c
with solution evaluated at z = L of
2
2i! (2) 2
4!
A2(L) =
A1L or |A2(L)|2 = 2 2 [ (2)]2|A1|4L2. (10)
n2 c
n2 c
Note that the SHG intensity scales as the square of the input intensity
and also as the square of the length L of the crystal.
2ω
2ω
dipole emitter
phased array of dipoles
Treatment of Second-Harmonic Generation – IV
For the general case of
k 6= 0, Eq. (8) can still be solved to yield
2
4!
|A2(L)|2 = 2 2 [ (2)]2 |A1|4L2 sinc2( k L/2)
(11)
n2 c
Note that k L must be kept much smaller than ⇡ radians in order
for efficient SHG to occur.
Second Harmonic Generation and Nonlinear Microscopy
Nonlinear Optical Microscopy
An important application of harmonic generation is nonlinear microscopy. . .
Microscopy based on second-harmonic generation in the configuration of a confocal
microscope and excited by femtosecond laser pulses was introduced by Curley et al.
(1992). Also, harmonic-generation microscopy can be used to form images of transparent (phase) objects, because the phase matching condition of nonlinear optics depends
sensitively on the refractive index variation within the sample being imaged (Muller et
al., 1998).
Boyd, NLO, Subsection 2.7.1
Caution!
Curley et al., not Curly et al.
How to Achieve Phase Matching: Birefringence Phase Matching
The phase matching condition k = 0 requires that
n1!1 n2!2 n3!1
+
=
where !1 + !2 = !3
c
c
c
These conditions are incompatible in an isotropic dispersive material.
However, for a birefringent material phase matching can be achieved.
ω
ordinary
ĉ
θ
k
2ω
extraordinary
Midwinter and Warner showed that there are two ways to achieve phase matching:
Positive uniaxial
(ne > n0)
Type I n03!3 = ne1!1 + ne2!2
Type II n03!3 = no1!1 + ne2!2
Negative uniaxial
(ne < n0)
ne3!3 = no1!1 + no2!2
ne3!3 = ne1!1 + no2!2
How to Achieve Phase Matching: Quasi Phase Matching
single domain crystal
periodically poled crystal
Λ
(2)
is periodically inverted to prevent reverse power flow.
(a) with perfect phase-matching
(b) with quasi-phase-matching
field amplitude
Sign of
(c) with a wavevector
mismatch
0
0
2
4
z / Lcoh
6
8
Additional Studies of Wave Propagation Effects
3. Mechanisms of Optical Nonlinearity
Typical Values of the Nonlinear Refractive Index
Mechanism
Electronic polarization
Molecular orientation
Electrostriction
Saturated atomic absorption
Thermal e↵ects
Photorefractive e↵ectb
a
(3)
1111
2
2
n2 a
(cm2/W)
(m /V )
Response time
(sec)
10 16
10 14
10 14
10 10
10 6
(large)
10 22
10 20
10 20
10 16
10 12
(large)
10 15
10 12
10 9
10 8
10 3
(intensity-dependent)
For linearly polarized light.
b
The photorefractive e↵ect often leads to a very strong nonlinear response. This response usually
cannot be described in terms of a (3) (or an n2) nonlinear susceptibility, because the nonlinear polarization does not depend on the applied field strength in the same manner as the other mechanisms
listed.
