Driven autoresonantthree-oscillator interactions

Oded Yaakobi1,2

Lazar Friedland2

Zohar Henis1

1 Soreq Research Center, Yavne, Israel.2 The Hebrew University of Jerusalem, Jerusalem, Israel.

Email: yaakobio@yahoo.com

O. Yaakobi, L. Friedland and Z. Henis, Phys. Rev. E (accepted).

Three waves interactions

• Plasma physics– Laser plasma interactions:

» Stimulated Brillouin Scattering (SBS)» Stimulated Raman Scattering (SRS)

• Nonlinear optics– Optical Parametric Amplifier/Generator (OPA/OPG)– Brillouin scattering, Raman scattering

• Hydrodynamics• Acoustic waves

321 Frequencies matching (energy):

Wave vectors matching (momentum): 321 kkk

Controlling three waves interactions is an important goal of both basic and applied physics research.

Three oscillators interactions

213233

312222

321211

xxxx

xxxx

xxxx

321

Three oscillators interactions

213233

312222

321211

xxxx

xxxx

xxxx

321

Research goal:

Study a control scheme of three oscillators interactions using an external drive.

d

kjjj

xxxx

kjkjxxxx

cos

,2,1,

213233

32

dttdd

321

td 3

])(Re[ jijj etAx

3 d t

Definitions:

Threshold phenomena

031.0

10

3 ,2 ,15

321

0155.0

0174.0

3 d

0 20 40 60 80

-30

-20

-10

0

10

20

30

40

x2

A2

Adiabatic approximation

])(Re[ jijj etAx

3

213

d

jj

Definitions: Approximated equations

0197.0

031.0

neglecting :jA

sin2

sin42

1

sin42

1

213333

3

AAAA

AAAA kjjjj

coscos2

cos2

33

2123

23

322

AA

AA

A

AA

j

kjj

213

33 t

Nonlinear frequency shift

Small nonlinear frequency shift

coscos

coscos

sinsin

sin

32

31

1

32

3

21

33

21

213

3

BB

BB

B

BB

B

BB

d

d

BB

BB

d

d

BBd

dB

BBd

dBk

j

jj

jjj AB

32

3214 t

Assumption:

Approximated equations:

Definitions:

8.1

1

0 20 40 60 80 100

10

20

30

40

50

60

70

80

B2

adiabatic

weak excitationsSmall nonlinear frequency shift

Range of validity

Autoresonant quasi steady state

)2 (mod

)2 (mod 0

consts

s

Assumptions:

Quasi steady state:

2

2

3

21

s

ss

B

BB

8.1

1

0 20 40 60 80 100

10

20

30

40

50

60

70

80

B2

adiabatic

weak excitations& asymptotic

Small nonlinear frequency shift & quasi steady state

213

3 d

-40 -20 0 20 40

-4

-2

0

2

4

Time,

1

Phas

e m

ism

atch

/

, /

/

/

4

3

2

Threshold analysis

213

21

sinBBBddB

ddB

Quasi-steady-state asymptotic result:

sin

1sin 21

3 BBd

dB

2

3sin

2sin

2

22

3

21

ss

s

ss

B

BB

1sin s

2

3

Constraint:

2

3

2

32

3

s0 s

Asymptotic phase mismatches:

Threshold analysis

d

kjjj

xxxx

kjkjxxxx

cos

,2,1,

213233

32

dttdd

321

td 3

12213

Dimensional equations:

Necessary condition:

2

3

Threshold analysis

5.1th

Necessary condition for autoresonant quasi steady state:

-2 -1 0 1 20

2

4

6

8

log10

()

th

th

= 1.5

Computed threshold (numerical)

-2 -1 0 1 20

2

4

6

8

10

log10

()

th

th

= 1.5

= 20 = -20 = 10 = -10 = 0

Linear frequencies mismatch

coscos

32

31

1

32

3

21

BB

BB

B

BB

B

BB

d

d

321

Dissipation3213

233 cos xxxxx d

3213 sinsin BBB

d

dB

2

22

3

32

d

Small nonlinear frequency shift:

Necessary condition:

0 20 40 60 80 1000

10

20

30

40

50

60

70

80

B2

asymptotic = 0

numerical = 0.3

analytic = 0.3

10

1

deBB djsj

3

Exponential decay: d

-40 -20 0 20 40

-4

-2

0

2

4

Time,

1

Phas

e m

ism

atch

/

, /

/

/

4

3

2

0 50 100 150 2000

1

2

Spec

tral

den

sity

Frequency

Weak excitations - Linearized

0 50 100 150 2000

1

2

Spec

tral

den

sity

Frequency

Weak excitations

12

2

Phase mismatches deviations

60 70 80 90 100-0.1

0

0.1

/

60 70 80 90 100-0.1

0

0.1

/

;

