INTRODUCTION TO SINGULAR NONLINEAR OPTICS … · INTRODUCTION TO SINGULAR NONLINEAR OPTICS ......

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.


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.



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)


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).


1. Singular optical beams out of laser cavities

HG01

LG01


1. Singular optical beams out of laser cavities


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).


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).


1. Optical vortices

l - integer called Winding number of the loop;Dislocation strength;Topological charge (TC).

∫∫ ∇==C

dRC

dl ϕπ

ϕπ 2

121


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).


2. Methods for generation of singular beams

Phys. Rev. Lett. 66, 1583 (1991).


-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



Optical vortex solitons do exist!!!

OVS all-optical guiding properties confirmed.

Experiment



Numerical

simulation


2. Generation of OVs by mode conversion

The solution of the paraxial wave equation depends on

Is the Gouy phase.

where

,



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


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


2. Generation of singular beams by micro-fabricated wedge

Appl. Phys. B86, 209 (2007).


2. Methods for generation of singular beams(Computer-generated holograms)

CGH etched directly on a laser mirror

Appl. Optics 46, 8583 (2007).


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).


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).



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).


Opt. Lett. 29, 1942-1944 (2004).



Opt. Lett. 32, 2025-2027 (2007).

Folded 4-f setup with purely reflective optics



Optics Express 13, 7599-7608 (2005).

2f-2f setup



[ ] ( ))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).



⎟⎠⎞

⎜⎝⎛= 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).



Opt. Lett. 33, 2970-2972 (2008).

Tilted prism compressor

TC=3



Opt. Lett. 31, 2042-2044 (2006).

Achromatic vortex “lens”

SPP for 9 keV X-rays: Opt. Lett. 27, 1752 (2002).


2. Method for generation of a femtosecond optical vortex

APS=axially symmetric polarizer

Optics Express 17, 14517-14525 (2009).


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



Appl. Phys. B62, 139-142 (1996).

100 200 300 400 500

-2

-1

0

1

2

Phas

e, ra

d

X, pixels



Appl. Phys. B62, 139-142 (1996).


3. All-optical interaction – physical idea.

Opt. Letters . 16, 438 (1991).


3. All-optical waveguiding by a bright spatial soliton.

Opt. Letters . 16, 793 (1991).


3. All-optical waveguiding by dark beams.

Opt. Letters . 17, 496 (1992).

pump probe pump probe


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).


4. 1-D dark (spatial) solitons – experiment vs. theory.

First result:

Dark spatial soliton in ZnSe

Opt. Letters . 16, 156 (1991).



First result:

Dark spatial soliton in ZnSe

Opt. Letters 16, 156 (1991).

beam width

peak irradiance

Soliton

constant



( )

( ) 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).


5. Quasi-2-D dark solitons and all-optical branching schemes.

Appl. Phys. B 69, 107 (1999).



J. Opt. Soc. Am. B 14, 2869 (1997).



J. Opt. Soc. Am. B 14, 2869 (1997).

X

Y

X

Y

X

Y

X

Y

X

Y


6. Optical vortex solitons – existence, soliton constant, stability.

Opt. Commun. 140, 77 (1997).

CmrI mOVBG ||2)( =



Phys. Rev. E 60, 6111 (1999).

? ||2)( CmrI mOVBG =

m=1 m=2

m=3 m=4



[ ]( ) [ ] 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).



Phys. Rev. E 60, 7518 (1999).


7. Ring dark solitary waves – how to rule their transverse dynamics.

Appl. Phys. B 63, 145 (1996).



Phys. Rev. E 52, 5517 (1995).



Physica Scripta 55, 68 (1997).

-R0 R00

I(r) a)

b)

c)

d)

f+

f+

f-

f-



Ch.1Ch.2 Ch.3 1+

1 1 2

Transverse coordinate

Proc. SPIE 4397, 191 (2001).


8. Dark beams with mixed phase dislocations – general characteristics.

J. Opt. Soc. Am. B 17, 2011 (2000).

Step-screw (ES) dislocation


8. Dark beams with mixed phase dislocations – general characteristics.

Step-screw (ES) dislocation

J. Opt. Soc. Am. B 17, 2011 (2000).


8. Dark beams with mixed phase dislocations – potential application.

Proc. SPIE 7747, 77471P (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!

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