tr−êng §¹i häc Vinh T¹p chÝ khoa häc, tËp XXXVII, sè 2A-2008 RAMAN STIMULATED SCATTERING IN THREE-DIMENSIONAL APPROACH Chu Van Lanh a, Dinh Xuan Khoaa, Ho Quang Quy b, Pham Thi Thuy
Trang 1tr−êng §¹i häc Vinh T¹p chÝ khoa häc, tËp XXXVII, sè 2A-2008
RAMAN STIMULATED SCATTERING
IN THREE-DIMENSIONAL APPROACH
Chu Van Lanh (a), Dinh Xuan Khoa(a),
Ho Quang Quy (b), Pham Thi Thuy Van (c)
Abstract In this paper we present a theory of Raman stimulated scattering in three-dimensional approach The intensity of Stokes waves is introduced and discussed in two limit conditions, there are transient limit and steady-state limit
I THREE-DIMENSIONAL MAXWELL-BLOCK EQUATIONS
We consider a collection of indentical atoms or molecules initially in ground states, contained in a pensil-shaped volume with length L and cross-sectional area
A The atomic positions are random, but fixed, and the average number density is N (atoms cm-3) A laser with electric field
) (
* ) (
) , ( )
, ( )
,
L
L L L
e t r E t
propagates through the volume in the z direction, which is parallel to the pencil axis As shown in Fig.1, an atom may absorb a laser photon at frequency ωL and scatters a photon at Stokes frequency ωS =ωL −ω31, ending up in the final state
3 We will treat the laser field mode as a classical electromagnetic wave and assume that it does not undergo depletion or any other back reaction from the medium On the other hand, the remaining modes of radiation field will be treated quantum mechanically, to allow for the spontaneous initiation of Raman scattering
As well as shown in previous works [1, 2, 3], we introduce a set of Maxwell-Block equations, describing Raman stimulated scattering in three-dimensional space:
) , ( ) , ( ˆ ) , ( ˆ ) , ( )
, (
) , ( ) , ( ˆ
2 )
, ( ˆ 1
* 1
) (
*
* 2 )
( 2
2 2 2
t r F t r E t r E ik t r Q t
r Q t
e t r Q t r E t c
k e
t r E t c
S L
z k t i L
S
z k t i S
S S S
S
ρ ρ
ρ ρ
ρ
ρ ρ
ρ
+
− Γ
−
=
∂
∂
∂
∂
=
∂
∂
−
∇
+
−
−
−
−
ω
(1)
where EˆL(rρ,t)
is the intensity operator of laser field with slowly-varying envelope approximation, E ˆS+( r ρ , t )
is the intensity operator of Stokes field with slowly-varying envelope approximation, dependent on frequency ωS, Q(rρ,t)
is the atomic-transition operator, which describes the relation between two states 1 and 3 (see Fig 1),
Qˆ
Γ is the term describing damping of Q(rρ,t)
at a collisional dephasing rate
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Γ,F(rρ,t)
is the quatum statistical Langevin operator describing the
collisional-induced fluctuations,
c
L
ω
= ,
c
k S ωS
= are the wave numbers of the laser field and Stokes field, respectively, and k1, k2 are the coupling constants given by:
c
k N k
d d k
S
S m L m m
m m
* 1 2
1 1
1 3 2 1
2
], 1 1
[ ω π
ω ω ω ω η
η
=
+
+
−
= − ∑
(2)
with c is the light velosity, and d ij =<i/dˆ/ j> is the atomic dipole matrix element
The atomic and Langevin operators have property:
)
( ) ( 2
) , ( ) , ( ˆ
) ( )
' 0 , ( ) 0 , ( ˆ
3 1
3 1
r r t t N t
r F t r F
r r N r
Q r Q
′
−
′
− Γ
=
>
′
′
<
′
−
=
>
′
<
− +
− +
ρ ρ ρ
ρ
ρ ρ ρ
ρ
δ δ
δ
(3)
Two important quantities presenting in the resolution of this set of equations
are: Raman gain coefficient
g 2k1k2 −1 E L(rρ,t)2
Γ
= (4) and Fresnel number
Φ =
L
A S
with λS is the wavelength of the Stokes field
1
3
m