LinearizationQuasi-steady-state:

2

3

21

2sin

2

3sin

2

2

s

s

s

ss

B

BB

ssjjsj BBB Expansion:

1

0

s

2

3

s

Assumptions:

coscos

coscos

sinsin

sin

32

31

1

32

3

21

33

21

213

3

BB

BB

B

BB

B

BB

d

d

BB

BB

d

d

BBd

dB

BBd

dBk

j

Exact equations:

Linearization

ss

sksj

BBd

Bd

BBd

Bd

213

3

Linearized equations:

22

43

22

2

22

2

d

d

d

d

321

3

12213

222 BBBd

d

B

BBBBB

d

d

s

ss

Differentiating with respect to :

0 50 100 150 2000

1

2

Spec

tral

den

sity

Frequency

Weak excitations - Linearized

0 50 100 150 2000

1

2

Spec

tral

den

sity

Frequency

Weak excitations

Numerical results comparison

WKB approximation

ieYX Re

22

43

22

2

22

2

d

d

d

d

22

43

2

2

M

X

02

2

MXXdd

dd

~

0~

~2

DYY

Y

d

d

d

di

22

22

2

~22

4~

3~

IMD

Od

d

d

d

~,

Y...10 YYY

envelope yingslowly var a is Y

00 DY nn O Y

WKB approximation

22

22

2,132

811

2

32~

:

frequencydependant linear 3~

frequencyconstant 3

2~

2

1

0~

~2

10

0 DYYY

d

d

d

di

First order terms satisfy:

Singular value decomposition

jk

dd

kHjk

Hj

HH11

vvuu

uvuvD 2221 D of aluessingular v real are jd

0 ,0 ,0 ,0 :solution trivial-non 2121 yydd

0 0 11 H1

H111 uDvvDu d,d

221 uuY 10 yy

00DY

0

2211 ydyd

conjugatecomplex transposeH

Singular value decomposition

0~

~2

10

0 DYYY

d

d

d

di

First order terms satisfy:

Multiplying with : H1v

0~

~2

~2 11

1

y

d

d

d

dy

d

dy1

H1

1H11

H1 uv

uvuv

10 uY 1y 0DvH1

Asymptotic form of matrices

:

02

03~

2

2

1

D

222

32

00~

D

frequencydependant linear 3~

frequencyconstant 3

2~

2

1

22

22

2

~22

4~

3~

IMD

0 ,0 DvDu H11

Quasi steady state stability

didi

eyey 21~

21

~

11 2

3

13

1~1

0~Re

1

~~

constant, ~

2111 yy

3~

; 3

2~21

Small deviations from the quasi-steady-state do not increase with time.

0 50 100 150 2000

1

2

Spec

tral

den

sity

Frequency

Weak excitations - Linearized

1

~1002

~

602

~

Quasi steady state stability

die

z 2~

2 Re~

3~

; 3

2~21

Small deviations from the quasi steady state do not increase with time.

diez 1

~

1 Re3

2

conditions initialby determined are and constants The 21 zz

0 50 100 150 2000

1

2

Spec

tral

den

sity

Frequency

Weak excitations - Linearized

0 50 100 150 2000

1

2

Spec

tral

den

sity

Frequency

Weak excitations

1

~1002

~

602

~

0 20 40 60 80 100

10

20

30

40

50

60

70

80

B2

adiabatic

weak excitations& asymptotic

Small nonlinear frequency shift & quasi steady state

e

sin2

sin42

1

sin42

1

213333

3

AAAA

AAAA kjjjj

coscos2

cos2

33

2123

23

322

AA

AA

A

AA

j

kjj

213

33 t

Large nonlinear frequency shift

)2 (mod

)2 (mod 0

consts

s

Assumptions:

233

21221

2

2

aB

aBB

s

ss

2321

23

321

38

83

2

e

0, 312 aa

ConclusionsControlling three oscillators interactions using

autoresonance is demonstrated.Analytic expressions for autoresonant time

dependent amplitudes are obtained.Conditions for autoresonant trapping are

analyzed in terms of coupling parameter, driving parameter, dissipation and linear frequencies mismatch.

The autoresonant quasi-steady state is linearly stable.

Outlook

Generalization of the theory to driven three-wave interactions is of interest.