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High resolution polarization spectroscopy in the strong saturation regime
Volume 29, number 1
OPTICS COMMUNICATIONS
April 1979
HIGH RESOLUTION POLARIZATION SPECTROSCOPY IN THE STRONG SATURATION REGIME H.-H. RITZE, V. STERT and E. MEISEL Zentralinstitut fiir Optik und Spektroskopie, Akademie der Wissenschaften der DDR, DDR - 1199 Berlin
Received 21 December 1978 The influence of the optical anisotropy induced by a circularly polarized saturating beam on a counterpropagating weak probe wave was studied in the case of strong saturation. Characteristic changes of the line shape were observed that can be used for a simple determination of saturation intensities.
1. Introduction When an inhomogeneously broadened transition is saturated by an intense monochromatic beam, for a weak counter-propagating probe wave of the same frequency we can observe a very narrow transmission peak (inverted Lamb dip) in the centre of the Doppler profile, see e.g. [11. In the polarization variant both beams are polarized in a different way. Here also the polarization of the weak wave, in general, is changed. The deviations from the initial polarization can be measured very sensitively; so the polarization detection technique allows us to obtain a much higher signal-tobackground ratio compared with the ordinary Lamb dip methods, further it is possible to get more informations about the parameters of the molecular transition. In the present work we will use a circularly polarized saturating beam and a linearly polarized probe [2]. (Regarding the variant with a linearly polarized saturating wave, see [ 3 - 5 ] . ) The corresponding scheme is outlined in fig. 1. Due to the optical anisotropy induced by the strong wave the probe polarization experiences a rotation of its plane (circular birefringence) and a small ellipticity (circular dichroism) in the vicinity of the line centre. The role o f these effects can be investigated measuring the angular dependence o f the intensity, transmitted through an analyzer. If the angle, by which the analyzer is rotated from the perpendicular position with respect to the initial probe polarization, is denoted by 0, for a single transition with the unsaturated absorption coefficient a0 the detected in-
beamsptitter tunable laser polarizer probe beamt
soturati beam n9 absorptio~cell n[ L
analyzerdetector
Fig. 1. Scheme of the polarization spectroscopic arrangement (L is the length of the absorption cell, I o the probe intensity before passing the cell) tensity I can be subdivided into three terms ( ~ is the detuning of the laser frequency from the transition frequency): I = I 0 e - s ° L { [ I + L ( ~ ) o~0L] sin20 +D(~2)aoL sin 20 + P ( ~ ) (a0L)2}.
(1)
[l+L(g2) a 0 L ] sin20 is maximal for 0 = n/2 and describes the inverted Lamb dip on the Doppler background. With decreasing 0 the signal also decreases and the antisymmetric contribution D(g2), caused by the birefringence, becomes more important. For perfectly crossed polarizers (0 = 0) we obtain a very small symmetric signal P ( ~ ) which is simultaneously effected by dichroism and birefringence, no background appears if we consider only ideal polarizers. If the saturation parameter X (averaged over the degenerate sublevels o f the transition) is small, the po51
Volume 29, number 1
OPTICS COMMUNICATIONS
larization effects can be calculated in a perturbation treatment [2]:
L(a)- -
l+d 1 e %L r2 y -- %L - X -r 2 + a - 2
D(~)
1
April 1979
M-I
M+I
+o(x),
M
P(a)
=
1 -e - a ° L
PgZ
_
_ ( 1 - d) 2 (1 e - a ° L ) 2 X2 . . . . . . - - i 6
-
--
(%L)2
-
P2 + o(X2). (2) r2 +a2
where P is the homogeneous half-width, we suppose, that all relaxation constants (phase and population relaxation) are equal. The parameter d, introduced by Wieman and tt/insch [2], characterizes the optical-induced anisotropy:
(2J- 1)(2J+3)
--~ "~ 2+9.]- - -
d
.(,_J
for J+-+ J transitions,
_J+l)
] _ 2J2+3 [ 2 ( 6 J 2 1)
(3) for J +-+ J - 1 transitions.