Quantum Mechanical Origin of the Nonlinear Optical Suesceptibility
(2)
ijk (!p
l
+ !q €, !q €, !p )
N
=
2 0 h2
(0)
ll
lmn
(a1 )
iln jnm kml
[(!nl !p !q ) inl ][(!ml !p ) iml ]
iln knm jml
+
[(!nl !p !q ) inl ][(!ml !q ) iml ]
kln inm jml
+
[(!mn !p !q ) imn ][(!nl + !p ) + inl ]
jln inm kml
+
[(!mn !p !q ) imn ][(!nl + !q ) + inl ]
(a1 )
(b2 )
jln knm iml
+
[(!ml + !p + !q ) + iml ][(!nl + !q ) + inl ]
ωq
ωp
n
m
m
ωp
ωq
l
l
n
n
(a )
2
l
l
n
n
m
m
(a' )
1
ωq
(a' )
2
ωp
l
(a2 )
+
kln jnm iml
+
[(!ml + !p + !q ) + iml ][(!nl + !p ) + inl ]
n
1
ωp
(b1 )
[(!nm + !p + !q ) + inm ][(!ml !q ) iml ]
l
l
(a2 )
jln inm kml
+
[(!nm + !p + !q ) + inm ][(!ml !p ) iml ]
kln inm jml
(a )
l
l
l
n n
ωq
l
n
n
(b )
1
m
(b 2 )
m
ωq
ωp
ωp
ωq
l
l
l
l
(b1 )
m
(b2 )
m
m m
b' )
2
b'
n
1
l
l
ωq
ωp
n ωp
ωq
l
l
Some Actual Z-Scan Data
Closed aperture
Open aperture
1.2
I0 = 0.176 GW/cm2
1.2
transmission
gold-silica composite
transmission
CS2
1.0
Chang, Shin
0.8
–20
1.1
1.0
n2 =
0.0373 cm2/GW
0
z (mm)
20
0.9
–125
Piredda
0
z (mm)
For closed aperture z-scan
ΔTpv = 0.406 ΦNL
where
ΦNL = n2 (ω/c) Ι0 L
M. Sheik-Bahae et al., IEEE J.
Quantum Electron. 26 760 (1990).
125
6. Self-Action Effects in Nonlinear Optics
Self-Action Effects in Nonlinear Optics
Self-action effects: light beam modifies its own propagation
tTFMGGPDVTJOH
tTFMGUSBQQJOH
tTNBMMTDBMFöMBNFOUBUJPO
d
Prediction of Self Trapping
n = n0
d
n = n0 + n
n = n0
radial profile of self
-trapped beam
Optical Solitons
Field distributions that propagate without change of form
Temporal solitons (nonlinearity balances gvd)
1973: Hasegawa & Tappert
1980: Mollenauer, Stolen, Gordon
Spatial solitons (nonlinearity balances diffraction)
1964: Garmire, Chiao, Townes
1974: Ashkin and Bjorkholm (Na)
1985: Barthelemy, Froehly (CS2)
1991: Aitchison et al. (planar glass waveguide
1992: Segev, (photorefractive)
Solitons and self-focussing in Ti:Sapphire
Diffraction-management
controls the spatial selffocussing
Dispersion-management
controls the temporal
self-focussing
Beam Breakup by Small-Scale Filamentation
Predicted by Bespalov and Talanov (1966)
Exponential growth of wavefront imperfections by four-wave mixing processes
exponential growth rate, /
Sidemode amplitude grows as
A1(z) = A1(0)ei z e z
2
q2
=
2
2k
q2
,
2k
= n2 ( / c)I0 , q = k
transverse wavevector, q/qmax
qmax = 2k
qmax
max =
k
Honeycomb Pattern Formation
Output from cell with a single gaussian input beam
At medium input power
at cell exit
in far field
At high input power
at cell exit
in far field
Quantum statistics?
Input power 100 to150 mW
Input beam diameter 0.22 mm
Sodium vapor cell T = 220o C
Wavelength = 588 nm
Bennink et al., PRL 88, 113901 2002.
Optical Radiance Limiter Based on Spatial Coherence Control
Controlled small-scale filamentation used to modify spatial degree of coherence
Alternative to standard appropches to optical power limiting
spatially
coherent
light in
spatially
incoherent
light out
n2 medium
1.6
1.2
increasing power
Schweinsberg et al., Phys. Rev. A 84, 053837 (2011).
Far-field dif
0.8
0.4
0
0
10
20
Incident pulse ener
30
4. Local-Field Effects in Nonlinear Optics
Local Field Effects in Nonlinear Optics – I
Recall the Lorentz-Lorenz Law
(linear optics)
E
E
P
loc
(1) − 1
N
α
ϵ
4
(1)
χ =
or (1)
= πN α.
4
1 − 3 πN α
ϵ +2 3
This result follows from the assumption that the field that acts on a
representative atom is not the macroscopic Maxwell field but rather
the Lorentz local field given by
Eloc = E + 43 πP
where
P = χ(1)E
We can rewrite this result as
ϵ(1) + 2
Eloc = LE where L =
is the local field factor.
3
The Lorentz Red Shift
See, for instance, H. A. Lorentz, Theory of Electrons, Dover, NY (1952).
Observation of the Lorentz Red Shift
Maki, Malcuit, Sipe, and Boyd, Phys. Rev. Lett. 68, 972 (1991).
