Investigation of the Influence of Mode Polarizations on Mode-Crossing Resonances

T O M A S F E L L M A N Accelerator Laboratory, Department of Physics, University of Helsinki, P.O. Box 43, FIN-O0014 Helsinki, Finland

The magneto-optic effect called mode-crossing is investigated. The cross- ings are produced with a polarization-stabilized two-mode 633-nm He- Ne laser. The beam from the laser interacts with a thermal neon gas absorber placed outside the laser. The absorber is in a tunable longi- tudinal magnet ic field. The investigations are focused on the behavior of the crossing signals when the laser mode polarizations are varied. Both linearly and circularly polarized modes are used. The results are compared with conclusions that are drawn from couplings between the Zeeman sublevels in J = 1 to J = 0 sys tems. It is found that the number and the strength of mode-crossings can be controlled by selections of the mode polarizations.

Index Headings: Mode-crossing; Magneto-optics; Neon.

INTRODUCTION

Many methods to obtain information about atoms and molecules are based on magneto-optic effects. In these methods the interaction between magnetically manipu- lated samples and light is investigated. The work per- formed by Hanle ~ was the beginning of this kind of spec- troscopy. A review of the early measuring techniques can be found in the book by Mitchell and Zemansky. 2 With lasers, nonlinear features of magneto-optic effects can be investigated. Mode-crossing, zero-field level-crossing, and forward scattering are different variations of these effects. Mode-crossings are resonances that are created when a multimode laser field interacts with atoms the energy levels of which are Zeeman split in a magnetic field. The resonance called zero-field level-crossing (stimulated Hanle effect) is analogous to the mode-crossing reso- nances, but it is created at zero magnetic field.

Early investigations on the mode-crossing and the zero- field level-crossing effects with laser sources were pre- sented by Feld 3 and Dumont. 4 Feld described the deter- mination of Land6 g-values in oxygen using the mode- crossing effect. In Ref. 4 a mostly theoretical study of the mode-crossing effect is presented. The interest following the pioneering works on the mode-crossing and the zero- field level-crossing phenomena is well covered in Refs. 5-8. Graubner and Hermann 5 determined the widths of some excited levels in neon using an internal cell config- uration. St~hlberg has, with various co-workers, 6-8 done investigations on the 633-nm neon transition using mode- crossing, zero-field level-crossing, and magnetic mode locking. One common feature in all these investigations is that Brewster window lasers have been used. With this kind of laser, all the laser modes have linear and parallel polarizations. The polarization properties of all the modes are always changed in the same way if some optical com- ponent is inserted in the laser beam. Thus, it is not pos-

Received 28 N o v e m b e r 1994; accepted 12 April 1995.

sible for the modes to have different polarizations. Con- sequently, it is not possible to investigate how the mode polarizations influence the crossing resonances. In the papers mentioned above, 3-s either the influence of the mode polarizations has not been considered or the modes have had parallel linear polarizations.

In this investigation it is the influence of the mode polarizations on the crossings that has been studied. To the best of our knowledge, this is the first time results have been reported that deal with this aspect of the mode- crossing and zero-field level-crossing phenomena. In the experiment reported in this paper, a low-power polariza- tion-stabilized two-mode 633-nm He-Ne laser has been used to induce the crossing signals. The modes from the laser have linear and orthogonal polarizations. With op- tical components in the laser beam, it is possible to obtain linearly polarized modes that are parallel to each other, as well as modes with different circular polarizations. Thus it is possible to investigate how the mode polari- zations affect the crossing resonances. The investigations are performed in an external neon gas absorber, and the mode spacing is large so that the crossing signals are well resolved. The line shapes of the resonances are not stud- ied.

The paper is organized as follows. First, we describe the experimental setup. Then the coupling rules for the mode-crossing resonances are discussed. Thereafter the measurements and the experimental results are presented, and they are compared with the presented discussion on the coupling rules. The last part consists of a discussion of the results with some concluding remarks.