( J is the rotational quantum number of a resonant level). From (3) it is seen that the polarization technique can be used to distinguish P, R from Q branches and to determine the degeneracy of molecular levels, i f J is not too large. So we identified a Q(5) transition in ammonia [61. In the low pressure region (some mTorr in the absorption cell) it can be possible to achieve such high saturation effects, where the perturbation theory is no longer valid. In sect. 2 we will study tire change of the polarization signal for stronger saturation. In sect. 3 we will given the results of our experiments confirming qualitatively our theoretical investigations.
2. Theory of polarization line shape for strong saturation Some years ago much work was done regarding the effects of power broadening and dynamic Stark splitting on the absorption coefficient of a probe wave in two- and three-level systems [7 10]. We will apply these theoretical methods to the calculation of polarization effects of transitions with arbitrary degeneracy. Orientating the spatial quantization axis along the di52
M÷2
Fig. 2. Sublevels and subtransitions important for the interaction with the light beams if the molecule is pumped in the state M. rection of light propagation we assume that after pumping of the ground sublevel with the quantum number M the strong field (s) saturates the M *-+ M + 1 transition. The probe polarization can be decomposed into two components rotating in the same (+) and the opposite ( ) sense compared to the saturating field. If we neglect any saturation effects of the probe, the investigation of the four-level system illustrated in fig. 2 is sufficient for our calculations. Assuming plane waves we can find the time-independent equations for the density matrix elements in the Wigner representation. The influence of the strong field can be considered exactly. Then the macroscopic dipole moment with respect to the probe colnponents can be calculated after integration over all possible velocities of the molecules in the Doppler limit. These standard procedures are explained in more detail, e.g. in [8]. The saturation-dependent absorption coefficient ~(-+M)and refractive index n(+M) of the two circularly polarized probe components are defined as follows: +
+
a(M ) = Re A (m),
n(-+M) = (C/2Co)hn A~M),
(4)
where +
"i-
(5)
A(M ) = CtO(M)
and
X/i~-XM(1 +3X/-i+XM+4i~2/F ) A The saturation parameter XM of the M ~ M + I subtransition is expressed as XM = I/~M,M+I[2 l(S)/h2F2 ,
(7)
Volume 29, number 1
OPTICS COMMUNICATIONS
where l(S) is the intensity of the saturating beam. The quantity #i./is the transition dipole nroment between the lower sublevel with the magnetic q u a n t u m number i and the upper one]. Now we must average the expressions (4) and (7) over the magnetic sublevels assuming that the pumping of the ground state is an isotropic process. We get
M
April 1979
saturating field. On the other hand, tile transition moments of tire opposite polarization component increase with raising magnetic quantum numbers. As the result of this behaviour after summation over M the absorption dip of the ( ) component is somewhat smaller and narrower than the corresponding (+) one. So for sufficient saturation in the line centre an additional mininmm can arise in the frequency dependence of the dichroism. These specific saturation phenomena
and L[O)
M
0.5
The quantity n + - n - describes the induced birefringence and ~+ ~ - the dichroism. We have numerically calculated the terms L ( ~ ) , D(~2) and P ( ~ ) for arbitrary saturation and rotational quantum numbers of the resonant levels assuming that the relation % L< I
(9)
is fulfilled. Now we will analyze some interesting saturation phenomena in P and R branches. Here the dipole moment I~M,M+II decreases with increasing M (see fig. 3). As a consequence the Bennett holes "burned" into the velocity distribution of the lower sublevels and the corresponding peaks of the upper levels become broader with decreasing M due to the power broadening of the
_•-
)(=20 X=15 ~X=IO
JL -12
-10
-8
-6
-4
2
-2
Z,
6
8
10
r
12
o(a)
(b)
,,-~#111/ -12
-10
~"
~
4
6
x=ls 8
10
12
J~"
ot ÷ - o c
P
\
\ ',
', ~
',,_,
~
"V4
n.\ f . ~ n*
n + - n-
/, \ \ ~ \~,:,s / \ \ \ \ , , , " x '~:'° -
",,(_,
Fig. 3. Level scheme of a P(2) transition. The induced nonequilibrium velocity distributions are outlined. The (+) component interacts with the strongly saturated transitions more effectively than the (-) one. On the right side the different shape of the corresponding absorption coefficients and refractive indexes is illustrated.
