- Email: [email protected]
Theory of laser beam propagation through a vapor cell in the collision-free optical pumping regime
15 February 2000
Optics Communications 175 Ž2000. 227–231 www.elsevier.comrlocateroptcom
Theory of laser beam propagation through a vapor cell in the collision-free optical pumping regime I.E. Mazets b
a,b,)
, L.B. Shifrin
c
a A.F. Ioffe Physico-Technical Institute, 194021 St. Petersburg, Russia Institut fur ¨ Experimentalphysik, Technische UniÕersitat ¨ Graz, 8010 Graz, Austria c St. Petersburg State Technical UniÕersity, 195251 St. Petersburg, Russia
Received 19 August 1999; received in revised form 9 November 1999; accepted 13 December 1999
Abstract We present a theory of a resonant laser light propagation through a vapor cell in a case characterized by the two key features: Ži. atoms undergo optical pumping into particular sublevels of their ground state, Žii. vapor density is low enough to avoid relaxation of these sublevels caused by interatomic collisions. q 2000 Elsevier Science B.V. All rights reserved. PACS: 32.80.Bx; 42.25.Bs Keywords: Optical pumping; Light propagation
Many atomic isotopes being a subject of laser spectroscopic studies have ground state hyperfine structure, and a resonant excitation of these isotopes by a laser radiation leads to optical pumping w1x. In a case of an optically thick medium, optical pumping results in a bleaching of the medium and in a breakdown of the usual exponential light absorption law even for light intensities far below the limit of the optical transition saturation. To provide reliable quantitative measurements in optically dense gaseous samples, one needs a theory of laser light propagation in the regime of optical pumping. Such a theory was developed in numerous works w2–4x. However, a common feature of all these studies is an assumption
)
Corresponding author. E-mail: [email protected]
of large longitudinal relaxation which, together with radiative processes, establishes a stationary distribution of atoms over both sensitive and insensitive to the laser radiation sublevels. Thus the medium response is calculated using the steady-state solution of the density matrix equations, and the light is approximated by a plane wave. However, this approach becomes invalid in the case of light propagation through an uncoated cell containing only a low-pressure vapor of an absorbing substance and no buffer gas. Atoms entering the region of interaction with the resonant light are totally thermalized with respect to their translational and internal degrees of freedom. They move across the region by straight trajectories, since the interatomic collision rate is negligible. An atom crossing the laser beam is quickly pumped into the non-interacting state and does not contribute to the absorption afterwards. It is obvious that the medium absorption
0030-4018r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 0 0 . 0 0 4 5 8 - 2
228
I.E. Mazets, L.B. Shifrinr Optics Communications 175 (2000) 227–231
coefficient must be spatially non-uniform under these conditions – the probability for a photon to be absorbed is sufficiently higher at the beam periphery than near the beam axis. Moreover, the absorption coefficient is non-local, since it is determined by the number of atoms not pumped out of the resonant sublevels and, hence, still participating in the absorption process. This number varies from point to point inside the laser beam, strongly depending on the intensity distribution in the beam edges. So the transverse profile of the beam becomes a key dynamical parameter of the problem. A transverse beam inhomogeneity is essential for the beam self-focusing w5x. In theoretical studies of optical pumping, a beam profile variation has also been considered with respect to the self-focusing problem w6x, but again the atomic motion has been assumed to be collision-dominated, and the steadystate atomic density matrix has been used. To our knowledge, a straightforward theory of light propagation through a collisionless gas in the optical pumping regime is still lacking. In the present paper we fill up this gap. Despite the complexity of the problem, the solution obtained under reasonable assumptions is quite transparent from the physical point of view. As a starting point for our theoretical considerations, we choose a model of a medium consisting of atoms with an effective three-level scheme of excitation. The first level is associated with the ground state component called ‘interacting state’. This state is coupled by the laser radiation to the optically excited state. The latter state decays to the first level as well as to the level which is not affected by the laser radiation Ž‘non-interacting state’.. The full width of the excited state is 2g , and the partial probability of a decay to the non-interacting state is n Ž0 - n - 1.. The temperature of the gas is determined by the temperature of the cell walls. In particular, the probability to find the atom in a given internal state immediately after collision with the wall is determined by the Boltzmann’s distribution. We denote the equilibrium number density of atoms in the first level by N1 and the total number density by N. Thermalization of the internal atomic degrees of freedom due to wall collisions is essential since it prevents accumulation of all the atoms contained in
the cell into the non-interacting state. However, we assume that the laser beam diameter is smaller than that of the cell. Therefore, the wall collisions are separated in space and time from the optical pumping process and, hence, the atoms are not in a steady state in the region of absorption. Naturally, the recoil effect due to photon scattering is neglected because of great difference between the velocities of recoil Ž; 10 cmrs. and thermal motion Ž; 10 5 cmrs.. If n were equal to zero, the propagation equation would read
E Ez
I Ž x , y, z . s yN1 s ² b Ž Õz . :I Ž x , y, z . ,
Ž 1.
