EFFICIENT PARAMETRIC AMPLIFICATION IN
DOUBLE SYSTEMS IN THE ABSENCE OF TWO-PHOTON
MAXIMAL COHERENCE
A.D. Wilson-Gordon, H. Shpaisman and H. Friedmann,Department of Chemistry, Bar-Ilan University,
Ramat Gan 52900, Israel
2
Two-level system: pump and probe
1
2
1 2
•If self-focusing and diffraction balanced, Gaussian pump propagates as a spatial soliton
•If pump-induced cross focusing of probe balanced by diffraction, the weak probe propagates as if it is a spatial soliton
•If pump is sufficiently intense, radiation generated at FWM frequency
•Parametric amplification between probe and FWM via the pump
•Process occurs over many diffraction lengths: EIPM important
3
Self-focusingNonlinear refractive index
Thus
Focusing obtained when
Self-focusing obtained when laser is detuned to the blue!
0 2n n n I
2n dn dI
0dn dI
4
Coherent Population Trapping(CPT)
1 2
3
31 32
• Fields are equally intense
• Population trapped in lower levels
• Two-photon coherence is maximal:
• Zero absorption
2 121 11 22 4| |
5
Electromagnetically Induced Transparency( EIT)
• Pump and probe fields• Population optically pumped into state |2>• Two-photon coherence is small
1 2
3
31 32
6
Double System
• Several possible configurations
• Highly efficient FWM when CPT occurs (Harris)
1 2
3
41 3231 42
4
7
Model • Three or four beams with Gaussian transverse intensity
profile (GTIP)• Beams copropagate• Assume steady-state• Compare results with plane-wave beams• Cases studied:• CPT with maximal coherence , either initially or on propagation• Four identical beams with 0 and phase• Three strong fields• Two strong fields • Incoherent pumping from state 2 to 4 (not shown here)• Raman detuning (not shown here)
8
Maxwell-Bloch Equations2' ' '4ij ij ijT
ijD
i iV V Lz L
2 2 2 2 2 2/ (1/ ) / (1/ ) /T
'ijV
DL
ijL
'ij
Transverse radial coordinate Direction of propagationInteraction lengthDiffraction lengthDensity matrix elementRabi frequency for transition
z
j i
9
Bloch Equations11 13 31 14 41 31 13 41 14 12 11 21 22 31 33 41 44
22 23 32 24 42 32 23 42 24 12 11 21 22 32 33 42 44
33 31 13 32 23 13 31 23 32 31 32 33 43 44
44
( ' ' ' ' ) ,( ' ' ' ' ) ,( ' ' ' ' ) ( ) ,
i V V V Vi V V V Vi V V V V
41 14 42 24 14 41 24 42 41 42 43 44 24 44 22
21 23 31 24 41 31 23 41 24 21 21 21
31 31 11 32 21 31 33 41 34 31 31 31
32 32 22 31 12
( ' ' ' ' ) ( ) ( ),' ( ' ' ' ' ) ( ) ' ,' ( ' ' ) ( ) ' ,' ( '
i V V V V ri V aV V aV ii V V V V ii V V
32 33 42 34 32 32 32
41 41 11 42 21 31 43 41 44 41 41 41
42 42 22 41 12 32 43 42 44 42 42 42
43 41 13 42 23 13 41 23 42 43 43
* ' ) ( ) ' ,' ( * ' ' ) ( ) ' ,' ( ' ' ) ( ) ' ,' ( ' * ' ' * ' ) (
V a V ii V a V V V ii V aV aV V ii V a V V a V i
43) '
10
Notation31 32 42 41
*24 4 2
*
exp( ), is initial relative phase,is longitudinal decay rate from state ,
is total decay rate from state ,
0.5( ) is transverse decay rate,
is r
kl
i
kl k l kl k l
kl
a ik l
i
r
24
21 21 3
ate of phase-changing collisions,is rate of incoherent pumping from state 2 4 ,
' is one-photon detuning from resonance; (3,4), (1, 2),
' exp[ ( )],
' exp{ [(
ij ij ij
ij ij ij ij ij
r
i j
i t k z
i
1 32 31 32 31 32
43 43 41 31 41 31 41 31
) ( ) ( )]},' exp{ [( ) ( ) ( )]}.
t k k zi t k k z
11
Multiphoton Resonance Condition
31 32 42 41
31 32 41 42 21
41 31 42 32 43
0 31 32 42 41
0
, two-photon detuningor ,
0, initial phase mismatchk k k k k
12
Analytical Solution
Real part – refraction Imag part – absorption FWM
Effect of phase: when =0, a=1; when =, a=-1
(1) (3)
(1) (3)31 31 31 31 32 24 41
(1) (3)32 32 32 32 31 14 42
(1) (3)41 41 41 41 42 23 31
(1) (3)42 31 42 42 41 13 32
' ' ' ,
' ,
' * ,
' * ,
' .