..9_
-12
-10
-8
-6
-4
-2
2
4
6
8
10
12
F"
Fig. 4. Calculated line shape of the terms L (s2) (a), D (I2) (b) and P(s2) (c) for a transition between the momentum quantum numbers 20 ,--. 21 for five different saturation parameters 53
Volume 29, number 1
OPTICS COMMUNICATIONS
can also lead to a special structure of the birefringence curve as it is shown in fig. 3. As an example let t,s look at the x-dependence of the computed line shape for a R(20) or P(21) transition (see figs. 4a, b, c): If 0 = rr/2, we observe the inverted Lamb dip which increases and broadenes with increasing intensity l(S). For very high saturation the width is proportional to the square root of the saturation parameter. Due to tile d y n a m i c Stark effect, even for X--* oo we obtain a finite probe absorption, here L(~2 = 0) approaches ~ 0 . 6 2 3 . Although the saturation resonance is not a korentzian, the profile does not show significant particularities (fig. 4a). Rotating the analyzer we can observe the antisymmetric term D(~2). We can see that for larger saturation parameters the line shape deviates frtm~ a dispersion curve: with raising X in the line centre the first derivative decreases, for X ~ 15 it is zero and then changes its sign (fig. 4b). For a crossed configuration (0 = O) in the case X > 4 in the centre an additional dip appears whose depth increases with increasing intensity of the saturating beam (fig. 4c). If we define the two characteristical saturation parameters X1 and X2 as follows: aD(~21Xl )] i ch~ it~ =0
-
-
OS~2
=0,
(10)
tz=O
Table 1 Dependence of the characteristical saturation parameters, defined by (10), on the rotational quantum numbers. For P(I) and R (0) lines no special structure in the line centre appears Transition 0~l
~ 2~ 3~ 4~ 5~
x1 -
-
5.12 4.25 4.12 4.08 4.06
10 *--, II
15.21
4.04
20 ~ 21
15.08
4.03
50 ~ 51
15.04
4.03
100 ~ 101
15.03
4.02
54
2 3 4 5 6
x2
273.8 21.24 17.22 16.17 15.74
1
we see from the table 1 that with increasing degeneracy of the molecular transition these values rapidly converge to the limits 15 and 4, respectively, for J 2> 10 the deviations are 1% and smaller.
3. Experimental
results
In our experiments we used an arrangelnenl similar to that outlined in fig. 1. Utilizing a line-tunable ( ' 0 2 lase~ emitting in the 10~m region we investigated tile polarization line shape of selected transitions in tile ~'3 band ill SF 6. The circular polarization of tile saturating beam was achieved with the help of a Fresnel rhomb. The strong beam was modulated by a 1 kllz chopper, and a lock-in amplifier was used to delecl only thal part of tile signal corresponding to the saturated absorption or dispersion. Both laser beams were extended to a beam diameter of 4 cm, only the cemral part of tile initial nearly Gaussian intensily profile was taken, so we could work approximately with a rectangular profile. (For gaussian intensity disl ributions Ihe results of sect. 2 must be modified by integration over the spread of possible saluration parametms, see e.g.
IIll.)
~)2P(~IX2)I =0,
April 1979
In fig. 5 tire nreasured strong saturation polarization line shapes are given for the P(33) A 1 line of SF 6. Varying the laser o u t p u t and observing the line shape we were able to fix the characteristic saturation parameters ( 1 0 ) X I and X2 with an accuracy of ~5~,~. ~[he asymmetry o f the line shape in fig. 5a can be explained by the unnegligible c o n t r i b u t i o n of the term L(XZ)sin20 for 0 = 3 ° . Due to the finite extinction values of our polarizers we obtained a relatively high background of about 2 X 10 -5 10 in tire crossed configuration (0 = 0). Although the observed experimental line shapes are in a good qualitative agreement with the predicled ones of sect. 2, we can see sonle minor quantitative discrepancies. The measured relation X1 was 3. I . (Tile calculated value Xl/X2 is 3.75.) But these deviations are understood if we notice that for p = 8 mTorr we have significant smearing of our spectral profile which arises from the finite crossing angle of 10' between the counterpropagating beams and from the unresolved magnetic hyperfine structure which is spread over a range of about 60 kHz [12]. Furthermore, the finite c o n t r i b u t i o n of the inverted Lamb dip at 0 = 0 {background due to real polarizers) leads to an increase of
IX2
Volume 29, number 1
OPTICS COMMUNICATIONS 1 MHz
X =3/,,1
4. Discussion
X= 1.8 X 1
(a) 1 MHz
X= 0.5 X 2
X= 1.75X 2
April 1979
X=t,% 2
(b) Fig. 5. Observed line shapes for two saturation parameters if 0 = 3° (a) and three different intensities ifO = 0 (b). The investigated transition was the P(33) A 1 line of the v 3 band of SF6, the pressure wasp = 8 mTorr. the observable value X2. Finally for the used absorption cell of 120 cm length the relation (9) is not fulfilled very well, so our results are influenced by the inhomogeneity of the saturation parameter due to the absorption of the saturating beam along its direction of propagation. Now our aim is to consider all the enumerated effects in a numerical calculation o f the expected line profiles. Let us note that recently Schieder [13] published results about a special polarization variant using a Jamin interferometer. Investigating molecular iodine with the help o f a dye laser the author observed similar saturated dispersion and absorption line shapes as we found (fig. 5a, b). We suppose that those results are caused by the same reasons discussed above in sect. 2.