where I is the monochromatic light intensity which is assumed to be below the limit of saturation of the optical transition, z is the light propagation direction and x, y are the transversal coordinates, s is the photon absorption cross-section at the line center, b Ž Õz . is the line shape. We assume, for the sake of simplicity, that the laser frequency is exactly in resonance with the optical transition Žit this case the self-focusing is absent w5x.. Then
b Ž Õz . s
1 1 q Ž kÕzrg .
2
,
k and v s ck are the laser radiation wavenumber and frequency, respectively. The angular brackets in Eq. Ž1. and hereafter denote averaging over the Maxwellian velocity distribution Ž2 p Õth2 .y 3r2 expwyŽ Õ x2 q Õ y2 q Õz2 .rŽ2 Õth2 .x. Of course, the light beam diffraction is neglected in Eq. Ž1., since even for the light beam with radius of order of 1 mm the characteristic diffraction length Ž; 1 m. far exceeds the cell length L. Eq. Ž1. must be supplied with a boundary condition, I Ž x, y,0. s I0 Ž x, y .. Because the inhomogeneous broadening parameter h s kÕth rŽ pr2 g . is much greater than unity, Eq. Ž1. reduces to the elementary equation
'
E Ez
I Ž x , y, z . s y
N1 s
h
I Ž x , y, z . ,
I.E. Mazets, L.B. Shifrinr Optics Communications 175 (2000) 227–231
from which the well-known Beer’s law follows: I s I0 expŽyt .. The quantity
229
the intensity does not depend on y. The explicit form of Eq. Ž3. now reads
E t s zrl
Ž 2.
Ez
is called optical density. Here l s hrŽ N1 s . is the Beer’s length. Now we turn to the case of optical pumping Ž n ) 0.. Let us consider a certain spatial point Ž x, y, z . inside the laser beam. We should take into consideration the probability P1 for an atom to remain in the first state during its flight from the bulk of the gas, where the radiation is absent, to the point Ž x, y, z .. Thus Eq. Ž1. must be generalized as follows:
IŽ x, z. `
s yN1 s
Hy`
H0 dÕ
x
I Ž x , y, z . s yN1 s ² b Ž Õz . P1 :I Ž x , y, z . .
Ž 3.
P1 s exp y
=
"v
0
Hy` I Ž x y Õ t , y y Õ t , z y Õ t . d t x
y
z
.
P1 s exp y
nsb Ž Õz . "v
`
H dx Õ "v x
Hy`d t I Ž x y Õ t , y y Õ t , z . x
y
.
Ž 5. Now we consider certain specific geometries of the problem. The one, most convenient for analytical calculations, is a plane beam configuration, where
I Ž xX , z .
x
x
H dx Õ " v y`
X
I Ž xX , z .
/
/
IŽ x, z. .
For a further simplification, we assume an additional symmetry: I Ž x, z . s I Žyx, z .. Now, integrating over x from 0 to `, we get the following equation for the quantity uŽ z . s Ž " v .y1H0`d x I Ž x, z . proportional to the total radiation power per unit length in y-direction: sy
N1
exp yÕz2r Ž 2 Õth2 .
`
H dÕ n y`
H0 dÕ
(2p Õ
2 th
Õ x exp yÕ x2r Ž 2 Õth2 .
`
=
z
x
(2p Õ
2 th
= 1 y exp Ž y2 nsb Ž Õz . u Ž z . rÕ x . .
