The Effect of Axial Magnetic Fields on Gas Lasers

Alan Corney

An expression is obtained for the intensity of polarized light transmitted through a laser amplifier andlinear polarizer, as a function of the strength of an axial magnetic field. The result is compared with anexperiment reported by Hotz in which it was claimed that the natural width of the laser transition wasmeasured. The effect of applying a magnetic field parallel to the axis of a laser having plane-parallelwindows is also considered. The variation in the frequency of beats between oppositely circularlypolarized modes, as a function of magnetic field, is obtained. The result shows that an experiment per-formed by Culshaw and Kannelaudsupposed.

Introduction

Recently, experiments have been reported" inwhich the effects of applying a variable magneticfield parallel to the axis of a laser or laser amplifierhave been studied. In particular Culshaw and Kan-nelaud' studied the variation with magnetic field ofthe low-frequency beat between the oppositely cir-cularly polarized oscillations of a He-Ne laser operat-ing on the 1.153-,4 transition, and Hotz' has measuredthe single-pass gain of a laser amplifier operating onthe 3 .3 9 -A neon transition, as a function of the appliedmagnetic field. In each case the results were inter-preted as due to the coherence effects which arisewhen the states of a single atom are correlated throughthe application of a magnetic field. Coherence effectsof this type are well known in level-crossing experi-ments3 and in optical double-resonance experiments, 4

where the main factor which decides the width of thesignal is the natural widths of the levels involved.

The purpose of this paper is to derive theoreticalexpressions to explain the experimental results, andto show that the effects are more properly attributedto the coherence of phase which exists between anincident light wave and the waves scattered by dif-ferent atoms in the forward direction. In this casethe width of the signals, in the limit of small scattering,would be determined by the Doppler width of thescattering medium rather than by the natural widthof a given optical transition.

Theory of Gain Narrowing

Consider an electromagnetic wave of frequencyk, incident on a medium which contains atoms having

The author is at the Joint Institute for Laboratory Astrophysicsof the National Bureau of Standards and the University ofColorado, Boulder, Colorado.

Received 10 June 1965.

cannot be interpreted as a level-crossing effect as was previously

an optical transition frequency ko close to k. Forthe sake of simplicity the optical transition involvedwill be considered to exhibit the classical Zeemaneffect. This is approximately true for the 3.39-Atransition where the g values are known to be almostequal. If a magnetic field is applied to the mediumparallel to the direction of propagation, then onlyright and left circularly polarized waves are able topropagate with no change in polarization. The com-plex wave vectors, Ki, for these two eigenwaves areobtained from the components of the dielectric tensorof the medium, E,:

(1)-,2 = = n2k2 2 ==

where no are the complex refractive indices associatedwith the opposite senses of polarization. The prob-lem of the propagation of a linearly polarized waveincident on the medium is solved by first decomposingthe wave into a combination of the two eigenwaves,and then recombining the eigenwaves after they havetraveled a distance through the medium, taking intoaccount the diff erencesin phase and amplitude produced.

It is shown in ref. 5 that the components of thepolarizability tensor at the frequency k for a singleatom under the influence of a static field H along theaxis of quantization are given by

h [(ko-k)-i(r/2) i ]j<< 1, (2)

where P is the reduced matrix element of the electricdipole operator; r = natural width of the transition;and = yH, where y = the common gyromagneticratio of the levels.

The bulk polarizability of the medium is obtainedby summing the contributions from all atoms able totake part in the transition, taking into account theDoppler distribution of atomic resonance frequencies,

January 1966 / Vol. 5, No. 1 / APPLIED OPTICS 127

ko, about the center frequency, ki, I

dN = N(k0 )dko -/2

N= ^ l/2 exp [-(k, - ko )'/A'] dko,

where

2A(1n2)'/2 = the full width at half-maximum of the dopplerdistribution, and

N = difference in population density between the up-per and lower levels.

Thus the components of the polarizability tensor ofthe medium are given by

a, f cA dN

N IP12 f+ exp(- 2) d,-t-(X + i)

where

= (ko -k)/A

= [(k i c, )],

y = r/2A.

The results of the integration can be expressed in termsof the plasma dispersion function, Z(x,y) tabulated byFried and Conte6 giving:

- = IP2_ P2Z(X, ) << 1-

We thus obtain expressions for the complex refractiveindices of the medium for right and left circularlypolarized light:

= = 1 + 47 rNP . Z(x., ). (3)

From Eqs. (1) and (3) we obtain an expression forthe complex wave vectors of the eigenwaves:

K, ck :: I +2,rNIPI'.Zx, ) 4hA

To solve the problem of the propagation of a wavethrough the medium let us assume that the wave istraveling parallel to the z axis of a rectangular co-ordinate system. The incident wave is assumed to belinearly polarized in the direction (i + j), where i and jare unit vectors parallel to the x and y axes, respectively.Then the electric field in the medium is described by acombination of the two eigenwaves:

E = a -(i + ii) exp[i(K+z - k)]A/2... ~~~~~~~1

+ b 2 (i - ij) exp[i(K-z - kt)]. (5)

The amplitudes a and b are determined by the condi-tion that at the first face of the medium, z = 0, Emust be linearly polarized in the direction (i + ij) andhave amplitude Eo. This yields

1b = ~2(1 ±i) Eo. (6)

When the optical frequency of the incident wavecoincides with the center of the atomic resonancefrequency we have x = + o/A, and the symmetryproperties of the real and imaginary parts of the plasmadispersion function give

Zr(WC/AY) = -Z(-/AY),

Zi(co/Ay) = +Zi(-w/AY) (7)

Then using Eqs. (4), (6), and (7) we can write Eq. (5)as

E = E/2{(1 - i)(i + i) exp(ibz/2)+ (1 + i)(i - ij) exp(-i4z/2)}

X exp[i(K+t + K-7)z/2] exp[K'zl, (8)

where 4' = (K+ I - K_ ) and superscripts r and i denotethe real and imaginary parts of K,.