We have found theoretically and experimentally that in the regime of strong saturation, when power broadening and dynamic Stark splitting must be taken into account, there exist significant changes of the line shape detected under (nearly) crossed polarizers compared with the perturbation treatment. We could accurately fix such intensities of the strong beam where additional maxima or minima are created in the registrated line profile of the probe. So the polarization method allows us to determine the relations between the saturation intensities o f different P and R branch transitions in a simple way. This rechnique should be also valuable for the measurement of pressure dependences o f saturation intensities. The absolute values o f the saturation parameters X could be determined considering the special beam geometry, etc. Our calculations showed that for 1 ~< X ~< 50 the J-dependence of the line shapes is relatively weak. However for X ~> 500 this dependence is essentially stronger (cx l/J), so the regime of very strong saturation should be more useful for the determination of not too high J-values than the small saturation limit in which the polarization effects are proportional to l/J 2 (3). Q branch transitions were investigated numerically. According to (3) for X '~ 1 in the limit of very large momentum quantum numbers the optically induced anisotropy vanishes. But the different role of dynamic Stark splitting for (+) and ( - ) components (cf. (5) and (6)) leads to a remaining birefringence and dichroism with a frequency dependence similar to that of highly saturated P and R lines. We have also studied theoretically the case of a difference between the various relaxation constants. A detailed analysis o f these effects would be beyond the scope of the present work, we only note that the characteristic saturation parameters Xl and X2 ( I 0 ) depend on the relations between the relaxation constants. Finally it can be shown that in the variant of a linearly polarized saturating beam stronger saturation effects can also lead to additional dips or peaks of the resonance signal this will be studied in a subsequent paper.
55
Volume 29, number 1
OPTICS COMMUNICATIONS
References [ 1 ] V.S. Letokhov and V.P. Chebotayev, Nonlinear laser spectroscopy, Springer Series in Optical Sciences, Vol. 4 (Springer Berlin, Heidelberg, New York, 1977). [2] C. Wieman and T.W. H/insch, Phys. Rev. Lett. 36 (1976) 1170. [3] M. Sargent III, Phys. Rev. A14 (1976) 524. J41 J.C. Keller and C. Delsart, Optics Comm. 20 (1977) 147; J. Appl. Phys. 49 (1978) 3662. I51 R. Fischer, V. Stert and E. Meisel, Appl. Phys. 17 (1978) 151. J6] V. Stert, R. Fischer, E. Meisel and H.-H. Ritze, Kvantovaya Electronica (Soy. Journal) 4 (1977) 2620.
56
April 1979
[71 tt. Paul, Fortschritte der Physik 22 (1974) 1. [8] E.V. Baklanov and V.P. Chebotayev, Zh. Eksp. Teor. l.iz. 60 (1971) 552. [9] S. Haroche and F. tiartmann, Phys. Rev. A6 (1972) 1280. [10] N. Skribanowitz, M.J. Kelly and M.S. Feld, Phys. Rev. A6 (1972) 2302. J 11 ] B. Couillaud, A. Ducasse and A. Dienes, Appl. Phys. 16 (1978) 359. [12] Ch. J. Bord~, M. Ouhayoun and J. Bord6, J. Mol. Spectr. 73 (1978) 344. [13] R. Schieder, Optics Comm. 26 (1978) 113.
