Non- paraxiality and

femtosecond optics

Lubomir M. Kovachev

Institute of Electronics, Bulgarian Academy of Sciences Laboratory of Nonlinear and Fiber Optics

Nonlinear physics. Theory and Experiment. V 2008

......2

1

0

Akz

Ai

Paraxial optics of a laser beam

Solution in (x, y, z) space

yxyxyx

yx

yx dkdkykxikkkkikkAzyxA )exp()exp()2/)(exp()0,,(),,( 022

,

Analytical solution for initial Gaussian beam

Initial conditions - Gaussian beam

200

20

220 );2/)(exp()0,,( rkzryxAyxA diff

diffdiff zizr

yx

zizAzyxA

/12

)(exp

/1

1

4),,(

20

22

0

2

220

22

220

42

/12

)(exp

/1

1

16 diffdiff zzr

yx

zzAA

z=0z=zdiff

Numerical solution using FFT technique. Paraxial optics. Laser beam on 800 nm (zdiff=k0r0

2= 7.85 cm; r0= 100µm)

Initial condition

z=0 z=1/3 z=2/3 z=1;zdiff=7.85 cm

)2/)(exp()0,,(),,( 0221 kkkikkAFzyxA yxyx

);2/)(exp()0,,( 220 yxAyxA

z=0 z=1/3 z=2/3 z=1=z diff

Phase modulated (by lens) Gaussian beam

)),(2exp(),()0,,( 00 yxidiyxAyxA )()(/(),( 222 yxafSyx eff

a-radius of the lens, f- focus distanced0- thickness in the centrum

Seff- effective area of the laser spot

nm800

f=200 cm

a=1,27 cmSeff=0.2

Paraxial optics of a laser pulse. From ns to 200-300 ps time duration

Dimensionless analyze:

;'xrx 'yry '0zzz

......'' 2

2

t

AA

z

Ai

dissp

diff

z

z

kt

rk

"/20

20

02.0

1085.7,/103";1~;330~ 140

2310

cmkcmsekkmmrfst

In air, gases and metal vapors t0>100-200 fs ; β<<1 - Negligible dispersion.

....2

"

2

12

2

0

termsnlt

AkA

kz

Ai

1/ 02

0 zrk2

00 rkz

Nonlinear paraxial optics

Nonlinear paraxial equation:

;xAA x

Initial conditions:

1) nonlinear regime near to critical γ~ 1.2

2) nonlinear regime γ=1.7

AAAz

Ai

22

)2/2/exp()0,,( 22 yxzyxAx

• 1) nonlinear regime near to critical γ~ 1.2

2) Nonlinear regime γ=1.7

References

1. A. Braun, G. Korn, X. Liu, D. Du, J. Squier, and G. Mourou, "Self-channeling of high-peak-power femtosecond laser pulses in Air, Opt. Lett. 20, 73-75, 1995.2. E. T. Nibbering, P. F. Curley, G. Grillon, B. S. Prade, M. A. Franco, F. Salin, and A. Mysyrowich, "Conical emission from self-guided femtosecond pulses", Opt. Lett, 21, 62, 1996.3. A. Brodeur, C. Y. Chien, F. A. Ilkov, S. L. Chin, O. G. Kosareva, and V. P. Kandidov, "Moving focus in the propagation of ultrashort laser pulses in air", Opt. Lett., 22, 304-306, 1997.4. L. Wöste, C. Wedekind, H. Wille, P. Rairroux, B. Stein, S. Nikolov, C. Werner, S. Niedermeier, F. Ronnenberger, H. Schillinger, and R. Sauerbry, "Femtosecond Atmospheric Lamp", Laser und Optoelektronik 29, 51 , 1997.5. H. R. Lange, G. Grillon, J.F. Ripoche, M. A. Franco, B. Lamouroux, B. S. Prade, A. Mysyrowicz, E. T. Nibbering, and A. Chiron, "Anomalous long-range propagation of femtosecond laser pulses through air: moving focus or pulse self-guiding?", Opt. lett. 23, 120-122, 1998.

Non-collapsed regime of propagation of fsec pulses

Nonlinear pulse propagation of fsec optical pulsesThree basic new experimental effects

1. Spectral, time and spatial modulation

2. Arrest of the collapse

3. Self-channeling

Extension of the paraxial model for ultra short pulses and single-cycle pulses ?