ij ij ij
V a V V V
V a V V V
V a V V V
V a V V V
13
CPT and Maximal Two-Photon Coherence
• When CPT exists, no absorption or focusing or defocusing occurs since (1)=0
• Phase-matching unimportant• Maximum FWM occurs within a
propagation distance less than diffraction length
• This is completely different from a two-level system
14
CPT and Maximal Two-Photon Coherence
31 32 42 41
31 32 41 423
8; V 8; V 1; V 0.001; 4; 1.66 10 ;NL D
V
L L
15
Transverse Intensity Profile
16
Comparison: GTIP’s and PW’s
17
CPT and Maximal Two-Photon Coherence
31 32 42 41
31 32 41 424
8; V 8; V 8; V 0.001; 4; 100; 1.66 10 ;NL D
V
L L
18
Transverse Intensity Profile
19
Comparison: GTIP’s and PW’s
20
Onset of CPT vs. Maximum Conversion
• For detuning , CPT exists at the outset. Maximum conversion of 87% occurs at 0.047.
• For detuning , CPT occurs at 0.1, whereas maximum conversion of 73% occurs at 0.009.
• Thus, it is possible to get efficient conversion before CPT, without focusing, defocusing or ring formation
41 42 100
41 42 10
21
CPT and Maximal Two-Photon Coherence
31 32 42 41
31 32 41 424
8; V 8; V 8; V 0.001; 4; 10; 1.66 10 ;NL D
V
L L
22
Focusing
• Focusing can occur before CPT established• Beams blue-detuned• Nonlinear length sufficiently long• Maximum FWM can still occur within a
short propagation distance• Phase-matching still unimportant
23
Three strong lasers
31 42 32 41
31 32 41 423
8; V 8; V 8; V 0.001; 4; 1.52 10 ;L D
V
L L
24
Initial gain at FWM frequency
-50
5
00.05
0.10
5
10
z/Ld
|V31
| Am
plitu
de
-50
5
00.05
0.10
5
10
z/Ld
|V32
| Am
plitu
de
-50
5
00.05
0.10
5
10
z/Ld
|V41
| Am
plitu
de
-50
50
0.050.10
5
10
z/Ld
|V42
| Am
plitu
de
gain
25
Focusing
-50
5
00.05
0.10
5
10
z/Ld
|V31
| Am
plitu
de
-50
5
00.05
0.10
5
10
z/Ld
|V32
| Am
plitu
de
-50
5
00.05
0.10
5
10
z/Ld
|V41
| Am
plitu
de
-50
50
0.050.10
5
10
z/Ld
|V42
| Am
plitu
deFocusing
Focusing
26
Maximum focusing and conversion
-50
5
00.05
0.10
5
10
z/Ld
|V31
| Am
plitu
de
-50
5
00.05
0.10
5
10
z/Ld
|V32
| Am
plitu
de
-50
5
00.05
0.10
5
10
z/Ld
|V41
| Am
plitu
de
-50
50
0.050.10
5
10
z/Ld
|V42
| Am
plitu
de
Focusing
FocusingFocusing
Focusing
27
Comparison: GTIP’s and PW’s
0 0.1 0.20
5
10
15TP
W o
n-ax
is a
mpl
itude
Z/Ld
|V31||V32|
0 0.1 0.20
5
10
TPW
on-
axis
am
plitu
de
Z/Ld
|V42||V41|
0 0.1 0.24
6
8
10
PW
am
plitu
de
Z/Ld
|V31||V32|
0 0.1 0.20
2
4
6
8
PW
am
plitu
de
Z/Ld
|V42||V41|
Max
Max
28
Two strong lasers: strong-weak-strong –weak configuration
31 42 32 41
31 32 41 423
4; V 4; V 0.1; V 0.001; 4; 1.66 10 ;L D
V
L L
29
Two strong lasers: strong-weak-strong –weak configuration
31 42 32 41
31 32 41 423
4; V 4; V 0.1; V 0.001; 4; 1.66 10 ;L D
V
L L
30
Two strong lasers: strong-weak-strong –weak configuration
31 42 32 41
31 32 41 423
4; V 4; V 0.1; V 0.001; 4; 1.66 10 ;NL D
V
L L
31
Focusing on propagation
32
Phase dependence
• When field at FWM frequency absent or small at outset, phase is unimportant
• When field at FWM frequency present at outset, dramatic phase effects can be obtained
33
Four Strong Fields: phase 0 vs. 31 42 32 41
31 32 41 423
4; V 4; V 4; V 4; 4; 1.11 10 ; NL D
V
L L
34
Zero PhaseNo Change on Propagation: CPT
35
Four strong fields: phase 31 42 32 41
31 32 41 423
4; V 4; V 4; V 4; 4; 1.11 10 ; = ;NL D
V
L L
36
Phase Focusing on propagation: no CPT
37
Conclusions
• Efficient FWM in double lambda systems can be obtained even before CPT occurs
• It is obtained at short propagation distances• Focusing can be obtained by blue one-photon
detuning• Often accompanied by ring formation