OPTICS COMMUNICATIONS
April 1979
HIGH RESOLUTION POLARIZATION SPECTROSCOPY IN THE STRONG SATURATION REGIME H.-H. RITZE, V. STERT and E. MEISEL Zentralinstitut fiir Optik und Spektroskopie, Akademie der Wissenschaften der DDR, DDR - 1199 Berlin
Received 21 December 1978 The influence of the optical anisotropy induced by a circularly polarized saturating beam on a counterpropagating weak probe wave was studied in the case of strong saturation. Characteristic changes of the line shape were observed that can be used for a simple determination of saturation intensities.
1. Introduction When an inhomogeneously broadened transition is saturated by an intense monochromatic beam, for a weak counter-propagating probe wave of the same frequency we can observe a very narrow transmission peak (inverted Lamb dip) in the centre of the Doppler profile, see e.g. [11. In the polarization variant both beams are polarized in a different way. Here also the polarization of the weak wave, in general, is changed. The deviations from the initial polarization can be measured very sensitively; so the polarization detection technique allows us to obtain a much higher signal-tobackground ratio compared with the ordinary Lamb dip methods, further it is possible to get more informations about the parameters of the molecular transition. In the present work we will use a circularly polarized saturating beam and a linearly polarized probe [2]. (Regarding the variant with a linearly polarized saturating wave, see [ 3 - 5 ] . ) The corresponding scheme is outlined in fig. 1. Due to the optical anisotropy induced by the strong wave the probe polarization experiences a rotation of its plane (circular birefringence) and a small ellipticity (circular dichroism) in the vicinity of the line centre. The role o f these effects can be investigated measuring the angular dependence o f the intensity, transmitted through an analyzer. If the angle, by which the analyzer is rotated from the perpendicular position with respect to the initial probe polarization, is denoted by 0, for a single transition with the unsaturated absorption coefficient a0 the detected in-
beamsptitter tunable laser polarizer probe beamt
soturati beam n9 absorptio~cell n[ L
analyzerdetector
Fig. 1. Scheme of the polarization spectroscopic arrangement (L is the length of the absorption cell, I o the probe intensity before passing the cell) tensity I can be subdivided into three terms ( ~ is the detuning of the laser frequency from the transition frequency): I = I 0 e - s ° L { [ I + L ( ~ ) o~0L] sin20 +D(~2)aoL sin 20 + P ( ~ ) (a0L)2}.
(1)
[l+L(g2) a 0 L ] sin20 is maximal for 0 = n/2 and describes the inverted Lamb dip on the Doppler background. With decreasing 0 the signal also decreases and the antisymmetric contribution D(g2), caused by the birefringence, becomes more important. For perfectly crossed polarizers (0 = 0) we obtain a very small symmetric signal P ( ~ ) which is simultaneously effected by dichroism and birefringence, no background appears if we consider only ideal polarizers. If the saturation parameter X (averaged over the degenerate sublevels o f the transition) is small, the po51
Volume 29, number 1
OPTICS COMMUNICATIONS
larization effects can be calculated in a perturbation treatment [2]:
L(a)- -
l+d 1 e %L r2 y -- %L - X -r 2 + a - 2
D(~)
1
April 1979
M-I
M+I
+o(x),
M
P(a)
=
1 -e - a ° L
PgZ
_
_ ( 1 - d) 2 (1 e - a ° L ) 2 X2 . . . . . . - - i 6
-
--
(%L)2
-
P2 + o(X2). (2) r2 +a2
where P is the homogeneous half-width, we suppose, that all relaxation constants (phase and population relaxation) are equal. The parameter d, introduced by Wieman and tt/insch [2], characterizes the optical-induced anisotropy:
(2J- 1)(2J+3)
--~ "~ 2+9.]- - -
d
.(,_J
for J+-+ J transitions,
_J+l)
] _ 2J2+3 [ 2 ( 6 J 2 1)
(3) for J +-+ J - 1 transitions.
( J is the rotational quantum number of a resonant level). From (3) it is seen that the polarization technique can be used to distinguish P, R from Q branches and to determine the degeneracy of molecular levels, i f J is not too large. So we identified a Q(5) transition in ammonia [61. In the low pressure region (some mTorr in the absorption cell) it can be possible to achieve such high saturation effects, where the perturbation theory is no longer valid. In sect. 2 we will study tire change of the polarization signal for stronger saturation. In sect. 3 we will given the results of our experiments confirming qualitatively our theoretical investigations.