Ž 7. The integral H0`d s s expŽys 2r2. w1 y expŽyjrs .x can be approximated, better than with a 15% accuracy, by the rational function jrŽ 2rp q j . having the necessary asymptotics at j 0 and j `. Such an approximation, together with the use of the condition
™
'
0
X
x
dz
A presence of nonlocality in z could make the problem almost intractable. However, for relatively low atomic densities, a characteristic scale of intensity change in z-direction Žthe Beer’s length. is much greater than the product of the thermal velocity of atoms and the average duration of an optical pumping cycle. It allows us to make a significant simplification of Eq. Ž4.:
nsb
Ž 6.
d uŽ z .
Ž 4.
2 th
nsb
ž
nsb Ž Õz .
2 th
(2p Õ
ž
qexp y
P1 is determined by a time integral of optical pumping rate values experienced by an atom during its motion along a straight trajectory:
(2p Õ
=b Ž Õz . exp y
E Ez
exp yÕz2r Ž 2 Õth2 .
exp yÕ x2r Ž 2 Õth2 .
`
=
dÕz
ns u Ž z . rÕth < h 2
™
Ž 8.
Žthe opposite case is out of the scope of the present paper, because it takes place for intensities large
I.E. Mazets, L.B. Shifrinr Optics Communications 175 (2000) 227–231
230
enough to saturate the optical transition., allows us to reduce Eq. Ž7. to the following simple form du u sy Ž 9. '1 q Au . dt Here uŽ z . s uŽ z .ruŽ0. and A s '2p ns J0 RrÕth .
`
H0 d x I Ž x ,0. rI Ž 0,0. .
Ž 11 .
Eq. Ž9. is formally identical to the differential equation describing light propagation in a two-level medium taking into account saturation of the optical transition w7x and can be easily solved in the form of quadrature: ln
'A q 1 y 1 ' 'A q 1 q 1 q 2 A q 1 'Au q 1 y 1 ' y ln 'Au q 1 q 1 y 2 Au q 1 s t .
ž
/
ž
I Ž r ,0 . s Imax exp y Ž rrR 0 .
2
.
Ž 14 .
Ž 10 .
The optical density t is defined by Eq. Ž2., as usual. Also we introduce the photon flux density in the innermost part of the beam at the medium entrance, J0 s I Ž0,0.rŽ " v ., and the effective transversal size of the beam, Rs
allow simple analytical solution. We solved this integro-differential equation numerically. The initial condition was taken in the form of a Gaussian incident beam,
/
The total incident power is, obviously, U0 s p R 20 Imax . After numerical results had been obtained, we tried to fit them by a formula like Eq. Ž12.. The dimensionless transmitted intensity H0` d r 2 p r =I Ž r, z .rU0 was taken as the variable uŽ z .. The question is, which quantity should be used as R in Eq. Ž10. Žthe definition J0 s ImaxrŽ " v . was adopted in the case of axial symmetry.. With the choice R s 2rp R 0 an excellent agreement between the results of numerical integration and the analytical approximation is obtained for various sets of the system parameters. It is illustrated in Fig. 1. A situation when different magnetic sublevels of a given hyperfine component are excited simultaneously is of a great practical interest. The degeneracy of the hyperfine component possessing the total an-
'
Ž 12 .
In a real experiment one often deals with an axially symmetric laser beam. In this case I s I Ž r, z ., where a cylindric system of coordinates Ž r, w , z . is introduced. Similarly to Eq. Ž6., we derive the following equation: E IŽ r, z. Ez ` N1 s 2p sy da dÕz 3r2 0 y` Ž 2p Õth .
H
`
=
H0 dÕ
r
Õr exp y Ž Õr2 q Õz2 . r Ž 2 Õth2 .
=b Ž Õz . exp y `
H0 d z I ž (r
=
H
2
nsb Ž Õz . " v Õr
q z 2 y 2 rz cos a , z
/
IŽ r, z. .