...

...2

"

2

12

2

0

ionization

termsnlt

AkA

kz

Ai

Expectations:Self-focusing to be compensated by plasma induced defocusing or high order nonlinear terms - Periodical fluctuation of the profile.

Experiment: 1) No fluctuations - Stable profile2) Self- guiding without

ionization

Arrest of the collapse and self-channeling

in absence of ionization

G. Méchian, C. D'Amico, Y. -B. André, S. Tzortzakis, M. Franco, B. Prade, A. Mysyrowicz, A. Couarion, E. Salmon, R. Sauerbrey, "Range of plasma filaments created in air by a multi-terawatt femtosecond laser", Opt. Comm. 247, 171, 2005.

G. Méchian, A. Couarion, Y. -B. André, C. D'Amico, M. Franco, B. Prade, S. Tzortzakis, A. Mysyrowicz, A. Couarion, R. Sauerbrey, "Long range self-channeling of infrared laser pulse in air: a new propagation regime without ionization", Appl. Phys. B 79, 379, 2004.

C. Ruiz, J. San Roman, C. Mendez, V.Diaz, L.Plaja, I.Arias, and L.Roso, ”Observation of Spontaneous Self-Channeling of Light in Air below the Collapse Threshold”, Phys. Rev. Lett. 95, 053905, 2005.

Self-Channeling of Light in Linear Regime ??(Femtosecond pulses)

Saving the Spatio -Temporal Paraxial Model – linear and nonlinear X waves??1) X-waves - J0 Bessel functions – infinite energy2) X-waves - Delta functions in (kx, ky) space.

Experiment: 1. Self-Channeling is observed for spectrally - limited (regular) pulses 2. “Wave type” diffraction for single- cycle pulses.

)2/)(exp()0,,(),,( 0221 kkkikkAFzyxA yxyx

Something happens in FS region??

Wanted for new model to explain:

3. Spectral, time and spatial modulation

4. Arrest of the collapse

5. Self-channeling

Three basic new nonlinear experimentally confirmed effects:

1. Relative Self -Guiding in Linear Regime.

2. “Wave type” diffraction for single - cycle pulses. Optical cycle ~2 fs ; pulses with 4-8 fs duration

1. L. M. Kovachev, "Optical Vortices in dispersive nonlinear Kerr-type media", Int. J. of Math. and Math. Sc. (IJMMS) 18, 949 (2004).

2. L. M. Kovachev and L. M. Ivanov, "Vortex solitons in dispersive nonlinear Kerr type media", Nonlinear Optics Applications, Editors: M. A. Karpiez, A. D. Boardman, G. I. Stegeman, Proc. of SPIE. 5949, 594907, 2005.

3. L. M. Kovachev, L. I. Pavlov, L. M. Ivanov and D. Y. Dakova, “Optical filaments and optical bullets in dispersive nonlinear media”, Journal of Russian Laser Research 27, 185- 203, 2006

4. L.M.Kovachev, “Collapse arrest and wave-guiding of femtosecond pulses”, Optics Express, Vol. 15, Issue 16, pp. 10318-10323 (August 2007).

5. L. M. Kovachev, “Beyond spatio - temporal model in the femtosecond optics”, Journal of Mod. Optics (2008), in press.

Non-paraxial model

),( trE

),(ˆ rE

Introducing the amplitude function of the electrical field

and the amplitude function of the Fourier presentation of the electrical field

tzkitrAtrE 00exp),(),(

tirAtzkitirE 0000 exp),(ˆexp)exp(),(ˆ

The next nonlinear equation of the amplitudes is obtained:

dtirAAkktrAkz

trAiktrA nl ))(exp(),(ˆˆ)()(),(

),(2),( 00

2222

00

Convergence of the series: I. Number of cycles; II. Media density:

..)()(2

1)()()()( 2

0"2

0'2

022 kkkk

"''2)( "2 kkkk vkkkk /1';......'2)'( 2

SVEA in laboratory coordinate frame

AAvnk

t

A

vkk

vA

k

v

t

AAnvkn

z

Av

t

Ai

220

2

2

20

"0

0

2

202 2

1

222

or

V. Karpman, M.Jain and N. Tzoar, D. Christodoulides and R.Joseph,N. Akhmediev and A. Ankewich, Boyd……

AAvnk

t

Avk

t

A

vz

AA

k

v

t

AAnvkn

z

Av

t

Ai

220

2

2"0

2

2

22

2

0

2

202 22

1

22

SVEA in Galilean coordinate frames

AAvnk

zt

Av

t

A

vkk

v

z

AkvA

k

v

t

Ai

220

2

2

2

20

"02

23

0 2''2

'