2. Theory of polarization line shape for strong saturation Some years ago much work was done regarding the effects of power broadening and dynamic Stark splitting on the absorption coefficient of a probe wave in two- and three-level systems [7 10]. We will apply these theoretical methods to the calculation of polarization effects of transitions with arbitrary degeneracy. Orientating the spatial quantization axis along the di52
M÷2
Fig. 2. Sublevels and subtransitions important for the interaction with the light beams if the molecule is pumped in the state M. rection of light propagation we assume that after pumping of the ground sublevel with the quantum number M the strong field (s) saturates the M *-+ M + 1 transition. The probe polarization can be decomposed into two components rotating in the same (+) and the opposite ( ) sense compared to the saturating field. If we neglect any saturation effects of the probe, the investigation of the four-level system illustrated in fig. 2 is sufficient for our calculations. Assuming plane waves we can find the time-independent equations for the density matrix elements in the Wigner representation. The influence of the strong field can be considered exactly. Then the macroscopic dipole moment with respect to the probe colnponents can be calculated after integration over all possible velocities of the molecules in the Doppler limit. These standard procedures are explained in more detail, e.g. in [8]. The saturation-dependent absorption coefficient ~(-+M)and refractive index n(+M) of the two circularly polarized probe components are defined as follows: +
+
a(M ) = Re A (m),
n(-+M) = (C/2Co)hn A~M),
(4)
where +
"i-
(5)
A(M ) = CtO(M)
and
X/i~-XM(1 +3X/-i+XM+4i~2/F ) A The saturation parameter XM of the M ~ M + I subtransition is expressed as XM = I/~M,M+I[2 l(S)/h2F2 ,
(7)
Volume 29, number 1
OPTICS COMMUNICATIONS
where l(S) is the intensity of the saturating beam. The quantity #i./is the transition dipole nroment between the lower sublevel with the magnetic q u a n t u m number i and the upper one]. Now we must average the expressions (4) and (7) over the magnetic sublevels assuming that the pumping of the ground state is an isotropic process. We get
M
April 1979
saturating field. On the other hand, tile transition moments of tire opposite polarization component increase with raising magnetic quantum numbers. As the result of this behaviour after summation over M the absorption dip of the ( ) component is somewhat smaller and narrower than the corresponding (+) one. So for sufficient saturation in the line centre an additional mininmm can arise in the frequency dependence of the dichroism. These specific saturation phenomena
and L[O)
M
0.5
The quantity n + - n - describes the induced birefringence and ~+ ~ - the dichroism. We have numerically calculated the terms L ( ~ ) , D(~2) and P ( ~ ) for arbitrary saturation and rotational quantum numbers of the resonant levels assuming that the relation % L< I
(9)
is fulfilled. Now we will analyze some interesting saturation phenomena in P and R branches. Here the dipole moment I~M,M+II decreases with increasing M (see fig. 3). As a consequence the Bennett holes "burned" into the velocity distribution of the lower sublevels and the corresponding peaks of the upper levels become broader with decreasing M due to the power broadening of the
_•-
)(=20 X=15 ~X=IO
JL -12
-10
-8
-6
-4
2
-2
Z,
6
8
10
r
12
o(a)
(b)
,,-~#111/ -12
-10
~"
~
4
6
x=ls 8
10
12
J~"
ot ÷ - o c
P
\
\ ',
', ~
',,_,
~
"V4
n.\ f . ~ n*
n + - n-
/, \ \ ~ \~,:,s / \ \ \ \ , , , " x '~:'° -
",,(_,
Fig. 3. Level scheme of a P(2) transition. The induced nonequilibrium velocity distributions are outlined. The (+) component interacts with the strongly saturated transitions more effectively than the (-) one. On the right side the different shape of the corresponding absorption coefficients and refractive indexes is illustrated.