Ž 13 . The right hand side of Eq. Ž13. has a structure which, unlike the case of a plane beam, does not
Fig. 1. Logarithm of the transmitted intensity versus optical density in a case of axially symmetric beam for different sets of parameters Ss Ž n , kR 0 , s Imax rg . Žnumerical results.. 1. As 1.50, squares: Ss Ž0.5, 2P10 4 , 0.01., circles: Ss Ž0.25, 4P10 4 , 0.01.; 2. As9.01, up triangles: Ss Ž0.5, 2P10 4 , 0.06., stars: Ss Ž0.5, 4P10 4 , 0.03.; 3. As18.92, down triangles: Ss Ž0.7, 2P10 4 , 0.09., crosses: Ss Ž0.7, 6P10 4 , 0.03.. The thermal velocity Õth s133g r k corresponds to a sodium vapour at 400 K. Solid lines represent an analytical approximation described in the text. Dashed line shows the Beer’s law, for comparison.
I.E. Mazets, L.B. Shifrinr Optics Communications 175 (2000) 227–231
gular momentum F is g F s 2 F q 1. Hence, the equilibrium number density of atoms in this hyperfine component is NF s Ng FrÝ F g g F g Žthe sum is taken over all the components of the ground state, N is the total atomic density.. Such a quantity will replace N1 in further considerations. The index j is used to distinguish between all possible transitions excited by the laser radiation with well-defined polarization, i.e., sj is the photon absorption crosssection for the j-th transition, and n j is the probability for the atom to decay into any of the non-interacting states after the excitation on this transition. Taking a sum over all the transitions and dividing by the degeneracy g F , we get the averaged cross-section
ss
1 gF
Ý sj .
Ž 15 .
sj
1
gF s
(1 q A u
du s yu Ý j
(
(
u
sy dt
,
Ž 16 .
A eff
(
,
where A eff s '2p neff s J0 RrÕth , and 1
1 s
(n
eff
Acknowledgements
gF
Ý j
(
This work is supported by the Austrian Science Foundation under the project P 12 894–PHY. We are grateful to Prof. L.Windholz, Dr. E.Korsunsky, and all the members of the Quantum Optics Group of the Institut fur ¨ Experimentalphysik, TU Graz, for helpful discussions and continuous support.
References
j
where A j s '2p n j sj J0 RrÕth . If the intensity is large enough, 1 q uA j f uA j for all j’s. In this case the sum in Eq. Ž16. can be taken easily, and the propagation equation reads du
In other words, for a sufficiently large intensity a multilevel medium can be approximated by a threelevel one, with the number density of atoms in the interacting state N1 s NF outside the beam, the absorption cross-section defined by Eq. Ž15., and the probability of optical pumping in a single resonant fluorescence cycle given by Eq. Ž17.. Of course, such an approximation does not take into account a fine effect of redistribution of atoms, due to upper state relaxation, not only between the states affected by the light Ž‘interacting states’. and non-interacting ones, but also within the manifold of the interacting states. However, this effect does not change the main features of the process.
j
Defining the optical density as t s zrl, where l is the length of a e-fold decrease of intensity of a light beam weak enough to provide the Beer’s law applicability, we obtain that t s NF s zrh. To take into account various channels of transitions, we must substitute Eq. Ž9. by the following one: dt
231
sj sn j
.
Ž 17 .
w1x W. Happer, Rev. Mod. Phys. 44 Ž1972. 169. w2x M.B. Gornyi, D.L. Markman, B.G. Matisov, Pis’ma Zh. Tekhn. Fiz. 7 Ž1981. 1356; Optika i Spektrosk. 55 Ž1983. 36; Zh. Prikl. Spektrosk. 40 Ž1984. 110. w3x G.G. Adonts, V.M. Arutunyan, D.G. Akopyan, K.V. Arutunyan, M.V. Slobodskoy, J. Phys. B 22 Ž1989. 1103. w4x M.E. Wagshul, T.E. Chupp, Phys. Rev. A 40 Ž1989. 4447. w5x Y.R. Shen, ‘Principles of Non-Linear Optics’, NY: Wiley, 1984. w6x D.G. Akopyan, A.Zh. Muradyan, in: G.V. Klement’yev ŽEd.., Optical Orientation of Atoms and Molecules, N 2, Leningrad: A.F. Ioffe Phys. -Tech. Inst. Publishing Ž1990., p. 163 Žin Russian.. w7x L. Allen, J.H. Eberly, ‘Optical Resonance and Two-Level Atoms’, NY: Wiley, 1975.