Fig.1 An atom initially in its ground state 1 is driven by a laser
field with frequency ωL, which is not necessarily in near resonance
any intermediate state m Raman scattering at frequency ωS =ωL
-ω31 may accur, living the atom in the final state 3
Trang 3tr−êng §¹i häc Vinh T¹p chÝ khoa häc, tËp XXXVII, sè 2A-2008
II PARAXIAL SOLUTION OF MAXWELL-BLOCK EQUATIONS
In this approximation, we consider the laser field inside pensil-shape dispersionless medium depends only on the local time variable, i.e τ = t – z/c, and thus a laser pulse whose leading edge is at t = z/c leaves the atomic operator Q(rρ,t)
unperturbed for τ < 0 Thus equations (3) should be extended to read
)
( 2
) , ( ) , ( ˆ
) ( )
0 , ( ) 0 , ( ˆ
1
3 1 τ τ δ τ
τ
δ τ
τ
′
− Γ
=
>
′
<
′
−
=
>
=
′
=
<
− +
− +
N r
F r F
r r N r
Q r
Q
(6)
The initial value for the Stokes field ˆ+( 0 , τ )
S
E is given at the input face of the medium, z=0, for all time t This means that backward Stokes emission is explicitly ingnored
We will consider only the case that not Stokes wave is externally incident on the medium, and so we have for the initial field
<E ˆS−( 0 , τ ' ) E ˆS+( 0 , τ " ) >= 0, (7) i.e., the vacuum fluctuations are not detected with a photodetecter To resolve the set of set of Maxwell-Block equations, we consider the reflection will be from outside face of medium cylinder, because of that the dispersion is ignored inside medium cylinder It means that the limit condition in face of medium cylinder is ignored and shows that the Fresnel number Φ =A/λL is of the order of unity With above consideration, the set of equations (1) will be resolved by Laplace transform, and the operation of Stokes field in three-dimension is found out:
) , ( ˆ
0
3 3
τ τ
τ τ
τ τ
τ
′
′
′
′
′
′ +
′
′
′
+
r F r
r H d r d r
Q r r K r d r
, (8) where K(r,r′,τ) is the Kernels integration, given by:
( )
( )=∫ ( )′ ′
′
−
′
−
′
−
′
−
−
′
−
=
′
′
′
−
′
−
−
′
−
=
′
− Γ
Γ
∗
τ
τ τ τ
τ τ
τ
τ τ ρ
ρ τ
π τ
τ
τ ρ
ρ π
τ τ
0
2
2 1 2
1 0
2
*
* 2
2 2
1 0
2
* 2
) ( 4
2
exp 2
,
,
,
) ( 4
2
exp 2
)
,
'
,
(
d E
P
z z P P k k I z z
k i z
z
e E k k r
r
H
z z P k k I z z
k i z
z
e E k k
r
r
K
L
S L
S
S L
S
ρ
ρ
ρ
ρ
,(9)
andρ is the radial vector (x,y); I0(x) is the zero-order Bessel function, r’ in integration (9) changes in the excited by laser beam cylinder The solution is obtained with an approximation that the laser beam propagates along z- axis under
Trang 4C V Lanh , D X Khoa, H Q Quy, P T T Van RAMAN STIMULATED , Tr 37-42
approximation leads that the factor (z-z )-1 is replaced by the factor r − r′−1and then the divergence around of point z’ = z This divergence is limited when pay attention that excited scattering around of point z’ = 0 is better than one around of point z’ = z
III NON-PARAXIAL SOLUTION OF MAXWELL-BLOCK EQUATIONS
In this case, we consider the pump laser is constant in certain interval of time, i.e EL(r,τ ) = AL (0 ≤ τ ≤ τL) So equations (1) are rewritten:
q( )r t (i ) ( )q r t ik A A r t f( )r t
*
− Γ
+
−
=
∂
∂
2
*
* 2 2
2 2 2
t r q t
A c
k t r A t
∂
=
∂
∂
−
∇
where
A S(r,t)=E S+(r,t)e−i(ωS t−K S z)
( , ) ( , ) e i( S t K z)
e t r Q t r
q = − ω − (11)
( )r,t F( )r,t e i( S t K S z)
By the Laplace transform, from equation (10) we have:
( )
4
0
,
exp exp
2
,
2 1 2
2 1 0 0
2 1 2
2 1 0
2 2 1 3
* 2
+
′
−
−
′
−
′
−
′
′
′
+
+
′
−
−
′
−
′