EXPERIMENTAL

In Fig. 1 the experimental setup is schematically shown. It is basically the same as that in the work by Sthhlberg, 8 but another type of laser has been used. The laser is a polarization-stabilized two-mode 633-nm He-Ne laser, the mode spacing of which is 670 MHz. The maximum two-mode output power that the laser can provide for the experiment is approximately 0.5 mW. The principle of the stabilization method is based on the linear and or- thogonal polarization properties of the modes. A detailed description of the method can, for instance, be found in the paper by Niebauer et al. 9 The long-term frequency stability of this laser is a few MHz and the intensity sta- bility is approximately 0.1%. These characteristics of the laser make it very suitable for the studies of mode-cross- ing properties.

The neon absorber discharge cell is 50 cm long and its diameter is 4 mm. The gas in the cell consists of 91% 2°Ne and 9% 22Ne. Directly on the cell there is a one-layer modulation solenoid for the magnetic modulation of the

1126 Volume 49, Number 8, 1995 0003-7028/95/4908-112652.00/0 APPLIED SPECTROSCOPY © 1995 Society for Applied Spectroscopy

sample gas. The main longitudinal magnetic field is de- rived from a multilayer 60-cm-long solenoid. The beam from the laser is slightly focused into the absorber cell. The diameter of the beam waist is approximately 0.5 ram. In front of the cell, a polarizer or a U4 plate can be inserted in order to change the polarizations of the laser modes. After the cell, the signal is detected directly in the laser beam by a photodiode. Lock-in detection is used to ex- tract the signal.

C O U P L I N G RULES

Explanations of the mode-crossing effect can be found in the works by Feld 3 and Dumont . 4 To be able to induce the crossings, one must fulfill certain requirements; these concern the number, the frequency, and the polarization properties of the laser modes. In this part of the paper, these requirements are discussed.

To obtain mode-crossing signals, one subjects the sam- ple to a longitudinal magnetic field. In this field, the level or levels with J larger than zero are Zeeman split, and A- or V-type three-level subsystems are created. The mode- crossing effect requires at least two modes in the laser field. Let us consider a two-mode laser interacting with atomic J = 1 to J = 0 systems. We denote the mode frequencies by fl~ and ft2 and the Larmor frequency by WL- Assuming that the modes interact simultaneously with the atomic ensemble that has the velocity component u in the beam direction, we obtain

~ l : COo + k l v - - O~L ( 1 )

and

~2 = o~o + k2v + o~L (2)

where ki is equal to f2i/c, and w0 is the transition center frequency. In our case (compare also with Ref. 8) the mode-crossing condition becomes

~2 - ~, = 200L. (3)

Next we show in which velocity ensemble the mode- crossings occur. We assume that the modes are symmet- rically tuned to both wings of a symmetric absorber Doppler profile. Cancellation of the Larmor frequency from Eqs. 1 and 2 leads to

f~2 + fir - 2OOo v = ( 4 )

k 2 -t- kl Using the assumption that the modes are symmetrically placed under the Doppler profile shows that mode-cross- ings are induced in the velocity ensemble constituting v = 0. This ensemble has the largest number of atoms. Thus, for instance, if the two-mode frequencies are separated so much that they interact with atoms far on the wings of the absorbing Doppler profile, the mode-crossing sig- nals can, in principle, be larger than those signals origi- nating from single-mode interactions with the atoms. These single-mode-induced zero-field level-crossings are discussed below.

Next we consider the polarization requirements. In Fig. 2 we show how two linearly polarized modes couple to a J = 1 to J = 0 system. The two modes are assumed to be symmetrically positioned on the absorbing profile. In a magnetic field, a level with J = 1 is split into three

Stabilized two-mode

He-Ne laser

FIG. I.