Optics Communications 175 Ž2000. 227–231 www.elsevier.comrlocateroptcom
Theory of laser beam propagation through a vapor cell in the collision-free optical pumping regime I.E. Mazets b
a,b,)
, L.B. Shifrin
c
a A.F. Ioffe Physico-Technical Institute, 194021 St. Petersburg, Russia Institut fur ¨ Experimentalphysik, Technische UniÕersitat ¨ Graz, 8010 Graz, Austria c St. Petersburg State Technical UniÕersity, 195251 St. Petersburg, Russia
Received 19 August 1999; received in revised form 9 November 1999; accepted 13 December 1999
Abstract We present a theory of a resonant laser light propagation through a vapor cell in a case characterized by the two key features: Ži. atoms undergo optical pumping into particular sublevels of their ground state, Žii. vapor density is low enough to avoid relaxation of these sublevels caused by interatomic collisions. q 2000 Elsevier Science B.V. All rights reserved. PACS: 32.80.Bx; 42.25.Bs Keywords: Optical pumping; Light propagation
Many atomic isotopes being a subject of laser spectroscopic studies have ground state hyperfine structure, and a resonant excitation of these isotopes by a laser radiation leads to optical pumping w1x. In a case of an optically thick medium, optical pumping results in a bleaching of the medium and in a breakdown of the usual exponential light absorption law even for light intensities far below the limit of the optical transition saturation. To provide reliable quantitative measurements in optically dense gaseous samples, one needs a theory of laser light propagation in the regime of optical pumping. Such a theory was developed in numerous works w2–4x. However, a common feature of all these studies is an assumption
)
Corresponding author. E-mail: [email protected]
of large longitudinal relaxation which, together with radiative processes, establishes a stationary distribution of atoms over both sensitive and insensitive to the laser radiation sublevels. Thus the medium response is calculated using the steady-state solution of the density matrix equations, and the light is approximated by a plane wave. However, this approach becomes invalid in the case of light propagation through an uncoated cell containing only a low-pressure vapor of an absorbing substance and no buffer gas. Atoms entering the region of interaction with the resonant light are totally thermalized with respect to their translational and internal degrees of freedom. They move across the region by straight trajectories, since the interatomic collision rate is negligible. An atom crossing the laser beam is quickly pumped into the non-interacting state and does not contribute to the absorption afterwards. It is obvious that the medium absorption
0030-4018r00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 0 0 . 0 0 4 5 8 - 2
228
I.E. Mazets, L.B. Shifrinr Optics Communications 175 (2000) 227–231
coefficient must be spatially non-uniform under these conditions – the probability for a photon to be absorbed is sufficiently higher at the beam periphery than near the beam axis. Moreover, the absorption coefficient is non-local, since it is determined by the number of atoms not pumped out of the resonant sublevels and, hence, still participating in the absorption process. This number varies from point to point inside the laser beam, strongly depending on the intensity distribution in the beam edges. So the transverse profile of the beam becomes a key dynamical parameter of the problem. A transverse beam inhomogeneity is essential for the beam self-focusing w5x. In theoretical studies of optical pumping, a beam profile variation has also been considered with respect to the self-focusing problem w6x, but again the atomic motion has been assumed to be collision-dominated, and the steadystate atomic density matrix has been used. To our knowledge, a straightforward theory of light propagation through a collisionless gas in the optical pumping regime is still lacking. In the present paper we fill up this gap. Despite the complexity of the problem, the solution obtained under reasonable assumptions is quite transparent from the physical point of view. As a starting point for our theoretical considerations, we choose a model of a medium consisting of atoms with an effective three-level scheme of excitation. The first level is associated with the ground state component called ‘interacting state’. This state is coupled by the laser radiation to the optically excited state. The latter state decays to the first level as well as to the level which is not affected by the laser radiation Ž‘non-interacting state’.. The full width of the excited state is 2g , and the partial probability of a decay to the non-interacting state is n Ž0 - n - 1.. The temperature of the gas is determined by the temperature of the cell walls. In particular, the probability to find the atom in a given internal state immediately after collision with the wall is determined by the Boltzmann’s distribution. We denote the equilibrium number density of atoms in the first level by N1 and the total number density by N. Thermalization of the internal atomic degrees of freedom due to wall collisions is essential since it prevents accumulation of all the atoms contained in
the cell into the non-interacting state. However, we assume that the laser beam diameter is smaller than that of the cell. Therefore, the wall collisions are separated in space and time from the optical pumping process and, hence, the atoms are not in a steady state in the region of absorption. Naturally, the recoil effect due to photon scattering is neglected because of great difference between the velocities of recoil Ž; 10 cmrs. and thermal motion Ž; 10 5 cmrs.. If n were equal to zero, the propagation equation would read
E Ez
I Ž x , y, z . s yN1 s ² b Ž Õz . :I Ž x , y, z . ,
Ž 1.
