driven autoresonant three-oscillator interactions oded yaakobi 1,2 lazar friedland 2 zohar henis 1 1...
Post on 21-Dec-2015
215 views
TRANSCRIPT
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: [email protected]
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
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:
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.