Polarizer or

M4-plate

Magnetic field

sweep

0 0 0 0 0 0 0 0 0 0 0 0 0 0 oaoooooooooooooooooooo

Neon absorber } ~

Solenoid

I

S c h e m a t i c d i a g r a m o f t h e e x p e r i m e n t a l s e t u p .

sublevels (m = - 1 , 0, + 1). In Fig. 2 the lower level b has J = 1 and the upper level a has J = 0. The coupling between the m b ---- - - 1 and ma = 0 is achieved with a a+ circularly polarized field and the coupling between mb = + 1 and m, = 0 with a a circularly polarized field. If there are two laser modes that are linearly polarized, both of them can be considered as a superposition of cr + and a- circularly polarized components. When the magnetic field is increased so that the Zeeman detunings fulfill the condition of Eq. 3, appropriate polarizations couple to the sublevels, and mode-crossing resonances are created. This pattern can occur for the two "r ight" and "left" longitudinal magnetic field directions, as shown in Fig. 2. One of the resonances is created with the a~ and ~ri- components and the other one with the a f and Crl + components.

For circularly polarized modes there are different pos- sibilities. In Fig. 3 we show the case where the modes have opposite circular polarizations. There are two pos- sibilities, depending on the circular polarizations of the modes. In Fig. 3 the case is shown where the lower-fre- quency mode 1 is o ~- polarized and the higher-frequency mode 2 is a + polarized. Thus, for an appropriate Zeeman tuning (with the B field to the "right") the g2 ~ and th!~ ~ components can couple to the A system. When the B field is to the "left", the cry- and the ~ components cannot couple, as shown at the bot tom of Fig. 3. For the other case where the lower-frequency mode 1 is a + polarized and the higher-frequency mode 2 is o- polarized, a similar explanatio D shows that there is a mode-crossing coupling when the B field direction is to the "left".

I f both modes have the same circular polarization, no mode-crossings are created. The reason is that if, for ex- ample, the modes are a + polarized, there is coupling be- tween the levels m b = - - 1 and m, = 0 but not simulta- neously between the levels mb = + 1 and m, = 0. For the other possibility with both modes o polarized, there is only coupling between the levels m b = + 1 and m, = 0.

At zero magnetic field, a resonance called the zero-field level-crossing can be induced. For observation of this resonance, one linearly polarized mode is enough. The ~+ and a- polarized components of the mode couple the m b = - - 1 and m~ = + 1 levels to the rn, = 0 level, re- spectively. I f there is more than one mode, they all give their contributions to the zero-field level-crossing signal. With circularly polarized modes, no zero-field level- crossing is created because the modes couple to only one of the transitions from mb = 1 or - 1 to rn, = 0.

APPLIED SPECTROSCOPY 1127

m=O

m=.l

B J

m=O

m=

m=+l

a

b

a

b

FIG. 2. This figure shows how two linearly polarized laser modes in- duce a mode-crossing resonance in a Jb = 1 to J. = 0 system. Two directions "right" and "left" of the longitudinal magnetic field are con- sidered.

> v

RESULTS

The transition used in this work is the well-known 633- nm neon transition between the 2P4 and 3s2 levels. The J value for the lower level is 2 and for the upper level 1. In a magnetic field the levels are split into 5 and 3 mag- netic sublevels, respectively. The Land6 g-factor for the 2p4 level is 1.301, and for the 3s2 level it is 1.295, so that the Zeeman splittings of both levels are practically equal.Z°,J~ It can be shown that the contribution from the V-type configuration is very small compared with that from the A-configurations.5 We can approximate the whole

=+1

B >

m=O

m

~/ ~ m-+1

m=-I

a

b

-..-)

m= 0

m=-I m=O m=+l

a

b

FIG. 3. Schematic diagrams of the cases with two laser modes having opposite circular polarizations. Details are explained in the text.

system with just one A-type configuration. This benefit makes the transition very useful for magneto-optic in- vestigations.