where I is the monochromatic light intensity which is assumed to be below the limit of saturation of the optical transition, z is the light propagation direction and x, y are the transversal coordinates, s is the photon absorption cross-section at the line center, b Ž Õz . is the line shape. We assume, for the sake of simplicity, that the laser frequency is exactly in resonance with the optical transition Žit this case the self-focusing is absent w5x.. Then
b Ž Õz . s
1 1 q Ž kÕzrg .
2
,
k and v s ck are the laser radiation wavenumber and frequency, respectively. The angular brackets in Eq. Ž1. and hereafter denote averaging over the Maxwellian velocity distribution Ž2 p Õth2 .y 3r2 expwyŽ Õ x2 q Õ y2 q Õz2 .rŽ2 Õth2 .x. Of course, the light beam diffraction is neglected in Eq. Ž1., since even for the light beam with radius of order of 1 mm the characteristic diffraction length Ž; 1 m. far exceeds the cell length L. Eq. Ž1. must be supplied with a boundary condition, I Ž x, y,0. s I0 Ž x, y .. Because the inhomogeneous broadening parameter h s kÕth rŽ pr2 g . is much greater than unity, Eq. Ž1. reduces to the elementary equation
'
E Ez
I Ž x , y, z . s y
N1 s
h
I Ž x , y, z . ,
I.E. Mazets, L.B. Shifrinr Optics Communications 175 (2000) 227–231
from which the well-known Beer’s law follows: I s I0 expŽyt .. The quantity
229
the intensity does not depend on y. The explicit form of Eq. Ž3. now reads
E t s zrl
Ž 2.
Ez
is called optical density. Here l s hrŽ N1 s . is the Beer’s length. Now we turn to the case of optical pumping Ž n ) 0.. Let us consider a certain spatial point Ž x, y, z . inside the laser beam. We should take into consideration the probability P1 for an atom to remain in the first state during its flight from the bulk of the gas, where the radiation is absent, to the point Ž x, y, z .. Thus Eq. Ž1. must be generalized as follows:
IŽ x, z. `
s yN1 s
Hy`
H0 dÕ
x
I Ž x , y, z . s yN1 s ² b Ž Õz . P1 :I Ž x , y, z . .
Ž 3.
P1 s exp y
=
"v
0
Hy` I Ž x y Õ t , y y Õ t , z y Õ t . d t x
y
z
.
P1 s exp y
nsb Ž Õz . "v
`
H dx Õ "v x
Hy`d t I Ž x y Õ t , y y Õ t , z . x
y
.
Ž 5. Now we consider certain specific geometries of the problem. The one, most convenient for analytical calculations, is a plane beam configuration, where
I Ž xX , z .
x
x
H dx Õ " v y`
X
I Ž xX , z .
/
/
IŽ x, z. .
For a further simplification, we assume an additional symmetry: I Ž x, z . s I Žyx, z .. Now, integrating over x from 0 to `, we get the following equation for the quantity uŽ z . s Ž " v .y1H0`d x I Ž x, z . proportional to the total radiation power per unit length in y-direction: sy
N1
exp yÕz2r Ž 2 Õth2 .
`
H dÕ n y`
H0 dÕ
(2p Õ
2 th
Õ x exp yÕ x2r Ž 2 Õth2 .
`
=
z
x
(2p Õ
2 th
= 1 y exp Ž y2 nsb Ž Õz . u Ž z . rÕ x . .
