J. Phys. B: At. Mol. Phys. 20 (1987) 5939-5952. Printed in the UK

Forward scattering of a strong non-monochromatic laser light

W Gawlik and J Zachorowski Instytut Fizyki, Uniwersytet Jagiellonski, ul. Reymonta 4, 30-059 Krakow, Poland

Received 23 March 1987, in final form 24 June 1987

Abstract. The experiment of forward scattering of a resonant intense a n d broad-band (of the order of 1 G H z ) laser light on sodium atoms is described. The experimental results for the D, and D, sodium lines ( the 32S,,2-32P,,2 and 32S,,,-32P,,, transitions) are compared with those of the scattering of the light from a conventional spectral lamp and from a monochromatic laser. The observed differences are explained taking into account the Zeeman coherences created in the irradiated atoms a n d the concept of the ‘hyperfine uncoupling’ which is proposed for describing the interaction of a toms with broad-band laser light.

1. Introduction

In an experiment on resonant light scattering in the forward direction we study the part of the light which has interacted with atoms (or other scattering centres) but has changed neither frequency nor direction of propagation. To distinguish the scattered light from the transmitted light we exploit the fact that they can differ in phase and/or in polarisation and we use the technique introduced by Corney et a1 (1966), i.e. we place a cell with scattering atoms between crossed linear polarisers in a longitudinal magnetic field. In this set-up the light transmitted without scattering is blocked by the second polariser. The scattered light, on the other hand, may have different polarisation due to birefringence and dichroism induced by the magnetic field in the scattering medium and also, which is of particular interest to us, due to a possible coherence between atomic sublevels. It can, therefore, pass the second polariser and be detected. What reaches the detector is a coherent superposition of waves scattered by atoms in random positions. The reason is that in forward scattering all optical paths are equal and the frequency of the forward scattered light is not Doppler shifted with respect to the incident light. Thus, N scattering atoms yield the forward-scattered intensity proportional to N 2 . In lateral scattering this coherence is lost due to random positions of atoms and the signal is proportional to N (Laloe 1971).

We record the intensity of the forward-scattered light against the magnetic field intensity at a fixed resonant frequency of the laser. This dependence will be called the forward scattering signal.

The first papers on forward scattering dealt with the scattering of light from classical sources, i.e. spectral lamps (Fork and Bradley 1964, Corney et a1 1966, Kr6las and Winiarczyk 1972, Durrant and Landheer 1971). The signals obtained were broad curves of the inverted bell shape with zero value at zero magnetic field and with a width usually of the order of a hundred Gauss. These results were interpreted in a linear (first-order perturbation) theory in terms of birefringence and dichroism. Birefringence is responsible for the well known Faraday effect, i.e. rotation of a polarisation plane in the magnetic field. It results from the difference between the refraction indices for

0022-3700/87/215939 + 14$02.50 @ 1987 IOP Publishing Ltd 5939

5940 W Gawlik and J Zachorowski

U, and U- polarised light caused by the Zeeman effect. Dichroism reflects a correspond- ing difference in the absorption coefficients. The frequency dependence of both refraction and absorption coefficients of a gaseous medium is affected by the Doppler effect. The changes of the dichroic and birefringent contributions occur therefore when the magnetic field is varied over about several hundred Gauss, i.e. the product of the Doppler width and the difference of the giromagnetic factors of the involved states.

A number of interesting effects were found in these experiments like the strong coherence narrowing due to a multiple scattering (Hackett 1968, Kr6las and Winiarczyk 1971) and the ‘line crossing’, i.e. the forward scattering analogue to the ‘level crossing’ showing interference between light from different atoms (Hackett and Series 1970).

