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A possible explanation of the inverse Raman scattering in the threshold region of stimulated Raman scattering
Volume
4, number
A IN
POSSIBLE THE
OI’TICS COMMUNICATIONS
3
EXPLANATION
THRESHOLD
OF
REGION
OF
THE
INVERSE
STIMULATED
November
RAMAN
SCATTERING
RAMAN
SCATTERING
1971
A. LAU, M. PFEIFFER, P. GADOW, W. WERNCKE, K. LENZ and H. -J. WEIGMANN Zctllmli?~stitut jii?’ Optik md Spcktvoskopie dcr Dcutscl~cn Akadcwzic dcs- Wisscnscltaj~cn ZZLBerlin R{,YIt?i-Adlrrsiiqf, DDR
Received
5 July 1971
If inverse Raman scattering is investigated with laser intensities near the threshold region for stimulated Raman emission, new phenomena appear, which were first described by Durmartin and co-morkcrs. Here it is shown, that these phenomena can be explained as an “interference effect” by a simple model, coupling a laser, a Stokes and an anti-Stokes wave.
The absorption caused by molecular vibrational transitions during the interaction of an intense monochromatic radiation and a continuum with a molecular system, which appears in the anti-Stokes region relative to the laser frequency is called by Stoicheff and co-workers inverse Raman scattering (IRS) [l]. New phenomena appear when IRS is investigated with laser intensities in the threshold region of the stimulated Raman scattering (SRS). Results obtained for such a situation were first described by Dumartin et al. [2] for pure liquids. The main features of these phenomena are: (1) An intense absorption of the Stokes wave at frequencies near to the Stokes emission of the SRS process. Dumartin and co-workers call this absorption “stimulated IRS” because of its threshold behaviour. In order to explain this effect they suggest the development of a stimulated photon wave. (2) An emission at the corresponding antiStokes frequency, appears very often. We found similar features with mixtures near the threshold region of SRS; the picture depends on the magnitude of the gain values of the frequencies concerned. Fig. 1 demonstrates the phenomenon of anti-Stokes emission within the absorption band. Till now the classical treatment of IRS has been carried out by connecting a suitable wave for the anti-Stokes region and a laser wave. The equations are similar to those derived for the Raman amplifier [3]. In the two-wave approximation the effect of a Stokes absorption cannot be 228
Fig. 1. Appearance of a narrow emission line within the anti-Stokes absorption band (absorption of benzene produced by stimulated Raman in a “continuum”, emission of toluene).
explained, which probably had led the French group to the concept of a phonon wave. In this work we show that Stokes absorption at low laser powers can be explained by a three-wave model (Stokes, anti-Stokes and laser). In this case it is not necessary to introduce a photon wave to get a population inversion. The interference term appearing in the three-wave
Volume 4, number
approximation can account for the anti-Stokes emission which is sometimes observed together with the Stokes absorption. In the following discussion we assume the laser intensity to be constant. The stationary coupled wave equations for this case are [4] dAs/dz
November
OPTICS COMMUNICATIONS
3
= i/31, As +ip exp (i&z)ALAz
Is =
YIexp (2gIL
+ 25 I,
=
Y2
cos
1971
2) + Yi exp (- 2gIL 2)
(i-h
+
Aq) ,
ctY;
exp (2gIL 2) + ci Yi exp (- 2gIL 2)
, +2rI y2 cos (Kz + Acp + AK) ,
dAi/dz
= - ifl exp (- i*kz)(AL)2As
- ipILAl
.
(1)
IL is the laser intensity,
Ai are the amplitudes of the corresponding waves, /3 the relevant susceptibilities (dispersion of susceptibilities neglected) and Ak = 2kl- k, - k, the phase mismatch. The coupled equations (1) have solutions of the type As = all
A; = bZ1 exp (ill 2) +a22 exp (iZ2 z)} exp (- ihkz) with Zi to be calculated Z2-Ak(Z-p)
,
from the secular
(2) equation
= 0.
withAcp=cpl-p2, K=Kl-K2andA~=~1-~2. The values CLand Ki following from the secular equation depend only on the relation j?/Ak. For Ci the following relations are valid c2 2 1.
exp (ill 2) +a I2 exp (iZ2 2) ,
(3)
where s = NIL. _ _ With complex o(@ = ip +X) the Zi have the form Zi = (iSi+Ki). From (3) we get S2 = -SI. The positive expression (S2) we name
(7)
Cl s 1,
Cl 9 = 1 .