The experimental recordings have all been made with the apparatus described above. The pressure in the dis- charge cell was 50 Pa, and the discharge current was 6

1128 Volume 49, Number 8, 1995

<

C

I I I

-15 0 15 Solenoid current (A)

FIG. 4. An experimental derivative recording of the crossings with two orthogonally and linearly polarized modes. The total two-mode laser power was 0.43 roW.

mA. With these values for the pressure and the current, the background emission from the discharge has no sig- nificant influence on the signals. Since magnetic modu- lation and lock-in detection were used, all the recorded curves are the first derivatives of the signals from the interaction in the gas sample. The signals are recorded as functions of the solenoid current. From the corresponding magnetic field the magnetic detuning can easily be cal- culated. In our case 1 A corresponds to 102 MHz. Re- cordings have been made for different polarizations of the laser modes. The three cases described below have been considered.

Modes with Orthogonal and Linear Polarizations. In this case the laser beam is directly focused into the ab- sorber cell. In Fig. 4 a recording obtained with this con- figuration is shown. The total laser power was 0.43 mW and the mode intensities were approximately equal. The crossing resonances are well separated and clearly seen. Both laser modes have a ÷ and a polarized components so that there is one mode-crossing resonance on each side of the zero-current value of the curve. The distances from zero current value to both mode-crossings are equal. At the center of the curve at zero magnetic field there is a zero-field level-crossing resonance. Both linearly polar- ized modes create their own contribution to this reso- nance. The broad background in the curve is the deriv- ative of the Doppler profile. This linear contribution to the signal is due to the absorption of the laser beam.

Modes with Parallel and Linear Polarizations. In this case a polarizer is utilized in front of the absorber cell. The polarizer selects a projection of the modes so that they now have linear and parallel polarizations. With this configuration it is possible to freely select the ratio of the intensities of the modes that traverse the absorber. In Fig. 5 there are examples of measurements with different ra- tios. The total laser power in all cases is approximately 0.26 roW. In curve a the modes are equally strong so that the ratio between the intensities is 1:1. In the curves b, c, and d the intensity ratios are 1:4, 1:8, and 1:30, re- spectively. As can be seen from curve d, even when one of the modes is very small (the power of it is about 10 uW), it is possible to detect the mode-crossing signals.

a)

A

b) C

>,

c)

N d)

I I I

-15 0 15

Solenoid current (A) Fie. 5. Experimental derivative recordings of the crossings where the two modes have parallel linear polarizations. The curves constitute different ratios of the mode intensities. The ratios are (a) 1:1; (b) 1:4; (c) 1:8; and (d) 1:30. The total two-mode laser power in all cases was 0.26 mW.

Modes with Circular Polarizations. With a U4 plate we obtained circularly polarized modes. Since the two modes from the laser were linearly and orthogonally polarized, they got opposite circular polarizations after the k/4 plate. A plate rotation of 90 ° changes the circular polarizations so that the mode with a* polarization gets a polarization and vice versa for the other mode. In Fig. 6 we show representative recordings of the two possible cases where the M4 plate is in the directions that make the individual modes circularly polarized. In both curves the modes have opposite circular polarizations, but the k/4 plate is rotated 90 ° between the two recordings. The total laser power is 0.48 roW. In both curves there is only one mode- crossing. Depending on the orientation of the M4 plate, it is either on the left- or the right-hand side of the curve. The reason that we get only one crossing in each case is that the circularly polarized modes can only create these specific couplings, as explained above. The zero-field level-crossings have practically disappeared. The circu- larly polarized light cannot produce the coherent cou- plings between the sublevels. At zero magnetic field there is a trace of a weak signal that is in the opposite direction

APPLIED SPECTROSCOPY 1129

A

c

t . .

= m

.Q

<

c

O3

I I I

-15 0 15

Solenoid current (A) FIG. 6. Experimental recordings of the cases with two modes having opposite circular polarizations. The total two-mode laser power was 0.48 mW.

to the mode-crossing signals. The origin of this signal is probably due to some residual ellipticity in the laser modes.