With the advancement of laser techniques there appeared papers describing the forward scattering of light from c w single-mode dye lasers (Gawlik er a/ 1974a, Giraud-Cotton et a1 1982, Gawlik 1982). They described non-linear interaction between laser light and atoms. This added important new features to the effect: the laser light creates coherences between the magnetic atomic sublevels (the Zeeman or Hertzian coherences) and resonant population changes (the saturation resonances) (e.g. Cohen- Tannoudji 1977). They are subject only to homogeneous broadening and their relaxa- tion time is of the order of the level lifetimes. The coherence contributions show up in the signal as narrow resonances at zero magnetic field on the broad background due to the linear effect. As this background is almost negligible around the zero value of the magnetic field the forward scattering experiment appears to be a sensitive methad of observation of atomic coherences. Let us note here that the well known advantage of background-free detection in polarisation spectroscopy (Wieman and Hansch 1976) is based on very similar principles.

The aim of this paper is to analyse the case of scattering of spectrally broad light, i.e. of a linewidth comparable to the hyperfine structure, but strong enough to create atomic coherences. To our knowledge this case was considered only in one experimental (Gawlik et a/ 1974b) and one theoretical paper (Zakrzewski and Dohnalik 1983). As has been shown by Zakrzewski and Dohnalik the forward scattering could yield some additional knowledge of the mechanism of interaction of non-monochromatic light with atoms with respect to other experimental methods.

It should be noted that the influence of a finite laser bandwidth is described differently for the interaction with classical and laser sources. For classical light the normal procedure is to calculate a signal contribution due to each single frequency component of the light source and to integrate all these contributions over a spectral profile of the source. Such a procedure is valid as long as the interaction is linear. For many-photon processes, however, i.e. for interaction with a strong light, this procedure is no longer justified (see e.g. Eberly 1979). The problem which we study in this paper is closely related to the question of physical observables in the process of classical optical pumping investigated by Bouchiat (1969, Happer and Mathur (1967) and Laloe (1971). However, since these early papers were limited to weak light perturbations only, it is not a priori obvious that their conclusions could be extended to the case of strong light beams.

We study the forward scattering of pulsed, spectrally broad laser light on sodium atoms. The pulsed dye laser serves as a source of light which has appreciable spectral width and high peak intensity. The laser wavelength is tuned to be resonant with one of the two D lines: A D , = 589.6 nm (32S,/2-32P,,2) and AD, = 589.0 nm (32S,/2-32P3/2).

The sodium atoms have been chosen because many of the previous forward scattering experiments were performed with them. It was, therefore, easier to attribute

Forward scattering of non-monochromatic laser light 5941

specific features of the observed signals to those characteristics of the interaction which were different from the earlier studies. On the other hand, the experiments with sodium are relatively simple (high vapour pressure, efficient laser dyes available, etc).

We take advantage of the above-mentioned characteristics of the forward scattering as a sensitive probe of the coherence properties of atoms and perform an experiment similar to that with a narrow-band laser described by Gawlik et a1 (1974a). Surprisingly, there appears to be a dramatic difference between the results obtained with a non- monochromatic and a single-mode laser. In the first case we observe no contribution of the Zeeman coherence at the D, line frequency, while there is such a contribution for the D2 line, though of a much simpler form than in the case of a monochromatic excitation.

Below we present details of the experiment and its explanation. In particular, we show how the dependence of these coherences on the light intensity and bandwidth supports the concept of the 'hyperfine uncoupling' based on the competition between hyperfine coupling and interaction with light as also discussed recently by Ekert and Gawlik (1987).

2. Experiment

2.1. The light source

We used a tunable dye laser pumped by a nitrogen laser described by Bojara et a1 (1984). The N2 laser gave about 150 kW peak power of 337.1 nm radiation in 7 ns pulses of very good amplitude stability (a relative spread of k1.5%). In the present experiment its repetition frequency was locked to 50 Hz of the mains. The dye laser consisted of an oscillator and an amplifier (figure 1). The oscillator includes a 20 mm long dye cell, a two-prism beam expander which enlarged the beam horizontally by a factor of 20 and a diffraction grating (316 lines/", blaze angle 65'35') placed at an angle of about 80" to the beam. The oscillator cavity was closed by a fully reflecting plane A1 mirror placed near the grating and a 30% reflecting output mirror on the

expander

I J i i i I .