Ž 7. The integral H0`d s s expŽys 2r2. w1 y expŽyjrs .x can be approximated, better than with a 15% accuracy, by the rational function jrŽ 2rp q j . having the necessary asymptotics at j 0 and j `. Such an approximation, together with the use of the condition
™
'
0
X
x
dz
A presence of nonlocality in z could make the problem almost intractable. However, for relatively low atomic densities, a characteristic scale of intensity change in z-direction Žthe Beer’s length. is much greater than the product of the thermal velocity of atoms and the average duration of an optical pumping cycle. It allows us to make a significant simplification of Eq. Ž4.:
nsb
Ž 6.
d uŽ z .
Ž 4.
2 th
nsb
ž
nsb Ž Õz .
2 th
(2p Õ
ž
qexp y
P1 is determined by a time integral of optical pumping rate values experienced by an atom during its motion along a straight trajectory:
(2p Õ
=b Ž Õz . exp y
E Ez
exp yÕz2r Ž 2 Õth2 .
exp yÕ x2r Ž 2 Õth2 .
`
=
dÕz
ns u Ž z . rÕth < h 2
™
Ž 8.
Žthe opposite case is out of the scope of the present paper, because it takes place for intensities large
I.E. Mazets, L.B. Shifrinr Optics Communications 175 (2000) 227–231
230
enough to saturate the optical transition., allows us to reduce Eq. Ž7. to the following simple form du u sy Ž 9. '1 q Au . dt Here uŽ z . s uŽ z .ruŽ0. and A s '2p ns J0 RrÕth .
`
H0 d x I Ž x ,0. rI Ž 0,0. .
Ž 11 .
Eq. Ž9. is formally identical to the differential equation describing light propagation in a two-level medium taking into account saturation of the optical transition w7x and can be easily solved in the form of quadrature: ln
'A q 1 y 1 ' 'A q 1 q 1 q 2 A q 1 'Au q 1 y 1 ' y ln 'Au q 1 q 1 y 2 Au q 1 s t .
ž
/
ž
I Ž r ,0 . s Imax exp y Ž rrR 0 .
2
.
Ž 14 .
Ž 10 .
The optical density t is defined by Eq. Ž2., as usual. Also we introduce the photon flux density in the innermost part of the beam at the medium entrance, J0 s I Ž0,0.rŽ " v ., and the effective transversal size of the beam, Rs
allow simple analytical solution. We solved this integro-differential equation numerically. The initial condition was taken in the form of a Gaussian incident beam,
/
The total incident power is, obviously, U0 s p R 20 Imax . After numerical results had been obtained, we tried to fit them by a formula like Eq. Ž12.. The dimensionless transmitted intensity H0` d r 2 p r =I Ž r, z .rU0 was taken as the variable uŽ z .. The question is, which quantity should be used as R in Eq. Ž10. Žthe definition J0 s ImaxrŽ " v . was adopted in the case of axial symmetry.. With the choice R s 2rp R 0 an excellent agreement between the results of numerical integration and the analytical approximation is obtained for various sets of the system parameters. It is illustrated in Fig. 1. A situation when different magnetic sublevels of a given hyperfine component are excited simultaneously is of a great practical interest. The degeneracy of the hyperfine component possessing the total an-
'
Ž 12 .
In a real experiment one often deals with an axially symmetric laser beam. In this case I s I Ž r, z ., where a cylindric system of coordinates Ž r, w , z . is introduced. Similarly to Eq. Ž6., we derive the following equation: E IŽ r, z. Ez ` N1 s 2p sy da dÕz 3r2 0 y` Ž 2p Õth .
H
`
=
H0 dÕ
r
Õr exp y Ž Õr2 q Õz2 . r Ž 2 Õth2 .
=b Ž Õz . exp y `
H0 d z I ž (r
=
H
2
nsb Ž Õz . " v Õr
q z 2 y 2 rz cos a , z
/
IŽ r, z. .
