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INTRODUCTION TO SINGULAR NONLINEAR OPTICS
FSU-Jena, Abbe School of Photonics’2011
LECTURE 1: Linear vs. nonlinear optics. Optical solitons.
LECTURE 2: Singular optical beams. Dark optical solitons– physics and applications.
LECTURE 3: Interactions between optical solitons.
LECTURE 4: Polychromatic spatial solitons.
INTRODUCTION TO SINGULAR NONLINEAR OPTICS
Singular optical beams. Dark optical solitons– physics and applications.
FSU-Jena, Abbe School of Photonics’2011
Linear vs. nonlinear optics. Optical solitons. Singular optical beams.
1. Singular optical beams (1-D, quasi-2-D, 2-D, ring-shaped and mixed phase dislocations).
2. Methods for generation and quantitative measurement of their phase profile.3. All-optical waveguiding – physical idea and motivation.
4. 1-D dark (spatial) solitons – experiment vs. theory.5. Quasi-2-D dark solitons and all-optical branching schemes.
6. Optical vortex solitons – existence, soliton constant, stability.7. Ring dark solitary waves – how to rule their transverse
dynamics. 8. Dark beams with mixed phase dislocations – general
characteristics and potential applications.
FSU-Jena, Abbe School of Photonics’2011
FSU-Jena, Abbe School of Photonics’2011
1. Singular optical beamsPhase singularities occur very generally whenever there is an angle continuously
dependent on two or three spatial parameters.
The simplest function with a phase singularity isthe natural map from Cartesian space
to the complex plane
ψ(x, y) = x + iy = Rexp(iφ).
The phase is defined everywhere except the origin.
Example: The north and south poles lie at points where all lines intersect, so are not in any unique zone: it is a phase singularity of the positionof the hour hand on a watch.
N.B. The presence of the British Antarctic survey at the south pole has led to Greenwich Mean Time taken as standard at the south pole.
Nature, 403, 21 (2000)
FSU-Jena, Abbe School of Photonics’2011
1. Singular optical beams (the early history)
George Biddell
Airy
1838
William Whewel
18331836
WilliamRowan
Hamilton
1832
ProblemNameYearConical refraction:In a general anisotropic material there are two directions
(optic axes) where the speeds of the waves are the same. As functions of direction, the two speeds can be represented
by surfaces forming a double cone at each optic axis.Conclusion: Light is a transverse wave.
Tides in the oceans: Cotidal lines - wavefronts of the tide, regarded as a wave of 12h period.Conclusion: There must be “rotatory systems of tide-waves where the cotidal lines revolve around a point where there is no tide (‘amphidromy’).A. Defant, Physical Oceanography, vol. 2. Oxford: Pergamon, 1961.
(M. Berry, "Making waves in physics," Nature, vol. 403, p. 21 (2000))
Singularities are places where mathematical quantities become infinite, or change abruptly.
The rainbow:Earlier Descartes had understood the bright bow as Sun rays directionally focused by raindrops. Conclusions:* the rainbow is a particular example of a caustic, that is, a line where
light rays are focused;* caustics are singularities, where ray optics predicts infinite brightness; * wave physics softens the singularities; * precise mathematical description of this softening (rainbow integral).
FSU-Jena, Abbe School of Photonics’2011
1. Singular optical beams out of laser cavities
HG01
LG01
FSU-Jena, Abbe School of Photonics’2011
1. Singular optical beams out of laser cavities
FSU-Jena, Abbe School of Photonics’2011
1. Singular optical beams out of laser cavities
FSU-Jena, Abbe School of Photonics’2011
1. Singular optical beams with mixed phase dislocations
Step-screw (SS) dislocation Edge-screw (ES) dislocation
D. Neshev et al., Appl. Phys. B72, 849-854 (2001).
FSU-Jena, Abbe School of Photonics’2011
1. Beams with ring-shaped phase dislocations
Intensity Phase CGH
Kivshar and Yang, Phys. Rev. E 50, R40-R43 (1994).