Oscillator Spatial filter Amplifier

Figure 1. The dye laser set-up

5942 W Gawlik and J Zachorowski

other side of the cell. Our set-up is a sort of compromise between the grazing-incidence configuration (Littman and Metcalf 1978) and the Littrow arrangement with a prism expander. It offers higher efficiency than the grazing-incidence set-up and higher resolution than the Littrow one because the grating is used twice for each cavity round trip.

A quartz etalon was inserted into the expanded beam between the prisms and the grating used to narrow the beam spectrally. In the first measurements we made the cavity as short as possible in order to have several round trips during the pulse and a distinct mode structure to be able to generate a single mode. However, it was very difficult to obtain a reliable and stable single-mode operation; instead we had two or three modes within the envelope of the total spectral width which was 1.7 or 2.75 GHz depending on the instrumental width of the etalon used. The modes were spaced by about 1 GHz which corresponds to the cavity length of 15 cm. Such a multimode structure might introduce unnecessary complications, e.g. mode crossing effects. There- fore we repeated the experiment with a modified laser set-up. The cavity length was increased to 40 cm, which was enough for the mode structure not to be pronounced. To improve the efficiency the plane output mirror was replaced by a spherical one with R = 40 mm and a single lens (f= 40 mm) was added on the other side of the dye cell to form the parallel beam.

In both set-ups the light beam from the oscillator was spatially filtered and focused on the second dye cell which served as an amplifier. We monitored the amplitude and spectral properties of the beam with the help of a slowly scanned Fabry-Perot inter- ferometer of the spectral range of 5 GHz.

2.2. The scattering cell

The sodium atoms were contained in a 5 cm long, 2 cm diameter Pyrex cell heated to 125 "C. This temperature corresponds to an atomic density of 4 x 10" ~ m - ~ , which is low enough to neglect multiple scattering. The heater was wound with a non-magnetic coaxial wire (Thermocoax, Philips). The magnetic field along the laser beam direction was produced by a solenoid and a pair of Helmholtz coils (figure 2). The power supply for the solenoid allowed a sweep of current of a given polarity while the coils provided a constant magnetic offset in the opposite direction. In this way we obtained the total

Figure 2. The forward scattering experimental arrangement: P and A denote the crossed polariser and analyser; A, and A, are neutral-density attenuators.

Forward scattering of non-monochromatic laser light 5943

change of the field intensity from -8 to 72 mT (-80 to 720 G). Additional coils (not shown in figure 2) were used to compensate the local magnetic fields in the orthogonal directions. The cell was placed between crossed Glan polarisers (Carl Zeiss Jena). The measurements were performed with a set of neutral density filters which served two purposes: to attenuate the laser beam and to prevent the detector from saturating with strong signals. It was important to ensure that the signals were within the range of linearity of the photomultiplier while the light intensity incident at the cell was varied by several orders of magnitude. We could accomplish this by finding a proper set of filters and by commuting single filters from the position A2 before the multiplier to the position A, before the first polariser in such a way that the light intensity at the multiplier did not change very much.

2.3. The detection system

The detecting and averaging system was built as a single-crate CAMAC system equipped with an autonomous controller. The pulses of the scattered light were detected by an inexpensive photomultiplier, amplified, shaped, digitised and stored in a 5 12-channel memory (figure 3). To each channel a given magnetic field value was ascribed-the digital-to-analogue converter ( DAC) transformed the numbers of successive memory channels into a voltage which controlled the sweep of the solenoid power supply. The relevant measurement parameters (number of channels, number of pulses stored in each channel and number of sweeps) were set from the keyboard. To reduce the noise the signal was gated and, moreover, to eliminate the stroboscopic-like effects in the detection resulting from a possible interference between the laser pulses and the mains, the clock generator was synchronised to the mains frequency. Finally, stored results were presented on a display, or on an X - Y recorder, or were available in a digital form.