The first two terms of expression (7) for the Stokes and anti-Stokes waves, respectively describe the gain and loss of the waves. There is an additional interference term, which can be neglected at large gain values, but it may be important to IRS near the threshold region of stimulated Raman emission. The behaviour of solution (7) depends on the i;it/;l vtue; (71, V2, Aq) and on the relation @ Ak y with p the imaginary part of 6. its real part in the following being neglected). At constant geometrical copditions of the experiment (Ak = const) the dependence of the relative weights of the terms in (7) and of the effective gain (I.7 / = gIL) on y x 1~ is represented in fig. 2. We can distinguish three regions with different behaviour of the solutions (7)
s2 = - s1 = gIz ) with g the Raman gain coefficient. One of the terms in (2) describes a gain solution (Sl < O), the other a loss solution (S2 ? 0). To get information about the behaviour of the wave, the corresponding intensities have to be calculated from (2). We introduce the initial value parameters aIi = Yi exp (icpi) .
(4)
For a2i we get U2i =
Ciali
eXp
(iKi)
,
(5)
with Ci
eXp
(i Ki)
=
(Zi - a)/$
following from the secular equation. For the intensities we get
(6)
Fig. 2. Distinction of the three regions of solution of eq. (7). Region I: CI << 1, c2 >> 1, r2 << ~1, ClyI c< ~2~2. Stokes and anti-Stokes decoupled. Region II: region where interference term plays a role. Region III: Cl, c2 N 1. Prevailing of exponential gain for both Stokes and anti-Stokes.
229
Volume
4, number
OPTICS
3
November
COMMUNICATIONS
1971
A 7.
6
, --
Fig.
3. Solution
according
to (7),
representative
for region
depending on the value of y. These regions are spanned with rising laser intensity. (1) At low laser intensities (g11 - 0.1 cm-l) we have y = 0.1 (region I). In this case cl ‘i_ 1, c.2 ._ 1. From reasonable values for I,(O) and I,(O) we get Y2’
?’
.
Cl Yl ,i 9’2
.
The two waves (s. a) are nearly decoupled. In the anti-Stokes region one can see a strong absorption which only slightly depends on the initial conditions (1,(0)/I,(O)). This result corresponds to the two-wave interaction commonly used. Because of the small gain, the interference term plays a certain role and therefore a weak absorption is possible for the Stokes wave. strongly dependent on the initial values. (2) With increasing laser intensity we come to region II. In this region all ‘1. “2, cl ‘1, c2 v2 are nearly of the same order: this means that for small values of gIlz both absorption and emission are possible for the Stokes and antiStokes waves. Considering only the first two terms of (7) we see that for the Stokes wave. the term ~2,‘~l and for the anti-Stokes wave, the term c2~2/cl~l are responsible whether emission or absorption appears:
25, ‘Y1
230
Stokes
absorption emission
II.
(a) Stokes
intensity:
(b) anti-Stokes
intensity
c2y2 absorption anti-Stokes --5 1 emission cl ‘1 According to the initial conditions for the Stokes wave both cases can occur. According to c2/cl : 1 from the first two terms an anti-Stokes emission with Stokes absorption at the same time cannot appear. It can be shown that due to the interference term a Stokes absorption in this region can be accompanied by an anti-Stokes emission. The situation for this case is shown in figs. 3a and 3b. (3) A more increased laser intensity (y ) 2) causes the gain term to prevail for both the Stokes and anti-Stokes waves. This discussion shows, that in the region y 2 1 for IRS a Stokes absorption accompanied by an anti-Stokes emission can occur. This is just the same region where the strongest antiStokes emission (anti-Stokes cone) appears in the case of SRS.
REFERENCES [l] W. J. Jones and B. P. Stoicheff, Phys. Rev. Letters 13 (1964) 657. [2] S. Dumartin, B. Oksengorn and B. Vodar, Compt. Rend. Acad. Sci. (Paris) 261 (1965) 3767. [3] N.Bloembergen, ‘Am. J: Phys. 35 (1967).989. [d] h’. Bloembergen and Y. R. Shen, Phys. Rev. Letters 12 (1964) 50.1.