Ž 13 . The right hand side of Eq. Ž13. has a structure which, unlike the case of a plane beam, does not
Fig. 1. Logarithm of the transmitted intensity versus optical density in a case of axially symmetric beam for different sets of parameters Ss Ž n , kR 0 , s Imax rg . Žnumerical results.. 1. As 1.50, squares: Ss Ž0.5, 2P10 4 , 0.01., circles: Ss Ž0.25, 4P10 4 , 0.01.; 2. As9.01, up triangles: Ss Ž0.5, 2P10 4 , 0.06., stars: Ss Ž0.5, 4P10 4 , 0.03.; 3. As18.92, down triangles: Ss Ž0.7, 2P10 4 , 0.09., crosses: Ss Ž0.7, 6P10 4 , 0.03.. The thermal velocity Õth s133g r k corresponds to a sodium vapour at 400 K. Solid lines represent an analytical approximation described in the text. Dashed line shows the Beer’s law, for comparison.
I.E. Mazets, L.B. Shifrinr Optics Communications 175 (2000) 227–231
gular momentum F is g F s 2 F q 1. Hence, the equilibrium number density of atoms in this hyperfine component is NF s Ng FrÝ F g g F g Žthe sum is taken over all the components of the ground state, N is the total atomic density.. Such a quantity will replace N1 in further considerations. The index j is used to distinguish between all possible transitions excited by the laser radiation with well-defined polarization, i.e., sj is the photon absorption crosssection for the j-th transition, and n j is the probability for the atom to decay into any of the non-interacting states after the excitation on this transition. Taking a sum over all the transitions and dividing by the degeneracy g F , we get the averaged cross-section
ss
1 gF
Ý sj .
Ž 15 .
sj
1
gF s
(1 q A u
du s yu Ý j
(
(
u
sy dt
,
Ž 16 .
A eff
(
,
where A eff s '2p neff s J0 RrÕth , and 1
1 s
(n
eff
Acknowledgements
gF
Ý j
(
This work is supported by the Austrian Science Foundation under the project P 12 894–PHY. We are grateful to Prof. L.Windholz, Dr. E.Korsunsky, and all the members of the Quantum Optics Group of the Institut fur ¨ Experimentalphysik, TU Graz, for helpful discussions and continuous support.
References
j
where A j s '2p n j sj J0 RrÕth . If the intensity is large enough, 1 q uA j f uA j for all j’s. In this case the sum in Eq. Ž16. can be taken easily, and the propagation equation reads du
In other words, for a sufficiently large intensity a multilevel medium can be approximated by a threelevel one, with the number density of atoms in the interacting state N1 s NF outside the beam, the absorption cross-section defined by Eq. Ž15., and the probability of optical pumping in a single resonant fluorescence cycle given by Eq. Ž17.. Of course, such an approximation does not take into account a fine effect of redistribution of atoms, due to upper state relaxation, not only between the states affected by the light Ž‘interacting states’. and non-interacting ones, but also within the manifold of the interacting states. However, this effect does not change the main features of the process.
j
Defining the optical density as t s zrl, where l is the length of a e-fold decrease of intensity of a light beam weak enough to provide the Beer’s law applicability, we obtain that t s NF s zrh. To take into account various channels of transitions, we must substitute Eq. Ž9. by the following one: dt
231
sj sn j
.
Ž 17 .
w1x W. Happer, Rev. Mod. Phys. 44 Ž1972. 169. w2x M.B. Gornyi, D.L. Markman, B.G. Matisov, Pis’ma Zh. Tekhn. Fiz. 7 Ž1981. 1356; Optika i Spektrosk. 55 Ž1983. 36; Zh. Prikl. Spektrosk. 40 Ž1984. 110. w3x G.G. Adonts, V.M. Arutunyan, D.G. Akopyan, K.V. Arutunyan, M.V. Slobodskoy, J. Phys. B 22 Ž1989. 1103. w4x M.E. Wagshul, T.E. Chupp, Phys. Rev. A 40 Ž1989. 4447. w5x Y.R. Shen, ‘Principles of Non-Linear Optics’, NY: Wiley, 1984. w6x D.G. Akopyan, A.Zh. Muradyan, in: G.V. Klement’yev ŽEd.., Optical Orientation of Atoms and Molecules, N 2, Leningrad: A.F. Ioffe Phys. -Tech. Inst. Publishing Ž1990., p. 163 Žin Russian.. w7x L. Allen, J.H. Eberly, ‘Optical Resonance and Two-Level Atoms’, NY: Wiley, 1975.