Display

Preamp. Shaping amp

CAMAC

, I

Laser \ tr igger I

Magnetic f i e ld control \

Figure 3. The scheme of the detection a n d averaging system

3. Experimental results

The signals obtained for the D, and D2 lines are presented in figures 4 and 5, respectively. Each figure contains a set of curves for different light intensities. The intensities are

5944 W Gawlik and J Zachorowski

100 Yo I,

/ 10% I,

1 % I,

0.1 % Io

> 0 200 400 600

B I G )

Figure 4. The forward scattering signals, i.e. the scattered light intensity against magnetic field for the D, line (589.6 nm). The relative laser light intensity is given for each curve.

0 200 400 600 B ( G )

Figure 5. The forward scattering signals for the D, line (589.0 nm). The relative laser light intensity is given for each curve.

Forward scattering of non-monochromatic laser light 5945

represented by their relative values set by the density filters. From the shape of the curves obtained, and in particular from the power broadening of the D2 line coherence signals, the maximum Rabi frequency corresponding to 100% relative intensity can be estimated to be of the order of several GHz. Pulse power corresponding to 100% yo was of the order of 1 kW.

The forward scattering signals for the D1 line (figure 4) are essentially all the same for each light intensity. They are broad inverted bell-shaped curves similar to those obtained with spectral lamps. This leads us to the conclusion that in this case there is no coherence created even when the incident light has a maximal intensity. This is quite a different behaviour from that of the signals observed by Gawlik et a1 (1974a) with a monochromatic laser tuned to the D, line where the ground-state Zeeman coherences were easily visible.

For the scattering at the D2 line (figure 5) and for the lowest intensity (0.1% of Io) the signal again is a broad curve as in the cases of the D1 line and a classical light source. However, with the increase of the laser intensity, narrow structure appears near the zero magnetic field, reflecting creation of atomic coherences by the higher-order interaction with the light. These resonances increase their amplitude with the light intensity and are strongly power broadened.

Let us remark here that the results presented by Gawlik et a1 (1974b) were erroneously labelled: the D2 transition was assigned to the D, line. Taking this into account they agree with our present observations.

4. Calculations

The signal in the forward scattering experiment in an optically thin sample is given by the expression derived by Zakrzewski and Dohnalik (1983) (see also Durrant 1972):

where C is a constant, N the number of scattering atoms, d F m f F the matrix element of the atomic dipole moment matrix element between the Zeeman sublevels of the upper (F, m ) and lower (f, p ) states, vbmfF is the corresponding element of the density matrix of the ith atom and e is the analyser unit vector. The appearance of the two-atom interference terms in the signal results from the above-mentioned coherent character of the forward scattering. The ( ) symbol denotes the average over atomic statistic properties (the Maxwell distribution of atomic velocities in our case) and over the statistic properties of light.

In our analysis we aim at a general understanding of the investigated process rather than a precise quantitative agreement with the experimental data. The main objective of our calculations is to explain the dramatic qualitative difference in the scattering for the D , and D2 lines. Therefore, we have performed them in the simplest possible model with several rather strong assumptions. The first one is that the average in (1) decorrelates into two one-atom averages:

( U ' ( T ~ ) = (d~(+))( u * E ( - ) ) / ( E ( - ) E ' + ) ) (2) where E ( * ) denote positive and negative frequency parts of the electric field. It simplifies the calculations substantially; it is fully justified for the averages over the atomic part, but not generally true for the light statistics. However, as shown by Zakrzewski and Dohnalik (1983), the difference between the results with and without this approximation

5946 W Gawlik and J Zachorowski

is not very pronounced for the two particular models of the light field, i.e. for the diffusing phase and the chaotic field. The next simplification is that we calculate the density matrix elements in the stationary perturbation theory to the third order in E as it is the lowest order in which the influence of the Zeeman coherences can appear. We have further assumed that the phase diffusion model describes satisfactorily our laser light fluctuations. In this model the spectral properties of light are taken into account by modification of the transverse relaxation constant (Agarwal 1978, W6dkiewicz 1979, Eberly 1979)

y+T= Y + Y L (3) i.e. the relaxation constant y of the (TFmfp element is increased by the laser spectral width yL. With the decorrelation assumption (equation ( 2 ) ) we arrive at the expression for the signal in the form

I 1 2

where the atomic indices have already been omitted, uFmfp = ugirp + upifp is a sum of the first- and third-order terms and L'") = CN E l k d,,e*(u;[)) ( i , k = Fmfp) .