4, number
A IN
POSSIBLE THE
OI’TICS COMMUNICATIONS
3
EXPLANATION
THRESHOLD
OF
REGION
OF
THE
INVERSE
STIMULATED
November
RAMAN
SCATTERING
RAMAN
SCATTERING
1971
A. LAU, M. PFEIFFER, P. GADOW, W. WERNCKE, K. LENZ and H. -J. WEIGMANN Zctllmli?~stitut jii?’ Optik md Spcktvoskopie dcr Dcutscl~cn Akadcwzic dcs- Wisscnscltaj~cn ZZLBerlin R{,YIt?i-Adlrrsiiqf, DDR
Received
5 July 1971
If inverse Raman scattering is investigated with laser intensities near the threshold region for stimulated Raman emission, new phenomena appear, which were first described by Durmartin and co-morkcrs. Here it is shown, that these phenomena can be explained as an “interference effect” by a simple model, coupling a laser, a Stokes and an anti-Stokes wave.
The absorption caused by molecular vibrational transitions during the interaction of an intense monochromatic radiation and a continuum with a molecular system, which appears in the anti-Stokes region relative to the laser frequency is called by Stoicheff and co-workers inverse Raman scattering (IRS) [l]. New phenomena appear when IRS is investigated with laser intensities in the threshold region of the stimulated Raman scattering (SRS). Results obtained for such a situation were first described by Dumartin et al. [2] for pure liquids. The main features of these phenomena are: (1) An intense absorption of the Stokes wave at frequencies near to the Stokes emission of the SRS process. Dumartin and co-workers call this absorption “stimulated IRS” because of its threshold behaviour. In order to explain this effect they suggest the development of a stimulated photon wave. (2) An emission at the corresponding antiStokes frequency, appears very often. We found similar features with mixtures near the threshold region of SRS; the picture depends on the magnitude of the gain values of the frequencies concerned. Fig. 1 demonstrates the phenomenon of anti-Stokes emission within the absorption band. Till now the classical treatment of IRS has been carried out by connecting a suitable wave for the anti-Stokes region and a laser wave. The equations are similar to those derived for the Raman amplifier [3]. In the two-wave approximation the effect of a Stokes absorption cannot be 228
Fig. 1. Appearance of a narrow emission line within the anti-Stokes absorption band (absorption of benzene produced by stimulated Raman in a “continuum”, emission of toluene).
explained, which probably had led the French group to the concept of a phonon wave. In this work we show that Stokes absorption at low laser powers can be explained by a three-wave model (Stokes, anti-Stokes and laser). In this case it is not necessary to introduce a photon wave to get a population inversion. The interference term appearing in the three-wave
Volume 4, number
approximation can account for the anti-Stokes emission which is sometimes observed together with the Stokes absorption. In the following discussion we assume the laser intensity to be constant. The stationary coupled wave equations for this case are [4] dAs/dz
November
OPTICS COMMUNICATIONS
3
= i/31, As +ip exp (i&z)ALAz
Is =
YIexp (2gIL
+ 25 I,
=
Y2
cos
1971
2) + Yi exp (- 2gIL 2)
(i-h
+
Aq) ,
ctY;
exp (2gIL 2) + ci Yi exp (- 2gIL 2)
, +2rI y2 cos (Kz + Acp + AK) ,
dAi/dz
= - ifl exp (- i*kz)(AL)2As
- ipILAl
.
(1)
IL is the laser intensity,
Ai are the amplitudes of the corresponding waves, /3 the relevant susceptibilities (dispersion of susceptibilities neglected) and Ak = 2kl- k, - k, the phase mismatch. The coupled equations (1) have solutions of the type As = all
A; = bZ1 exp (ill 2) +a22 exp (iZ2 z)} exp (- ihkz) with Zi to be calculated Z2-Ak(Z-p)
,
from the secular
(2) equation
= 0.
withAcp=cpl-p2, K=Kl-K2andA~=~1-~2. The values CLand Ki following from the secular equation depend only on the relation j?/Ak. For Ci the following relations are valid c2 2 1.
exp (ill 2) +a I2 exp (iZ2 2) ,
(3)
where s = NIL. _ _ With complex o(@ = ip +X) the Zi have the form Zi = (iSi+Ki). From (3) we get S2 = -SI. The positive expression (S2) we name
(7)
Cl s 1,
Cl 9 = 1 .