For all the cases the strength of the resonances depends on the populations in the velocity groups that are in- volved, the degree of saturation that the fields manage to create, the degree of the field fluctuations, and the cou- pling configurations. From the experimental curves it is difficult to make conclusions about the strength of the resonances with each other. The reason is that the Dopp- ler background is different for the mode-crossings and for the zero-field level-crossings. The mode-crossings are lo- cated on the wings of the curve where the derivative of the Doppler background is almost horizontal. The zero- field level-crossing resonance is in the center of the curve where the derivative of the Doppler background is very steep. This factor affects the height of the zero-field level- crossing resonance so that it appears smaller than it would be on a horizontal background.

DISCUSSION

The main motivation for this investigation was to show the large influence the polarization properties of the laser modes have on the crossing resonances. In the works published before, this aspect has not been considered. In this work, attempts to draw any detailed conclusions from the line shapes have not been done. The results of this

investigation show very well separated mode-crossing sig- nals. From these signals, it would be possible to extract the different contributions to the signal known as the population effect and the Zeeman coherence effect. 7 How- ever, this effort would not give any new spectroscopic information on tfiis much utilized red He-Ne laser tran- sition.

One general feature observed in this investigation is that the mode-crossing resonances can be detected with very small laser powers. This pattern is particularly ap- parent in the case with two modes that have parallel linear polarizations. Even when one of the modes is only a small fraction of the other, the mode-crossings are clearly dis- tinguished. One conclusion of this observation is that when single-mode magneto-optic experiments are done, it is of great importance that the laser that is used really have only one well-defined mode. Some kind of multi- mode operation can otherwise affect the results.

In the last part of this investigation different configu- rations of circularly polarized modes were investigated. The orthogonal and linear polarizations of the modes, which the polarization-stabilized red He-Ne laser pro- vides, make it possible to produce opposite circular po- larizations. In the present work where opposite circular polarizations are utilized, the influence of the different circular polarizations is clearly demonstrated. Depending on how the U4 plate is orientated, different mode-cross- ings are detected. The results are in agreement with the conclusions that can be drawn from couplings to J = 1 to J = 0 systems. Finally this investigation also shows that a low-power polarization-stabilized two-mode He- Ne laser is well suited for this kind of spectroscopic mea- surement. The frequency and intensity were stable enough to give a good signal-to-noise ratio, and the laser power was also adequate for the measurements.

ACKNOWLEDGMENTS

The author wants to thank B. St~hlberg and S. Stenholm for valuable help and encouragement during this work. The financial support of the Magnus Ehrnrooth Foundation is gratefully acknowledged.

1. W. Hanle, Z. Phys. 30, 93 (1924). 2. A. C. G. Mitchell and M. W. Zemansky, Resonance Radiation and

Exited Atoms (Cambridge University Press, Cambridge, 1961). 3. M. S. Feld, "Laser Saturation Spectroscopy in Coupled Doppler-

Broadened Systems", in Proceedings of Esfahan Symposium 1971: Fundamental and Applied Laser Physics, M. S. Feld, A. Javan, and N. A. Kurnit, Eds. (Wiley, New York, 1973), p. 369.

4. M. Dumont, Phys. Rev. Lett. 28, 1357 (1972). 5. F. Graubner and G. Hermann, Z. Phys. A 289, 31 (1978). 6. B. Sthhlberg and K. Weckstrrm, Physica Scripta 22, 483 (1980). 7. B. St~hlberg, M. Lindberg, and P. Jungner, J. Phys. B: At. Mol.

Phys. 18, 627 (1985). 8. B. St~hlberg, Z. Phys. D 22, 391 (1991). 9. T. M. Niebauer, J. E. Failer, H. M. Godwin, J. L. Hall, and R. L.

Barger, Appl. Opt. 27, 1285 (1988). 10. G. Hermann, G. Lasnitschka, and A. Scharmann, Z. Phys. A 282,

253 (1977). 11. C. E. Moore, Atomic" Energy Levels, National Bureau of Standards

Circular 467 (U.S. Government Printing Office, Washington, D.C, 1949), Vol. 1, p. 77.

1130 Volume 49, Number 8, 1995