The linear terms have the form

= -p- I F m f p

a F m f p

where R F m f p = wFf+ ( mgF- pg,)B + kv - w - iT, 1 is the angular part of the dipole matrix element d F m f p = I F m f p 9 (9 being Ihe reduced matrix element), p is the Rabi frequency for the transition between the F and f states; wFf is the corresponding transition frequency; g F , gf are the Land6 factors for the F and f states, B is the intensity of the magnetic field and kv denotes the Doppler shift. We have to average the terms over the Maxwell distribution of longitudinal velocities v which leads to the plasma dispersion function Z (Fried and Conte 1961). The real part of 2 has an asymmetrical shape and describes atomic dispersion; the imaginary part is bell shaped and represents absorption. The first-order contributions to signal (4) are Doppler-broadened features centred at zero magnetic field. 'They reproduce well experimental signals obtained with weak light beams.

In the third order of perturbative calculations there are two kinds of contribution to

p' I F m f ' p l f ' p ' F ' m ' l F ' m ' F p C' w m m , - i Y e 0 F m f p Q f ' p ' F ' m '

and (5)

where ye and yg are the relaxation constants of the upper and lower levels respectively and C' is a constant. The contributions with m # m' and p # p' are due to the Zeeman coherences in the upper or lower level. The integration of the third-order contributions over the velocity distribution leaves the (CO,,. - i ye) and ( w c r p t - i yg) denominators unchanged and affects only the 0, so that

Forward scattering of non-monochromatic laser light 5947

where A = w,,. - i ye

z‘= [wFj f . - w + (m’g,.-p’gf,)B - i j ] / k u

or A = wcLcLL’ - i yg

z = [ w , f - w + ( m g , - p g f ) B - i j ] / k U

and U denotes the most probable atomic velocity. The denominators A and z - z‘ are responsible for the narrow resonances in the signal around zero magnetic field while the magnetic field dependence of the 2 functions in the numerator is responsible for the broad background similar to that given by the first-order terms.

For interpretation of our experiment we have to calculate all the possible first- and third-order terms taking into account the hyperfine structure (HFS) of the levels in the D, and Dz transitions. In this calculation we assume a linear Zeeman effect for all transitions and neglect small differences in the plasma dispersion functions for various HFS components. We also neglect hyperfine coherences (with F # F’ and f # f‘) because their contribution to the signal in the direct vicinity of the zero magnetic field is much less than that of the Zeeman coherences of sublevels within one HFS component. The calculations are performed for a resonant excitation, i.e. for the case when laser frequency w equals the centre-of-gravity frequency of the atomic transitions.

With these assumptions we obtain from ( 2 ) the third-order contributions to signal ( 4 ) which have the form L‘3’ = DY)- D?’, where for the D1 line

6 9 +

1 D ~ I = & { ~ ( ~ + - T H - j * H - j + 2 a F H - j 2 a i H - j

+ l 2 + -2a 7 H - j -2a i H - j

9 + 6 - 6 A T H - 3 j - 6 A i H - 3 j

12 + + 6 A + H - 3 j 6 A + H -3 j

- 2 1 + 6 a * H - 3 j - 6 a i H - 3 j

6 a T 4 H - 3 j - 6 a i 4 H - 3 j 1 - 4 1 +

1 1 ~ ~

6 a * 2 H - 3 j - 6 a i 2 H - 3 j

- 1

6 a + 2 H - 3 j - 6 a + H - 3 j

- 2 1 + 2 A * H - j - 2 A i H - j

1 - 1 - 6 A T 4 H - 3 j - 6 A T 4 H - 3 j 1

+ -

1 - 6 A + 2 H -3 j - 6 A T 2 H -3 j

- 1 2 A F H - j - 2 A T H - j (7 )