Local Field Effects in Nonlinear Optics – II
For the case of nonlinear optics, Bloembergen (1962, 1965) showed
that, for instance,
χ(3)(ω = ω + ω − ω) = N γ (3)|L(ω)|2[L(ω)]2.
where γ (3) is the second hyperpolarizability and where
ϵ(ω) + 2
L(ω) =
3
For the typical value n = 2, L = 2, and L4 = 16. Local field effects
can be very large in nonlinear optics! But can we tailor them for our
benefit?
We have been developing new photonic materials with enhanced NLO
response by using composite structures that exploit local field effects.
Nanocomposite Materials for Nonlinear Optics
)
i
2a
a
h
b
b
Fractal Str
e
Layer
a
b
- In each case, scale size of inhomo eneity << optical avele
-
s all optical properties, ch as n an
effective
r
)
s
, can be escribe by
Local Field Enhancement of the NLO Response
- Under very general conditions, we can express the NL
response as
2 (3)
2
(3)
eff
= fL L
where f is the volume fraction of nonlinear material and L is
the local-field factor, which is different for each material geometry.
- Under appropriate conditions, the product f L2 L 2 can exceed unity.
- For a homogeneous material
+2
3
L=
- For a spherical particle of dielectric constant
of dielectic constant h
3h
L=
m
+2
m
embedded in a host
h
- For a layered geometry with the electric field perpendicular to the
plane of the layers, the local field factor for component a is given by
L=
eff
1
a
eff
=
fa
a
+
fb
b
Enhanced NLO Response from Layered Composite Materials
A composite material can display a larger NL response than its constituents!
Quadratic EO effect
Alternating layers of TiO2 and
the conjugated polymer PBZT.
gold electrode
BaTiO3
signal generator
1 kHz
AF-30 in
polycarbonate
ITO
glass substrate
photodiode
polarizer
Measure NL phase shift as a
function of angle of incidence.
35% enhancement in
(3)
Fischer, Boyd, Gehr, Jenekhe, Osaheni, Sipe, and
Weller-Brophy, Phys. Rev. Lett. 74, 1871 (1995).
Diode Laser
(1.37 um)
dc voltmeter
lockin amplifier
3.2 times enhancement!
Nelson and Boyd, APL 74 2417 (1999)
Accessing the Optical Nonlinearity of Metals with Metal-Dielectric
Photonic Crystal Structures
Metals have very large optical nonlinearities but low transmission
Low transmission because metals are highly reflecting (not because they are absorbing!)
Solution: construct metal-dielectric photonic crystal structure
(linear properties studied earlier by Bloemer and Scalora)
E
Cu
SiO2
E
Cu
PC structure
80 nm of
copper (total)
bulk metal
80 nm of copper
T = 0.3%
T = 10%
z
0
L
Bennink, Yoon, Boyd, and Sipe, Opt.
Lett. 24, 1416 (1999).
Lepeshkin, Schweinsverg, Piredda, Bennink,
and Boyd, Phys. Rev. Lett. 93, 123902 (2004).
bulk Cu
M/D PC
nm
I = 500 MW/cm2
(response twelvetimes larger)
z
Gold-Doped Glass: A Maxwell-Garnett Composite
Red Glass Caraffe
Nurenberg, ca. 1700
Huelsmann Museum, Bielefeld
Developmental Glass, Corning Inc.
gold volume fraction approximately 10-6
gold particles approximately 10 nm diameter
• Composite materials can possess properties very different
from those of their constituents.
• Red color is because the material absorbs very strong in the
blue, at the surface plasmon frequency
Counterintuitive Consequence of Local Field Effects
Cancellation of two contributions that have the same sign
Gold nanoparticles in a reverse saturable absorber dye solution (13 μM HITCI)
normalized transmittance
1.3
SA
β<0
9
1.2
8
1.1
Increasing Gold
Content
7
1.0
< 0
6
0.9
0.8
2
0.6
1
-10
0
(3)
5
Im
4
β>0
3
0.7
0.5
Im
(3)
RSA
(3)
eff
2
= f L L
10
z (cm)
2
(3)
i
+
20
D.D. Smith, G. Fischer, R.W. Boyd, D.A.Gregory, JOSA B 14, 1625, 1997.
(3)
h
30
> 0
NLO in Plasmonics (Photonics Using Metals)
Is there an intrinsic nonlinear reponse to surface plasmon polaritons (SPPs)?