..9_
-12
-10
-8
-6
-4
-2
2
4
6
8
10
12
F"
Fig. 4. Calculated line shape of the terms L (s2) (a), D (I2) (b) and P(s2) (c) for a transition between the momentum quantum numbers 20 ,--. 21 for five different saturation parameters 53
Volume 29, number 1
OPTICS COMMUNICATIONS
can also lead to a special structure of the birefringence curve as it is shown in fig. 3. As an example let t,s look at the x-dependence of the computed line shape for a R(20) or P(21) transition (see figs. 4a, b, c): If 0 = rr/2, we observe the inverted Lamb dip which increases and broadenes with increasing intensity l(S). For very high saturation the width is proportional to the square root of the saturation parameter. Due to tile d y n a m i c Stark effect, even for X--* oo we obtain a finite probe absorption, here L(~2 = 0) approaches ~ 0 . 6 2 3 . Although the saturation resonance is not a korentzian, the profile does not show significant particularities (fig. 4a). Rotating the analyzer we can observe the antisymmetric term D(~2). We can see that for larger saturation parameters the line shape deviates frtm~ a dispersion curve: with raising X in the line centre the first derivative decreases, for X ~ 15 it is zero and then changes its sign (fig. 4b). For a crossed configuration (0 = O) in the case X > 4 in the centre an additional dip appears whose depth increases with increasing intensity of the saturating beam (fig. 4c). If we define the two characteristical saturation parameters X1 and X2 as follows: aD(~21Xl )] i ch~ it~ =0
-
-
OS~2
=0,
(10)
tz=O
Table 1 Dependence of the characteristical saturation parameters, defined by (10), on the rotational quantum numbers. For P(I) and R (0) lines no special structure in the line centre appears Transition 0~l
~ 2~ 3~ 4~ 5~
x1 -
-
5.12 4.25 4.12 4.08 4.06
10 *--, II
15.21
4.04
20 ~ 21
15.08
4.03
50 ~ 51
15.04
4.03
100 ~ 101
15.03
4.02
54
2 3 4 5 6
x2
273.8 21.24 17.22 16.17 15.74
1
we see from the table 1 that with increasing degeneracy of the molecular transition these values rapidly converge to the limits 15 and 4, respectively, for J 2> 10 the deviations are 1% and smaller.
3. Experimental
results
In our experiments we used an arrangelnenl similar to that outlined in fig. 1. Utilizing a line-tunable ( ' 0 2 lase~ emitting in the 10~m region we investigated tile polarization line shape of selected transitions in tile ~'3 band ill SF 6. The circular polarization of tile saturating beam was achieved with the help of a Fresnel rhomb. The strong beam was modulated by a 1 kllz chopper, and a lock-in amplifier was used to delecl only thal part of tile signal corresponding to the saturated absorption or dispersion. Both laser beams were extended to a beam diameter of 4 cm, only the cemral part of tile initial nearly Gaussian intensily profile was taken, so we could work approximately with a rectangular profile. (For gaussian intensity disl ributions Ihe results of sect. 2 must be modified by integration over the spread of possible saluration parametms, see e.g.
IIll.)
~)2P(~IX2)I =0,
April 1979
In fig. 5 tire nreasured strong saturation polarization line shapes are given for the P(33) A 1 line of SF 6. Varying the laser o u t p u t and observing the line shape we were able to fix the characteristic saturation parameters ( 1 0 ) X I and X2 with an accuracy of ~5~,~. ~[he asymmetry o f the line shape in fig. 5a can be explained by the unnegligible c o n t r i b u t i o n of the term L(XZ)sin20 for 0 = 3 ° . Due to the finite extinction values of our polarizers we obtained a relatively high background of about 2 X 10 -5 10 in tire crossed configuration (0 = 0). Although the observed experimental line shapes are in a good qualitative agreement with the predicled ones of sect. 2, we can see sonle minor quantitative discrepancies. The measured relation X1 was 3. I . (Tile calculated value Xl/X2 is 3.75.) But these deviations are understood if we notice that for p = 8 mTorr we have significant smearing of our spectral profile which arises from the finite crossing angle of 10' between the counterpropagating beams and from the unresolved magnetic hyperfine structure which is spread over a range of about 60 kHz [12]. Furthermore, the finite c o n t r i b u t i o n of the inverted Lamb dip at 0 = 0 {background due to real polarizers) leads to an increase of
IX2
Volume 29, number 1
OPTICS COMMUNICATIONS 1 MHz
X =3/,,1
4. Discussion
X= 1.8 X 1
(a) 1 MHz
X= 0.5 X 2
X= 1.75X 2
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X=t,% 2
(b) Fig. 5. Observed line shapes for two saturation parameters if 0 = 3° (a) and three different intensities ifO = 0 (b). The investigated transition was the P(33) A 1 line of the v 3 band of SF6, the pressure wasp = 8 mTorr. the observable value X2. Finally for the used absorption cell of 120 cm length the relation (9) is not fulfilled very well, so our results are influenced by the inhomogeneity of the saturation parameter due to the absorption of the saturating beam along its direction of propagation. Now our aim is to consider all the enumerated effects in a numerical calculation o f the expected line profiles. Let us note that recently Schieder [13] published results about a special polarization variant using a Jamin interferometer. Investigating molecular iodine with the help o f a dye laser the author observed similar saturated dispersion and absorption line shapes as we found (fig. 5a, b). We suppose that those results are caused by the same reasons discussed above in sect. 2.