5948 W Gawlik and J Zachorowski

and for the D2 line

175 + 75 T H - j * H - j 2 a r H - j

420

100 + + 50 + 2 a i H - j - 2 a T H - j - 2 a + H - j

420 +

+ ~ ( ~ 4 ~ ~ 3 j + 3 E , , 7 4 H - 3 j - 3 E 3 , 7 4 H - 3 j

+ 126 + 126

150 + 150 + 104115 + 104/15 3E2 ,F4H-3 j -3E2,+4H-3j 3 E 2 , T 4 H - 3 j - 3 E 2 , T 4 H - 3 j

3E3, r 4 H - 3j -3 E31 + 4 H -3j

+ 150 - 150

-6a 7 7 H - 3j -6a + 7 H - 3j +

2o 4 - +

- 100 6a + 4 H -3j

- -6a + 4 H - 3j

20 E , , F H - j -E32FH- j E 3 , r H - j

+

4

- 5 - - E 3 , 7 H - j E , , F H - j - E , , T H - j

- + 2o E , , + H - j -E , ,+H- j E z 0 * H - j

- 4 + 1/48 42 20 +

*H-iy,

In ( 7 ) and (8) d = - i C N Y p 3 / ( 1 2 k u ) , Y = Im(Z), H = pBB (pB being the Bohr magneton), j = 2ij, 2a and 2 A are hyperfine splittings of the 32Sl/2 and 32P1,2 states, respectively, and EFF, denotes separation between hyperfine sublevels F and F of the 32P3,2 state. As the coherence signals in the forward scattering do not depend very strongly on Y, we have used a rough approximation of Y by a simple, analytical formula:

Y(x+iy) = [ ~ " ~ / ( l + y ) ] exp[ -x* / (~+y)~] .

The cases of a monochromatic and/or a broad-band laser can be readily discussed with the help of ( 7 ) and (8) by substitution of an appropriate value of j = 2ij.

For a nearly monochromatic laser 7 = y<< wFF', wff, (2a, 2A, EFF,). Denominators of terms in brackets in (7) and (8) have different values and many of them are large enough to neglect the corresponding terms.

When the laser light is spectrally very broad we may have 73 w F F ' , Off, (2a, 2A, EFF,). In such a situation the denominators in the internal brackets are all nearly equal to 7. Thus, for the D, line all contributions in (7) which depend on the Zeeman coherences, i.e. those with denominators (* H - 3i y e ) , (7 H - i yJ, sum up to zero. For the D2 line, on the other hand, only those contributions in (8) cancel which are due to the Zeeman coherences in the ground state (denominators (* H - i yJ) .

Forward scattering of non-monochromatic laser light 5949

We argue below that this is not just an accidental situation that such a cancellation occurs for a broad-band excitation but that it is due to the general fact that an interaction with a broad-band laser can be satisfactorily described without taking into account the hyperfine coupling. It is important to note that such a compensation of the coherence contributions is possible only if various hyperfine components are excited

(a1 G 1.02 ( 6 ) 0 = 1.54

10 0 -1 0 -1 0 0 10

B ( G 1

Figure 6. The calculated signals of the forward scattering for the D, line for different values of the saturation parameter G = p2/2fr; ( a ) laser spectral width yL = 10 MHz; ( b ) yL = 10 GHz.

l u )

G = 1 .02

0.30

0.10

0.03

-10 0 -1 0 0 10 E ( G I

Figure 7. The calculated signals of the forward scattering for the D2 line for different values of the saturation parameter G = p2/2fr; ( a ) laser spectral width y L = 10 MHz; ( b ) yL = 10 GHz.