A. Dreischuh et al., Appl. Phys. B62, 139-142 (1996).
D. Neshev et al., Appl. Phys. B64, 429-433 (1997).
FSU-Jena, Abbe School of Photonics’2011
1. Optical vortices
l - integer called Winding number of the loop;Dislocation strength;Topological charge (TC).
∫∫ ∇==C
dRC
dl ϕπ
ϕπ 2
121
FSU-Jena, Abbe School of Photonics’2011
1. Optical vortices
Note:
In free space the Poynting vector gives the momentum flow.
Fof helical phase fronts, the Poynting vector has an azimuthal component.
That component produces an orbital angular momentum parallel to the beam axis.
Because the momentum circulates about the beam axis, such beams are said to contain an optical vortex.
Unlike spin angular momentum, orbital angular momentum is independent of the beam’s polarization.
Transitions forbidden by known selection rules in the electric and magnetic dipole approximation appear allowed (|ΔL| ≤ |l|+ 1 ≤ Li + Lf , ΔM = ±(l + s), ΔL + l is odd).
FSU-Jena, Abbe School of Photonics’2011
2. Methods for generation of singular beams
Phys. Rev. Lett. 66, 1583 (1991).
FSU-Jena, Abbe School of Photonics’2011
-4 -2 0 2 40,0
0,2
0,4
0,6
0,8
1,0
Black and grey solitons
|U|2
t/t0
0,0
0,5
1,0
1,5
2,0
Phas
e, r
ad
-4 -2 0 2 40,0
0,2
0,4
0,6
0,8
1,0
|U|2
t/t0
0,0
0,5
1,0
1,5
2,0
Phas
e, r
ad
ziaeathatzU2
)(sec),( =
zUietUUtzU202
00 )tanh(),( =
bright soliton
black soliton
grey soliton
-4 -2 0 2 40,2
0,4
0,6
0,8
1,0
|U|2
t/t0
0,0
0,5
1,0
1,5
2,0
Phas
e, r
ad
FSU-Jena, Abbe School of Photonics’2011
2. Methods for generation of singular beams
Optical vortex solitons do exist!!!
OVS all-optical guiding properties confirmed.
Experiment
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2. Methods for generation of singular beams
Numerical
simulation
FSU-Jena, Abbe School of Photonics’2011
2. Generation of OVs by mode conversion
The solution of the paraxial wave equation depends on
Is the Gouy phase.
where
,
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2. Generation of OVs by mode conversion
Phys. Rev. A 45, 8185 (1992).
Opt. Commun. 96, 123 (1993).
Opt. Commun. 143, 265 (1997).
Opt. Commun. 159, 13 (1999).
Opt. Commun. 165, 11 (1999).converter - 2/π
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2. Generation of OVs by mode conversion
FSU-Jena, Abbe School of Photonics’2011
2. Generation of OVs by mode conversion
FSU-Jena, Abbe School of Photonics’2011
2. Methods for generation of singular beams(OV out of a resonator)
Opt. Commun. 169, 115 (1999); Opt. Commun. 182, 205 (2000).
Near filed Far filed
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2. Generation of singular beams by micro-fabricated wedge
Appl. Phys. B86, 209 (2007).
FSU-Jena, Abbe School of Photonics’2011
2. Methods for generation of singular beams(Computer-generated holograms)
CGH etched directly on a laser mirror
Appl. Optics 46, 8583 (2007).
FSU-Jena, Abbe School of Photonics’2011
Femtosecond singular beams after single CGH (grating)
1-D
quasi-2-D
-100 -50 0 50 1000
50
100
150
200Cw FsI
y
cw
cw fs
fs
Opt. Lett. 29, 1942-1944 (2004).