1

2'2

"

2'

AAvnk

t

Avk

t

A

vz

AA

k

v

z

Av

t

Ai

220

2

2"0

2

2

22

2

0 22

1

2

Constants

"'":;"';";";";" 00000 ttttttzzzzzzyryxrxAAA

20 rkzdiffr "

2

0

k

tzdis

;00 zk ;2

0

22

z

r ;dis

dif

z

z

;2

1 2

02

22

0 Anrk

2

0220

2

2

01 2

1

22

1An

nvknA

00 vtz

Dimensionless parameters

0 0k z 1.

Determine number of cycles under envelope with precise 2π 1

22

20

r

z 2. Determine relation between transverse and

longitudinal initial profile of the pulse

dispdiff zz /3.Determine the relation between diffraction and dispersion length

0 0z vt2

1 "2

01 kvk

22

0 2 0

1

2k r n A

��������������4.

5.2

2 21 2 0

1

2n A

��������������

Nonlinear constant

Constant connected with nonlinear addition to group velocity

SVEA in dimensionless coordinates

AAt

A

t

A

z

AA

t

AA

z

A

t

Ai

2

2

2

2

2

2

22

2

1

22

AAz

A

zt

A

t

AA

z

AA

t

AA

t

Ai

2

2

22

2

22

22

1

2

'''2

''''2

Laboratory

Galilean

vtzztt ';' domainpsandns ......300200......;..12 domainfs....300200;...12

domainfs....15020;...12

Linear Amplitude equation in media with dispersion (SVEA)

2

2

12

2

2

2

2222

t

A

t

A

z

AA

z

A

t

Ai

2

22

1

2

2

2

122

'''2

'1

'2

z

A

zt

A

t

AA

t

Ai

Laboratory:

Galilean:

01

2

2

2

t

E

cE

tzkitrAtrE 00exp),(),(

2

2

2

2222

t

A

z

AA

z

A

t

Ai

Linear Amplitude Equation in Vacuum (VLAE)

2

2

200 2

1

2

11

t

A

ckA

kz

Ac

t

A

ci

In air5"2

01 10 kvk042

02 / zrkzz beam

diffpulsediff

Laboratory frame

0ˆ

1ˆ2ˆ

22

2

1222222

t

AAkkkk

t

Ai L

LzzyxL

Galilean frame

0ˆ

1ˆˆ

22

2

122222

t

AAkkk

t

Aki G

GzyxG

z

tk

ikkkAtkkkA zyxLzyxL1

2

22

11 1

ˆ

11exp)0,,,(ˆ),,,(ˆ

exp)0,,,(ˆ),,,(ˆzyxGzyxG kkkAtkkkA

tkkkkk

i zyxzz

12

21

2222

11 111

Solutions in kx ky kz space :

where zzyx kkkkk 2ˆ 22222

)',',,()',,,(ˆ tzyxAFtkkkA zyxG

),,,(),,,(ˆ tzyxAFtkkkA zyxL

Fundamental solutions of the linear SWEA

t

kikkkAFtzyxA zyxLL

12

2

21

2

1

1

1

ˆ

)1(1exp)0,,,(ˆ),,,(

t

kkkkkikkkAFtzyxA zyxzz

zyxGG1

2

21

2222

11

1

111exp)0,,,(ˆ),,,(

zzyx kkkkk 2ˆ 22222

t

kikkkAFtzyxA zyxLL

12

2

21

2

1

1

1

ˆ

)1(1exp)0,,,(ˆ),,,(

Fundamental linear solutions of SVEA for media with dispersion:

t

kkkkkikkkAFtzyxA zyxzz

zyxGG1

2

21

2222

11

1

111exp)0,,,(ˆ),,,(

Fundamental solutions of VLAE for media without dispersion:

1 1 2 2 2ˆ ˆ0, , , exp /L L x y zA F A k k k F i k t

1 1 2 2 2ˆ ˆ0, , , exp /G L x y z zA F A k k k F i k k t

5"201 10 kvk zzyx kkkkk 2ˆ 22222

Evolution of long pulses in air (linear regime, 260 ps and 43 ps)

Light source form Ti:sapphire laser, waist on level e-1 : mr 100

cmkcmk air /sec103;10.85.7 231"140

1) 260 ps: αδ2=1; β1=2.1X10-5

zzttv '';.....1

1'~1' zztt

43 ps (long pulse) αδ2=6; β1=2.1X10-5

Light Bullet (330 fs) α=785; δ2=1; β1=2.1X10-5

Light Disk (33 fs) α=78,5; δ2=100; β1=2.1X10-5

Analytical solution of SVEA (when β1<<1)and VLAE for initial Gaussian LB (δ=1) (Lab coordinate)

.)exp()exp()exp(

exp

2/exp2

1),,,(

222

2223

zyxzyx

zyx

zyx

dkdkdkzikyikxik

tkkki

kkktzyxAx

zz kk̂

.ˆ)(ˆexp()exp()exp(

ˆexp

2/ˆexp

2exp

2

1),,,(

222

222

2

3

zyxzyx

zyx

zyx

kddkdkizkiyikxik

tkkki

kkk

ztitzyxAx

22222 2)(ˆ zirizyxr

.ˆsinexp

)2/exp(ˆ1

2exp

2

1),,,(

0

22

3

rrr

rr

dkkrtki

kkr

ztitzyxAx

Analytical solution of SVEA (when β1<<1)and VLAE for initial Gaussian LB (δ=1)

rti

erfcriitrti

rti

erfcriitrti

ztir

itzyxA

ˆ2

2ˆ

2

1expˆ

ˆ2

2ˆ

2

1expˆ

2exp

ˆ2),,,(

2

2

2

22 2ˆ zirr

Gaussian shape of the solution when t=0.The surface |A(x,y=0,z; t=0) | is plotted.

Deformation of the Gaussian bullet with 330 fs time duration obtained from exact solution of VLAE. The surface |A(x,y=0,z; t=785) | is plotted. The waist grows by factor sqrt(2) over normalized time-distance t=z=785, while the amplitude decreases with A=1/sqrt(2).

Shaping of LB on one zdifpulse=k0

2r4/z0 length

785

785

zt

.)exp()exp()exp(

)(exp

2/exp2

1),,,(

2222

2223

zyxzyx

zyxz

zyx

dkdkdkzikyikxik

tkkkki

kkktzyxAx

Analytical solution of SVEA (when β1<<1)and VLAE for initial Gaussian LB (δ=1) (Galilean coordinate)

zz kk̂

.ˆ)(ˆexp()exp()exp(

ˆexp

2/ˆexp

2exp

2

1),,,(

222

222

2

3

zyxzyx

zyx

zyx

kddkdkitzkiyikxik

tkkki

kkk

ztitzyxAx

222 )(~ itzyxr

Analytical solution of SVEA (when β1<<1)and VLAE for initial Gaussian LB (δ=1) (Galilean coordinate)

rti

erfcriitrti

rti

erfcriitrti

ztir

itzyxA

~2

2~2

1exp~

~2

2~2

1exp~

2exp~2

),,,(

2

2

2

Analytical solution of SVEA (when β1<<1)and VLAE for initial Gaussian LB (δ=1) (Galilean coordinate)

222 )(~ itzyxr

Fig. 5. Shaping of Gaussian pulse obtained from exact solution of VLAEin Galilean coordinates. The surface A(x; y = 0; z=0; t= 785) is plotted.The spot grows by factor sqrt(2) over the same normalized time t = 785 while the pulse remains initial position z = 0, as it can be expected from Galilean invariance.

042

02 / zrkzz beam

diffpulsediff

Linear Amplitude equation in media with dispersion (SVEA).

2

2

12

2

2

2

2222

t

A

t

A

z

AA

z

A

t

Ai

2

22

1

2

2

2

122

'''2

'1

'2

z

A

zt

A

t

AA

t

Ai

Laboratory:

Galilean:

01

2

2

2

t

E

cE

tzkitrAtrE 00exp),(),(

2

2

2

2222

t

A

z

AA

z

A

t

Ai

Linear Amplitude Equation in Vacuum (VLAE). Analytical (Galilean invariant ) solution of 3D+1 Wave equation.