5950 W Gawlik and J Zachorowski

simultaneously. Combining together (4), (6), ( 7 ) and (8) we can obtain the forward scattering signals.

The signals were calculated for various laser spectral widths yL from 10 MHz to 10GHz and for various values of the effective saturation parameter defined as G =

G was changed from 0.01 to 1 (for G > 1 the third-order perturbative treatment becomes inadequate). The effective saturation parameter G is more appropriate for the discussion of the interaction than just the light intensity. For large T, G is proportional to the spectral energy density of the light beam. The sequence of curves calculated according to the above model for the D, line is presented in figure 6 for two values of the laser bandwidth: yL= 10 MHz (figure 5 ( a ) ) and y L = 10 GHz (figure 5 ( 6 ) ) . For a nearly monochromatic laser ( yL = 10 MHz) an increasing contribution from the Zeeman coherences is clearly seen in figure 6 ( a ) with an increased power, i.e. larger G. For the broad-band laser (figure 6 ( b ) ) , however, no such structure can be detected. For the D2 line the curves are presented in figure 7 . For the nearly monochromatic case (figure 7 ( a ) ) two types of resonance due to the Zeeman coherences can be recognised due to their different widths ye and y p . Thus the contributions due to the coherences in the upper and lower states can be easily distinguished. On the other hand, the signals obtained with the broad-band excitation (figure 7 ( b ) ) are apparently quite different and are affected only by the coherences in the upper state characterised by width y e .

p2/2Tr where r = Yeyg/(Ye+ yg).

5. Discussion

The observed and calculated dependence of the signals on the laser bandwidth, different for the D, and D2 lines, results from the cancellation of the coherence contributions discussed above.

One way to interpret this cancellation is in terms of a destructive interference of the coherence contributions which takes place when all the hyperfine components of a given transition are excited simultaneously by broad-band radiation. This can be illustrated with a simple diagram in figure 8 showing two contributions to the coherence

f

Figure 8. The diagram illustrating two contributions to the Zeeman coherence in a given f level of the lower state associated with different H F S transitions F - f and F'-f . I f the components F - f and F'-f are excited simultaneously, a destructive interference may occur leading to cancellation of the coherence.

Forward scattering of non-monochromatic laser light 595 1

between Zeeman sublevels of the lower HFS component f which are due to irradiation with linearly polarised light simultaneously resonant with the HFS components f - F and f-F’. A destructive interference between the f - F and f-F’ links results in a cancellation of the lower-state Zeeman coherence. A similar diagram may be drawn also for the upper-state coherences.

An alternative, more physically intuitive explanation is based on the concept of what we call the ‘hyperfine uncoupling’. The coherence time of light is inversely proportional to its spectral width 7, = 1/ yL and only during this time can the atomic coherences be created (see e.g. Cohen-Tannoudji 1977). This coherence time can be comparable to the characteristic time of the hyperfine coupling, which is the time of precession of the Z and J vectors in the semiclassical picture, equal to the inverse of the hyperfine structure constant A. If we have yL >> A the influence of the nucleus on an electronic shell is negligible on the timescale of the coherent interaction and the process of interaction could be described in the IJ, M,) states basis rather than in the IFM,) basis as F is no longer a good quantum numbert.

As the D, line corresponds to the 2S,/2-2P,,2 transition no coherence between IJ, M,) states can be created by the linearly, U-polarised light (figure 9 ( a ) ) ; in the D2 case, however, the upper state is 2P3/2 and the coherence in this state can be created (figure 9( b ) ) and indeed is in fact observed in our experiment. This picture is consistent with the preceding perturbative analysis where in the D, case all coherence contributions were cancelled out by the broad-band excitation, whereas in the D2 case some of the upper-state coherences were not completely cancelled out. Another argument support- ing our explanation is that a similar increase of the transverse relaxation constant, but brought about by the collisions with the buffer gas as in the forward scattering experiment with a narrow-band laser by Gawlik et a1 (1974a), leads to a similar disappearance of the narrow features in the signal.