FSU-Jena, Abbe School of Photonics’2011
2. Methods for generation of femtosecond singular beams
The challenge here is to impose the phase dislocation in each spectral componentof the short pulse and to keep the pulse width as short and undistorted as possible.
f fff
L LSLM
G G
( )))y,x(iexp(x
diexp
yxexp
f)y,x(E ϕ
ββπ
βσλπ ⎭⎬⎫
⎩⎨⎧
⎟⎟⎠
⎞⎜⎜⎝
⎛+
⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧ +−=′ 1121
20
22
222
Opt. Lett. 29, 1942-1944 (2004).
FSU-Jena, Abbe School of Photonics’2011
2. Methods for generation of femtosecond singular beams
In the Fourier plane :
In front of the CGH :
At the exit :
Fresnel diffraction integral :
⎪⎪
⎭
⎪⎪
⎬
⎫
⎪⎪
⎩
⎪⎪
⎨
⎧
⎟⎟⎠
⎞⎜⎜⎝
⎛
+⎟⎠⎞
⎜⎝⎛ −
−= 2
0
22
0
2
exp),(
πσλ
λ
λσ
f
ydfx
Ef
yxEff
ff
( ) ⎭⎬⎫
⎩⎨⎧
⎭⎬⎫
⎩⎨⎧ +−= x
di
yxf
EyxEβπ
βσπλ2expexp),( 2
0
22
220
( ) ( ))),(exp(112expexp),( 2
0
22
20 yxix
di
yxf
EyxE ϕββ
πβσπλ ⎭
⎬⎫
⎩⎨⎧
⎟⎟⎠
⎞⎜⎜⎝
⎛+
⎭⎬⎫
⎩⎨⎧ +−=′
( ) ( ) ( )[ ]∫∫ ⎭⎬⎫
⎩⎨⎧ −+−= 00
20
20exp/2exp),,( dydxyyxx
siE
sisisyxE ni λ
πλ
λπ
Opt. Lett. 29, 1942-1944 (2004).
FSU-Jena, Abbe School of Photonics’2011
Opt. Lett. 29, 1942-1944 (2004).
FSU-Jena, Abbe School of Photonics’2011
2. Methods for generation of femtosecond singular beams
Opt. Lett. 32, 2025-2027 (2007).
Folded 4-f setup with purely reflective optics
FSU-Jena, Abbe School of Photonics’2011
2. Methods for generation of femtosecond singular beams
Optics Express 13, 7599-7608 (2005).
2f-2f setup
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2. Methods for generation of femtosecond singular beams
[ ] ( ))y,x(iexpd/lsiexp)y,x(E)y,x(E diff ϕλλπ
π⎟⎠⎞
⎜⎝⎛ −−=′ 22
4
21
J. Opt. Soc. Am. B 23, 26-35 (2006).
IEEE J. Quant. Electron. 5, 454 (1969).
FSU-Jena, Abbe School of Photonics’2011
2. Methods for generation of femtosecond singular beams
⎟⎠⎞
⎜⎝⎛= 111
2exp1),( xd
iyxT ππ
⎟⎠⎞
⎜⎝⎛−= 222
2exp1),( xd
iyxT ππ
⎟⎠⎞
⎜⎝⎛−= 333
2exp1),( xd
iyxT ππ
{ }),(exp2exp1),( yxixd
iyxT ϕππ
⎟⎠⎞
⎜⎝⎛=
Transmission functions :
Field distribution in front of the CGH :
Output field distribution :
[ ] ( )),(exp/2exp),(1),( 224 yxidlsiyxEyxE diff ϕλ
λπ
π⎟⎠⎞
⎜⎝⎛ −−=′
( ) ( )[ ] 002
02
00023 exp),(
2exp2exp2exp1),( dydxyyxx
siyxE
si
six
di
dliyxE
⎭⎬⎫
⎩⎨⎧ −+−
⎟⎠⎞
⎜⎝⎛
⎟⎠⎞
⎜⎝⎛−⎟
⎠⎞
⎜⎝⎛−= ∫∫ λ
πλλπ
πλππ
J. Opt. Soc. Am. B 23, 26-35 (2006).
FSU-Jena, Abbe School of Photonics’2011
2. Methods for generation of femtosecond singular beams
Opt. Lett. 33, 2970-2972 (2008).