2 is106 times that of silica)
sp
polarized
Translation
stage
Au film
z
x
Laser
PD
6
y
Aperture
sp
HWP PBS
sp
sp
Rotation
TM stage
1.0
0
Kretschmann
dip
1
2
3
Ssp
Reflectance, R
0.5 TIR onset
increasing
power
Pi = 2.70 mW
Pi = 8.70 mW
Pi = 15.0 mW
Pi = 22.8 mW
Pi = 28.4 mW
0.1
0.05
41.5
42
42.5
43
Incidence angle, (deg.)
43.5
I. De Leon, Z. Shi, A. Liapis and R.W.Boyd, Optics Letters 39, 2274 (2014)
Slow Light in a Fiber Bragg Grating (FBG) Structure
(Can describe properties of FBGs by means of analytic expressions)
forward and backward waves
are strongly coupled
dispersion relation
slow light
stored energy (same as group index)
theory (Winful)
KL = 4
k
k
Bhat and Sipe showed that the
nonlinear coeficient is given by
detuning from Bragg frequency
where the slow-down factor S = ng/n
Improved Slow-Light Fiber Bragg Grating (FBG) Structure
Much larger slow-down factors possible
with a Gaussian-profile grating
Observation of (thermal)
optical bistability at mW
power levels
group index
approximately 140
H. Wen, M. Terrel, S. Fan and M. Digonnet, IEEE Sensors J. 12, 156-163 (2012).
J. Upham, I. De Leon, D. Grobnic, E. Ma, M.-C. N. Dicaire, S.A. Schulz, S. Murugkar, and R.W. Boyd,
Optics Letters 39, 849-852 (2014).
The other Lake Como
5. Slow and Fast Light
Controlling the Velocity of Light
“Slow,” “Fast” and “Backwards” Light
– Light can be made to go:
slow:
vg << c (as much as 106 times slower!)
fast:
vg > c
backwards: vg negative
Here vg is the group velocity: vg = c/ng ng = n +
(dn/d )
– Velocity controlled by structural or material resonances
absorption
profile
Review article: Boyd and Gaut ier, Science 326, 1074 (2009).
Slow and Fast Light Using Isolated Gain or Absorption Resonances
absorption
resonance
0
0
n
ng
gain
resonance
g
n
slow light
ng
slow light
fast light
fast light
ng = n +
(dn/d )
Light speed reduction
to 17 metres per second
in an ultracold atomic gas
|4〉 = |F =3, MF = –2 〉
|3〉 = |F =2, MF = –2 〉
60 MHz
Lene Vestergaard Hau*², S. E. Harris³, Zachary Dutton*²
& Cyrus H. Behroozi*§
ωc
* Rowland Institute for Science, 100 Edwin H. Land Boulevard, Cambridge,
Massachusetts 02142, USA
² Department of Physics, § Division of Engineering and Applied Sciences,
Harvard University, Cambridge, Massachusetts 02138, USA
³ Edward L. Ginzton Laboratory, Stanford University, Stanford, California
94305, USA
ωp
D2 line
λ = 589 nm
|2〉 = |F =2, MF = –2 〉
1.8 GHz
|1〉 = |F =1, MF = –1〉
30
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
a
25
20
–30
1.006
Refractive index
T = 450 nK
τDelay = 7.05 ± 0.05 μs
1.004
–20
–10
0
10
20
30
b
PMT signal (mV)
Transmission
NATURE | VOL 397 | 18 FEBRUARY 1999 |www.nature.com
L = 229 ± 3 μm
vg = 32.5 ± 0.5 m s–1
15
10
1.002
1.000
5
0.998
0.996
0
0.994
–30
–20
–10
0
10
20
30
–2
0
Probe detuning (MHz)
Note also related work by Chu, Wong, Welch, Scully, Budker, Ketterle, and many others
2
4
Time (
μs)
6
8
10
12
Goal: Slow Light in a Room-Temperature
Solid-State Material
Crucial for many real-world applications
We have identified two preferred methods for producing
slow light
(1) Slow light via coherent population oscillations (CPO)
(2) Slow light via stimulated Brillouin scattering (SBS)
Slow Light by Stimulated Brillouin Scattering (SBS)
S
=
L
-
dI S
=
dz
L
g=
S
z
gI LI S
2 2
e
nv c3 0 B
.
refractive index SBS gain
We often think of SBS as a pure gain process, but it also leads to a change in refractive index
Stokes gain
monochromatic
pump laser
anti-Stokes loss
slow light
The induced time delay is
Td
fast light
G
B
where G= g Ip L and
B
is the Brillouin linewidth
Okawachi, Bigelow, Sharping, Zhu, Schweinsberg, Gauthier, Boyd, and Gaeta Phys. Rev. Lett. 94, 153902 (2005).