Figure 9. The sodium fine-structure levels and allowed c transitions for the D, and D, lines. The created coherences are shown by wavy lines.

There is a correspondence between our conclusions on the role of hyperfine coupling in the strong-field interaction and those of a group of papers (Bouchiat 1965, Happer and Mathur 1967, Laloe 1971) dealing with the problem of observables in the process of optical pumping. These early papers, however, were limited to the weak perturbation which is appropriate for the case of optical pumping with conventional light sources. The coherences under investigation had to be introduced by an additional, different from the light beam, perturbation, e.g. by the R F field. It was, therefore, not a priori

t A similar line of thought can be applied to fast collisions (Percival and Seaton 1957)

5952 W Gawlik and J Zachorowski

obvious that the conclusions valid for the weak light perturbation could be extended to the case of strong light beams. Our analysis shows that this is indeed the case.

6. Conclusions

We have studied the forward scattering of broad-band laser radiation on atoms with hyperfine structure. Despite its simplicity the experimental method of forward scatter- ing is especially well suited for studying coherence properties of atoms. We have observed a behaviour of atomic coherences created by the scattered light which was significantly different from the case of a monochromatic excitation. This behaviour is explained in terms of the concept of hyperfine uncoupling. According to this hypothesis, as the influence of the nucleus during the time of coherent interaction with broad-band light is negligible, atoms interacting with intense, broad-band radiation should be described in the uncoupled, fine-structure states basis rather than in the basis of IFm,) states.

Acknowledgments

We are much indebted to Professor G W Series for the loan of some laser optics. This work was supported by the Polish Research Program CPBP 1.06.

References

Agarwal G S 1978 Phys. Rev. A 18 1490 Bojara A, Gawlik W, Grabski R and Zachorowski J 1984 Rev. Sci. Instrum. 55 166 Bouchiat M 1965 J. Physique 26 415 Cohen-Tannoudji C 1977 Frontiers in Laser-Spectroscopy, Proc. Les Houches Summer School 1975 ed R

Corney A, Kibble B P and Series G W 1966 Proc. R. Soc. A 293 70 Durrant A V 1972 J. Phys. E: At. Mol. Phys. 5 133 Durrant A V and Landheer B 1971 J. Phys. E: At. Mol. Phys. 4 1200 Eberly J 1979 Laser Snectroscopy IV ed H Walther and K Rothe (Berlin: Springer) p 80 Ekert A and Gawlik W 1987 Phys. Lett. l2lA 175 Fork R L and Bradley L C 1964 Appl. Opt. 3 137 Fried B D and Conte S D 1961 The Plasma Dispersion Function (New York: Academic) Gawlik W 1982 Phys. Lett. 89A 278 Gawlik W, Kowalski J , Neumann R and Trager F 1974a Opt. Commun. 12 400 - 1974b Phys. Lett. 48A 283 Giraud-Cotton S, Kaftandjian V P and Klein L 1982 Phys. Lett. 88A 453 Hackett R Q 1968 PhD Thesis Oxford University (unpublished) Hackett R Q and Series G W 1970 Opt. Commun. 2 93 Happer W and Mathur B S 1967 Phys. Rev. 163 12 Krdlas I and Winiarczyk W 1972 Acta Phys. Pol. 5 133 Laloe F 1971 Ann. Phys., Paris 6 5 Littman M G and Metcalf H J 1978 Appl. Opt. 17 2224 Percival I C and Seaton M S 1957 Proc. Phys. Soc. 53 644 Wieman C and Hansch T W 1976 Phys. Rev. Lett. 36 1170 Wddkiewicz K 1979 Phys. Rev. A 19 1686 Zakrzewski J and Dohnalik T 1983 J . Phys. B: At. Mol. Phys. 16 2119

Balian, S Haroche and S Liberman (Amsterdam: North-Holland) p 3