Tilted prism compressor
TC=3
FSU-Jena, Abbe School of Photonics’2011
2. Methods for generation of femtosecond singular beams
Opt. Lett. 31, 2042-2044 (2006).
Achromatic vortex “lens”
SPP for 9 keV X-rays: Opt. Lett. 27, 1752 (2002).
FSU-Jena, Abbe School of Photonics’2011
2. Method for generation of a femtosecond optical vortex
APS=axially symmetric polarizer
Optics Express 17, 14517-14525 (2009).
FSU-Jena, Abbe School of Photonics’2011
2. Quantitative measurement of the phase profiles of singular beams
-60 -40 -20 0 20 40 60-60
-40
-20
0
20
40
60
-60 -40 -20 0 20 40 60-60
-40
-20
0
20
40
60
-60 -40 -20 0 20 40 60-60
-40
-20
0
20
40
60
-60 -40 -20 0 20 40 60-60
-40
-20
0
20
40
60
Phase = 0 Phase = π/2 Phase = π Phase = 3π/2
-60 -40 -20 0 20 40 60-60
-40
-20
0
20
40
60
-60 -40 -20 0 20 40 60-60
-40
-20
0
20
40
60
-60 -40 -20 0 20 40 60-60
-40
-20
0
20
40
60
Step 1:
Step 2:
⎥⎦
⎤⎢⎣
⎡−−
=Φ),(),(),(),(atan),(
31
24
yxIyxIyxIyxIyx Phase ramp Quantitatively
reconstructed phase
FSU-Jena, Abbe School of Photonics’2011
2. Quantitative measurement of the phase profiles of singular beams
Appl. Phys. B62, 139-142 (1996).
100 200 300 400 500
-2
-1
0
1
2
Phas
e, ra
d
X, pixels
FSU-Jena, Abbe School of Photonics’2011
2. Quantitative measurement of the phase profiles of singular beams
Appl. Phys. B62, 139-142 (1996).
FSU-Jena, Abbe School of Photonics’2011
3. All-optical interaction – physical idea.
Opt. Letters . 16, 438 (1991).
FSU-Jena, Abbe School of Photonics’2011
3. All-optical waveguiding by a bright spatial soliton.
Opt. Letters . 16, 793 (1991).
FSU-Jena, Abbe School of Photonics’2011
3. All-optical waveguiding by dark beams.
Opt. Letters . 17, 496 (1992).
pump probe pump probe
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4. 1-D dark temporal solitons – experiment vs. theory.
First result:
Dark solitonpropagation in
optical fibers
Phys. Rev. Lett. 61, 2445 (1988).Prog. Quant. Electron . 19, 161 (1995).
FSU-Jena, Abbe School of Photonics’2011
4. 1-D dark temporal solitons – experiment vs. theory.
First result:
Dark solitonpropagation in
optical fibers
Phys. Rev. Lett. 61, 2445 (1988).Prog. Quant. Electron . 19, 161 (1995).
FSU-Jena, Abbe School of Photonics’2011
4. 1-D dark (spatial) solitons – experiment vs. theory.
First result:
Dark spatial soliton in ZnSe
Opt. Letters . 16, 156 (1991).
FSU-Jena, Abbe School of Photonics’2011
4. 1-D dark (spatial) solitons – experiment vs. theory.
First result:
Dark spatial soliton in ZnSe
Opt. Letters 16, 156 (1991).
beam width
peak irradiance
Soliton
constant
FSU-Jena, Abbe School of Photonics’2011
4. 1-D dark (spatial) solitons – experiment vs. theory.