×
+
′
−
− Γ +
−
′
−
′
−
′
=
∫
∫
c
z z c
r r r
r A k k I r
f
d
c
z z c
r r r r A k k I
r
q
c
z z c
r r i
r r c
A k k i r r
r d A k
k
t
r
A
L L
S L
L S
S
τ τ τ
τ
τ
τ ω
π
τ
(12)
In (12) there is a delay time
c
z z c
r
−
′
−
−
τ , describing the nature of laser propagating through medium Besides, in this solution there is not the divergence if z’→z, like in solution for the paraxial case That because of the factor
'
1
z
z− is
Trang 5tr−êng §¹i häc Vinh T¹p chÝ khoa häc, tËp XXXVII, sè 2A-2008
replaced by the Green function of
r
r− ′
1
argument This solution is genaralized for space-time description of the arbitrary laser field
IV STOKES INTENSITY IN THREE-DIMENSION
The average intensity of Stokes fied at output face of medium z = L is given [2,3,4]:
( )= < −( ρ τ ) (+ ρ τ )>
ω π τ
2
s S
η , (13)
with considering only the case that no Stokes wave is external incident on the medium, and so all atoms are in initial state
From (6), (7), (8), (9) and (13) we have:
2
,
0
2 3
1 2
3 1
′
′
′
′ Γ
+
′
τ
ρ
τ τ τ
τ ω
π τ
I S
Now we discuss in the transient limit and the steady-state limit
4.1 Stokes intensity in transient limit
When the scattering time τL is much less than the collisional dephasing time i.e [3],
gL
L <
from (8), (9) and (14) the intensity Stokes wave at output face z = L is
( ) ( )
( ) exp { [ 16 ( ) ] } 8
2 1
2 2
τ τ
τ π
τ
P
E A
This result is similar to one of the one-dimension, byond the factor Φ2
4.2 Stokes intensity in steady-state limit
When the time τL is much larger than the collisional dephasing time [3], i.e.,
Γ τL >> gL (17) from (8), (9) and (17) the intensity Stokes wave at output face z = L is
) 4
2
gL A
e I
gL SS
π
Φ
This result is similar to one of the one-dimension, byond the factor Φ2
V DISCUSSION
In three-dimensional approach, the Stokes intensities in two cases of limit
Trang 6C V Lanh , D X Khoa, H Q Quy, P T T Van RAMAN STIMULATED , Tr 37-42
dimension by factor Φ2 (Fresnel number), which relates to structure of medium It is true for the general case too, when the value of Γ τL is arbitrary, for that the Stokes intensity is given by
( ρ,τ ) 1 ( ,τ )
2
L I A
= (19) where I1S D(L,τ ) is the Stokes intensity when Φ=1(in one-dimentional approach) Have in mind that all results are found out for the initiation Raman scattering
at high-gains in the absence of an input Stokes field
REFERENCES
[1] M Trippenbach and Rzazewski, Stimulated Raman Scattering of Colored Chaotic Light, Optic Society of America, Vol 1, 671, 1984
[2] M G Raymer and L A Westling, Quantum theory of Stokes generation with a multimode laser, J Opt Soc Am.B, Vol.2, No.9, 1417, 1985
[3] Dinh Xuan Khoa, Chu Van Lanh and Tran Manh Hung, Intensity of stimulated Raman scattering under quantum theory view, Proc XXVIIth NSTP, Cualo, August 2-6, 2002
[4] D Homoelle et al, Conical three-photon-excited stimulated hyper-Raman scattering, Phys awRev A, 72, 011802-2, 2005
TóM TắT TáN Xạ RAMAN CƯỡNG BứC TRONG GầN ĐúNG BA CHIềU
Trong bài này chúng tôi giới thiệu lý thuyết tán xạ Raman cưỡng bức trong gần đúng ba chiều Cường độ sóng Stokes đã được tính toán và thảo luận trong hai trường hợp giới hạn là giới hạn thời gian ngắn và giới hạn thời gian dài
(a) Physical Department of Vinh University
(b) Institute for Applied Physics, MISTT
(c)Master student of Optical course 14 th