The first two terms of expression (7) for the Stokes and anti-Stokes waves, respectively describe the gain and loss of the waves. There is an additional interference term, which can be neglected at large gain values, but it may be important to IRS near the threshold region of stimulated Raman emission. The behaviour of solution (7) depends on the i;it/;l vtue; (71, V2, Aq) and on the relation @ Ak y with p the imaginary part of 6. its real part in the following being neglected). At constant geometrical copditions of the experiment (Ak = const) the dependence of the relative weights of the terms in (7) and of the effective gain (I.7 / = gIL) on y x 1~ is represented in fig. 2. We can distinguish three regions with different behaviour of the solutions (7)
s2 = - s1 = gIz ) with g the Raman gain coefficient. One of the terms in (2) describes a gain solution (Sl < O), the other a loss solution (S2 ? 0). To get information about the behaviour of the wave, the corresponding intensities have to be calculated from (2). We introduce the initial value parameters aIi = Yi exp (icpi) .
(4)
For a2i we get U2i =
Ciali
eXp
(iKi)
,
(5)
with Ci
eXp
(i Ki)
=
(Zi - a)/$
following from the secular equation. For the intensities we get
(6)
Fig. 2. Distinction of the three regions of solution of eq. (7). Region I: CI << 1, c2 >> 1, r2 << ~1, ClyI c< ~2~2. Stokes and anti-Stokes decoupled. Region II: region where interference term plays a role. Region III: Cl, c2 N 1. Prevailing of exponential gain for both Stokes and anti-Stokes.
229
Volume
4, number
OPTICS
3
November
COMMUNICATIONS
1971
A 7.
6
, --
Fig.
3. Solution
according
to (7),
representative
for region
depending on the value of y. These regions are spanned with rising laser intensity. (1) At low laser intensities (g11 - 0.1 cm-l) we have y = 0.1 (region I). In this case cl ‘i_ 1, c.2 ._ 1. From reasonable values for I,(O) and I,(O) we get Y2’
?’
.
Cl Yl ,i 9’2
.
The two waves (s. a) are nearly decoupled. In the anti-Stokes region one can see a strong absorption which only slightly depends on the initial conditions (1,(0)/I,(O)). This result corresponds to the two-wave interaction commonly used. Because of the small gain, the interference term plays a certain role and therefore a weak absorption is possible for the Stokes wave. strongly dependent on the initial values. (2) With increasing laser intensity we come to region II. In this region all ‘1. “2, cl ‘1, c2 v2 are nearly of the same order: this means that for small values of gIlz both absorption and emission are possible for the Stokes and antiStokes waves. Considering only the first two terms of (7) we see that for the Stokes wave. the term ~2,‘~l and for the anti-Stokes wave, the term c2~2/cl~l are responsible whether emission or absorption appears:
25, ‘Y1
230
Stokes
absorption emission
II.
(a) Stokes
intensity:
(b) anti-Stokes
intensity
c2y2 absorption anti-Stokes --5 1 emission cl ‘1 According to the initial conditions for the Stokes wave both cases can occur. According to c2/cl : 1 from the first two terms an anti-Stokes emission with Stokes absorption at the same time cannot appear. It can be shown that due to the interference term a Stokes absorption in this region can be accompanied by an anti-Stokes emission. The situation for this case is shown in figs. 3a and 3b. (3) A more increased laser intensity (y ) 2) causes the gain term to prevail for both the Stokes and anti-Stokes waves. This discussion shows, that in the region y 2 1 for IRS a Stokes absorption accompanied by an anti-Stokes emission can occur. This is just the same region where the strongest antiStokes emission (anti-Stokes cone) appears in the case of SRS.
REFERENCES [l] W. J. Jones and B. P. Stoicheff, Phys. Rev. Letters 13 (1964) 657. [2] S. Dumartin, B. Oksengorn and B. Vodar, Compt. Rend. Acad. Sci. (Paris) 261 (1965) 3767. [3] N.Bloembergen, ‘Am. J: Phys. 35 (1967).989. [d] h’. Bloembergen and Y. R. Shen, Phys. Rev. Letters 12 (1964) 50.1.