( )
( ) 2/10
22
02
2
2/||||
2/||||
cos
−=
=
≅
nEn
nEnxA
kA
NL
A
NLNL
θλ
λλ
Phys. Rev. Lett. 66, 1583 (1991). Opt. Letters 15, 783 (1990).
FSU-Jena, Abbe School of Photonics’2011
5. Quasi-2-D dark solitons and all-optical branching schemes.
Appl. Phys. B 69, 107 (1999).
FSU-Jena, Abbe School of Photonics’2011
5. Quasi-2-D dark solitons and all-optical branching schemes.
J. Opt. Soc. Am. B 14, 2869 (1997).
FSU-Jena, Abbe School of Photonics’2011
5. Quasi-2-D dark solitons and all-optical branching schemes.
J. Opt. Soc. Am. B 14, 2869 (1997).
X
Y
X
Y
X
Y
X
Y
X
Y
FSU-Jena, Abbe School of Photonics’2011
6. Optical vortex solitons – existence, soliton constant, stability.
Opt. Commun. 140, 77 (1997).
CmrI mOVBG ||2)( =
FSU-Jena, Abbe School of Photonics’2011
6. Optical vortex solitons – existence, soliton constant, stability.
Phys. Rev. E 60, 6111 (1999).
? ||2)( CmrI mOVBG =
m=1 m=2
m=3 m=4
FSU-Jena, Abbe School of Photonics’2011
6. Optical vortex solitons – existence, soliton constant, stability.
[ ]( ) [ ] Cmzzr
LzzzI
mOVDiff
BG ||2exp)0(/1
exp)0( 2)(2 ==
+
−=β
α
m 2 β LDiff m´
1 0.36 2.5 1(±0.04)
2 0.35 3.2 2(±0.08)
3 0.35 3.2 3(±0.05)
4 0.33 3.1 4(±0.01)
Phys. Rev. E 60, 6111 (1999).
FSU-Jena, Abbe School of Photonics’2011
6. Optical vortex solitons – existence, soliton constant, stability.
Phys. Rev. E 60, 7518 (1999).
FSU-Jena, Abbe School of Photonics’2011
7. Ring dark solitary waves – how to rule their transverse dynamics.
Appl. Phys. B 63, 145 (1996).
FSU-Jena, Abbe School of Photonics’2011
7. Ring dark solitary waves – how to rule their transverse dynamics.
Phys. Rev. E 52, 5517 (1995).
FSU-Jena, Abbe School of Photonics’2011
7. Ring dark solitary waves – how to rule their transverse dynamics.
Physica Scripta 55, 68 (1997).
-R0 R00
I(r) a)
b)
c)
d)
f+
f+
f-
f-
FSU-Jena, Abbe School of Photonics’2011
7. Ring dark solitary waves – how to rule their transverse dynamics.
Ch.1Ch.2 Ch.3 1+
1 1 2
Transverse coordinate
Proc. SPIE 4397, 191 (2001).
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8. Dark beams with mixed phase dislocations – general characteristics.
J. Opt. Soc. Am. B 17, 2011 (2000).
Step-screw (ES) dislocation
FSU-Jena, Abbe School of Photonics’2011
8. Dark beams with mixed phase dislocations – general characteristics.
Step-screw (ES) dislocation
J. Opt. Soc. Am. B 17, 2011 (2000).
FSU-Jena, Abbe School of Photonics’2011
8. Dark beams with mixed phase dislocations – potential application.
Proc. SPIE 7747, 77471P (2011).
FSU-Jena, Abbe School of Photonics’2011
8. Dark beams with mixed phase dislocations – potential application.
Phys. Rev. A 80, 053828 (2009).
FSU-Jena, Abbe School of Photonics’2011
8. Dark beams with mixed phase dislocations – potential application.
Phys. Rev. A 80, 053828 (2009).
FSU-Jena, Abbe School of Photonics’2011
8. Dark beams with mixed phase dislocations – potential application.
Phys. Rev. A 80, 053828 (2009).
There are so many open questions …
Thank you for your attention!