We have found theoretically and experimentally that in the regime of strong saturation, when power broadening and dynamic Stark splitting must be taken into account, there exist significant changes of the line shape detected under (nearly) crossed polarizers compared with the perturbation treatment. We could accurately fix such intensities of the strong beam where additional maxima or minima are created in the registrated line profile of the probe. So the polarization method allows us to determine the relations between the saturation intensities o f different P and R branch transitions in a simple way. This rechnique should be also valuable for the measurement of pressure dependences o f saturation intensities. The absolute values o f the saturation parameters X could be determined considering the special beam geometry, etc. Our calculations showed that for 1 ~< X ~< 50 the J-dependence of the line shapes is relatively weak. However for X ~> 500 this dependence is essentially stronger (cx l/J), so the regime of very strong saturation should be more useful for the determination of not too high J-values than the small saturation limit in which the polarization effects are proportional to l/J 2 (3). Q branch transitions were investigated numerically. According to (3) for X '~ 1 in the limit of very large momentum quantum numbers the optically induced anisotropy vanishes. But the different role of dynamic Stark splitting for (+) and ( - ) components (cf. (5) and (6)) leads to a remaining birefringence and dichroism with a frequency dependence similar to that of highly saturated P and R lines. We have also studied theoretically the case of a difference between the various relaxation constants. A detailed analysis o f these effects would be beyond the scope of the present work, we only note that the characteristic saturation parameters Xl and X2 ( I 0 ) depend on the relations between the relaxation constants. Finally it can be shown that in the variant of a linearly polarized saturating beam stronger saturation effects can also lead to additional dips or peaks of the resonance signal this will be studied in a subsequent paper.
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Volume 29, number 1
OPTICS COMMUNICATIONS
References [ 1 ] V.S. Letokhov and V.P. Chebotayev, Nonlinear laser spectroscopy, Springer Series in Optical Sciences, Vol. 4 (Springer Berlin, Heidelberg, New York, 1977). [2] C. Wieman and T.W. H/insch, Phys. Rev. Lett. 36 (1976) 1170. [3] M. Sargent III, Phys. Rev. A14 (1976) 524. J41 J.C. Keller and C. Delsart, Optics Comm. 20 (1977) 147; J. Appl. Phys. 49 (1978) 3662. I51 R. Fischer, V. Stert and E. Meisel, Appl. Phys. 17 (1978) 151. J6] V. Stert, R. Fischer, E. Meisel and H.-H. Ritze, Kvantovaya Electronica (Soy. Journal) 4 (1977) 2620.
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[71 tt. Paul, Fortschritte der Physik 22 (1974) 1. [8] E.V. Baklanov and V.P. Chebotayev, Zh. Eksp. Teor. l.iz. 60 (1971) 552. [9] S. Haroche and F. tiartmann, Phys. Rev. A6 (1972) 1280. [10] N. Skribanowitz, M.J. Kelly and M.S. Feld, Phys. Rev. A6 (1972) 2302. J 11 ] B. Couillaud, A. Ducasse and A. Dienes, Appl. Phys. 16 (1978) 359. [12] Ch. J. Bord~, M. Ouhayoun and J. Bord6, J. Mol. Spectr. 73 (1978) 344. [13] R. Schieder, Optics Comm. 26 (1978) 113.