Related results reported by Song, González Herráez and Thévenaz, Optics Express 13, 83 (2005).
Slow Light via Coherent Population Oscillations
E3,
+
saturable
medium
E1,
ng = n +
dn/d
b
+
measure
absorption
T2 << T1
2
ba
2
=
T
ba
2
=
1
T
1
a
absorption
profile
1/T1
1/T2
Want a narrow feature in absorption
profile to give a large dn/d
Ground state population oscillates at beat frequency (for < 1/T1).
Population oscillations lead to decreased probe absorption
(by explicit calculation), even though broadening is homogeneous.
Ultra-slow light (ng > 106) observed in ruby and ultra-fast light
(ng = –4 x 105) observed in alexandrite.
Slow and fast light effects occur at room temperature!
PRL 90,113903(2003); Science, 301, 200 (2003)
Slow and Fast Light in an Erbium Doped Fiber Amplifier
• Fiber geometry allows long propagation length
• Saturable gain or loss possible depending on
pump intensity
Advance = 0.32 ms
Output
Input
FWHM =
1.8 ms
Time
Fractional Advancement
0.15
6 ms
in
- 97.5 mW
- 49.0 mW
- 24.5 mW
- 9.0 mW
- 6.0 mW
- 0 mW
0.1
0.05
0
-0.05
out
-0.1
10
Schweinsberg, Lepeshkin, Bigelow, Boyd, and Jarabo, Europhysics Letters, 73, 218 (2006).
100
10 3
10 4
Modulation Frequency (Hz)
10 5
Observation of Backward Pulse Propagation
in an Erbium-Doped-Fiber Optical Amplifier
or
ISO
80/20
coupler
980 nm laser
Signal
1550
980
WDM
EDF
WDM
We time-resolve the propagation
of the pulse as a function of
position along the erbiumdoped fiber.
Procedure
• cutback method
• couplers embedded in fiber
G. M. Gehring, A. Schweinsberg, C. Barsi, N. Kostinski,
R. W. Boyd, Science 312, 985 2006.
3
normalized intensity
1550 nm laser
Ref
pulse is placed on a cw
background to minimize
pulse distortion
2
in
1
out
0
0
2
4
time (ms)
6
Observation of Superluminal and
“Backwards” Pulse Propagation
s!STrONGLY
counTerinTUITIvE
PHENOMENON
concEPTUALPrEDIcTION
LABorATorYrESULTS
ng negative
sBUTEnTIrELY
cONSISTenTWITH
ESTABLISHEDPHySICS
t=0
t=3
s0REDICTEDBY'ARRETT
AND-C#UMBER
AND#HIAO
t=6
t=9
t = 12
s/BSERVEDBYGEHring,
3CHWEINSBERG"ARSI
+OSTINSKIAND"OYD
3CIENcE,
t = 15
propagation distance
-1 -0.5
0
0.5
1x105
Normalized length |n g|Z (m)
Causality?
t4VQFSMVNJOBMWgD
BOECBDLXBSETWgOFHBUJWF
QSPQBHBUJPONBZTFFN
DPVOUFSJOUVJUJWFCVUBSFGVMMZDPNQBUJCMFXJUIDBVTBMJUZ
t5IFHSPVQWFMPDJUZJTUIFWFMPDJUZBUXIJDIQFBLPGQVMTFNPWFTJUJTOPU
UIFiJOGPSNBUJPOWFMPDJUZw
pulse amplitude
t*UJTCFMJFWFEUIBUJOGPSNBUJPOJTDBSSJFECZQPJOUTPGOPOBMZUJDJUZPGBXBWFGPSN
time
sBroad spectral content at points of discontinuity
sDISTURBANce moves at vacuum speed of light
see, for instance, R. Y. Chiao
8. Spontaneous and Stimulated Light Scattering
Experiment in Self Assembly
Joe Davis, MIT
Thank you for your attention!