In air5"2

01 10 kvk042

02 / zrkzz beam

diffpulsediff

2. Comparison between the solutions of Wave Equation and SVEA in single-cycle regime

222zyxr kkkk

tkikAF

tkkki

kkkAFtzyxA

rrL

zyx

zyxL

L

exp)0,(ˆ

)(exp

)0,,,(ˆ

),,,(

1

222

1

Evolution of Gaussian amplitudude envelope of the electrical field in dynamics of wave equation. Single – cycle regime

01

2

2

2

t

E

cE

),,,( tzyxAx

))2/(exp(),,(

)2exp(),,()0,,,(2220

00

zyxzyxA

izzyxAtzyxE

x

xx

tkikEF

tkkki

kkkEFtzyxEx

rrx

zyx

zyx

x

exp)0,(ˆ

exp

)0,,,(ˆ),,,(

1

222

1

t=3PiT=0

),,,( tzyxAx

Analytical solution of SVEA (when β1<<1) and VLAE for initial Gaussian LB in single-cycle regime (δ=1 and α=2).

rti

erfcriitrti

rti

erfcriitrti

ztir

itzyxA

ˆ2

2ˆ

2

1expˆ

ˆ2

2ˆ

2

1expˆ

2exp

ˆ2),,,(

2

2

2

Conclusion(linear regime)

1. Fundamental solutions k space of SVEA and VLAE are obtained

2. Analytical non-paraxial solution for initial Gaussian LB.

3. Relative Self Guiding for LB and LD (α>>1) in linear regime.

4. “Wave type” diffraction for single - cycle pulses (α~1-3) .

5. New formula for diffraction length of optical pulses is confirmed from analytical solution zdif

pulse=k02W4/z0

Nonlinear paraxial optics

Nonlinear paraxial equation:

;xAA x

Initial conditions:

1) nonlinear regime near to critical γ~ 1.2

2) nonlinear regime γ=1.7

AAAz

Ai

22

)2/2/exp()0,,( 22 yxzyxAx

1) nonlinear regime near to critical γ~ 1.2

2) Nonlinear regime γ=1.7

Nonlinear non-parxial regime.

AAt

A

z

AA

z

A

t

Ai

2

2

2

2

222

2

AAzt

A

t

AA

t

Ai

22

2

222

''2

''2

Laboratory frames

Galilean

Dynamics of long optical pulses governed by the non - paraxial equationNonlinear regime γ=2

(x,y plane) of long Gaussian pulse. Regime similar to laser beam.

;81

120

22

z

r

Dynamics of long optical pulses governed by the non - paraxial equation

Nonlinear regime γ=2

Longitudinal x, z plane of the same long Gaussian pulse. Large longitudinal spatial and spectral modulation of the pulse is observed.

;81

120

22

z

r

1/ Optical bullet in nonlinear regime γ=1.4. Arrest of the collapse. ;12

2/ OPTICAL DISK in nonlinear regime γ=2.25 NONLINEAR WAVEGUIDING.

1/ Long optical pulse: The self-focusing regime is similar to the regime of laser beam and the collapse distance is equal to that of a cw wave. The new result here is that in this regime it is possible to obtain longitudinal spatial modulation and spectral enlargement of long pulse.

2/ Light bullet: We observe significant enlargement of the collapse distance (collapse arrest) and weak self-focusing near the critical power without pedestal.

3/ Optical pulse with small longitudinal and large transverse size (light disk): nonlinear wave-guiding.

Conclusion - Nonlinear regime

Something happens in FS region??Wanted for new model to explain:

√ 3. Spectral, time and spatial modulation of long pulse

√ 4. Arrest of the collapse of light bullets

√ 5. Self-channeling of light disk

Three basic new nonlinear effects:

√ 1. Relative Self Guiding in Linear Regime of light disk.

√ 2. “Wave type” diffraction for single - cycle pulses.

Експеримент - 800 nm: Ti-Sapphire laser30 fs; 100 μm – леща: Мощност- 1.109 Wпикова мощност на импулса 1X1013 W/cm2 ~2-3 Pkr

H. Hasegawa, L.I. Pavlov, ....

z=0 z=12 zdiff