The component of the electric field which passesan analyzer at z = , having its axis parallel to thedirection of polarization of the incident wave, is

Ell = E (i + ). (9)

Using Eqs. (7)-(9) we obtain

Ell = Eo cos((l/2) exp[i(K+r + K- ) 1/2] exp[K1].

The intensity of the transmitted light is proportionalto I Ell 12 giving finally

I = o {cos[gZ,(/2Ay)] exp[gZj(w/A,y)]}', (10)

where o is the intensity of light incident on the firstface of the medium. The parameter = 27rN PI2kl/chA determines the relative gain, I/lo, in zero field.When the axis of the analyzer is perpendicular to theaxis of polarization of the incident wave, the observedintensity is given by an equation similar to Eq. (10)in which the cosine is replaced by a sine function.

When Eq. (10) is evaluated we obtain theoreticalcurves showing symmetric minima and secondary max-ima which agree qualitatively with the experimentalresults of Hotz.2 The full width at half-maximum,measured in terms of A, for these theoretical curves isplotted as a function of the gain parameter g for variousvalues of y = r/2A in Fig. 1. It is clear that the half-width decreases continuously as the inversion densityincreases, and for sufficiently high gain falls well belowthe value 2y which is the natural width of the transitionmeasured in units of A. The relation between thisnarrowing of signals observed in the forward-scatteredradiation and the coherence-narrowing7 observed inoptical-double resonance experiments is discussed indetail in ref. 5. To obtain the natural line-width fromobservations of this type it is first necessary to knowboth g and the doppler width, 2(1n2)"/ 2 , more ac-curately before Eq. (10) can be fitted to the results andthe value of y = r/2A obtained.

128 APPLIED OPTICS / Vol. 5, No. 1 / January 1966

1.0

0.9s

0.8

0.7F-C'

C

:3C

-c. _

-

LI

0.61-

0.41-

0.31-

0.2 -

0.- 1-

C'

0 1 2 3 4Gain Factor g

5 6 7

Fig. 1. Full half-width as a function of gain parameter, g, forcurves computed from Eq. (5) for given values of y.

Theory of Low-Frequency Beats

Consider a laser with plane-parallel end windowsand whose length is sufficiently short that at low powerlevels it operates on a single axial mode. Then justabove threshold the frequency of oscillation, k, isgiven by

(2L) n,(k) (11)

where m is a large integer; L is the spacing between thecavity mirrors; and nr(k) is the real part of the re-fractive index of the laser medium at frequency k.The laser medium is assumed to fill the space betweenthe mirrors. If an axial magnetic field is now appliedto the laser, the single degenerate mode is split intotwo oppositely circularly polarized modes which oscil-late approximately independently. Since the re-fractive index of the medium differs for opposite sensesof circular polarization, there is a frequency differencebetween the modes

Ak = k+ - k = (-) [nr+(k+)nr(k.)] (12)(2L/ nr+(k+ ),-(k- ) IFor low values of the applied field the frequencies

of the oppositely polarized modes are approximatelyequal, thus we can replace k+ and k- in Eq. (12) bythe laser frequency in zero field, k. Also both n,+(k)and n,-(k) are close to unity so we have

Ak - (- ) [nr+(k) - n(k)]. (13)

Using the expressions given for the refractive indices

in Eq. (3) we have

(MC\(2rN IP' \Ak \z2L) VhA ) [Zr(x+,y) - Z7 (x-,y)I. (14)

This low-frequency beat can be observed if the out-put of the laser is detected by the combination of alinear polarizer followed by a photocell. Culshaw andKannelaud' give experimental results showing thebeat frequency as a function of magnetic field of theHe-Ne laser transition, X = 1.153 li. The positionwhere the beat frequency becomes zero, observed to beat 16 G in one experiment and at 7 G in another, theyinitially interpreted as due to a mixing of the statesof an atom caused by the rf magnetic field which ac-companies the laser excitation. If this were the casethe position of the zero would remain fixed at a givenfield value close to 1/2 (o/7y), (o/,y) or /2 (O/Y),

4

where o is the rf frequency, 52.5 Me/sec, and -y isthe gyromagnetic ratio. None of these agrees witheither of the experimental field values.

However, by evaluating Eq. (14) for the case wherethe cavity resonance, k, does not match the atomicresonance, k, we see the reason for the occurrence of azero beat frequency at a finite field. The position ofthe zero is sensitive to the error setting (k - kl)/Aas shown in Fig. 2.

This simple theory predicts a zero beat at finitefields only for relatively large error settings, (k -kl)/A

> 1.0. However, a zero beat at finite fields can beobserved for much smaller values of the error setting,8

when the laser is operated above threshold. To ex-plain this observation it is necessary to take into ac-count the effect of frequency pushing in addition to thefrequency pulling effects already considered. Bygeneralizing Eq. (89) of ref. 9 we obtain a more com-plete expression for the low-frequency beat:

Ak ;z(AC) (2wNIP ) [Zr(+,y) - Z7(x-,y)]

+ Y (2L) ( I ) (N -NT) Z ( , )

x y+ x-X X+2 + 2y2 X_2 + 2y 2 9 (15)

where NT is the population inversion density at thresh-old. The second term in Eq. (15) gives the effectof frequency pushing and is seen to be zero at threshold.Above threshold it gives a contribution to Ak which isopposite in sign to that produced by the first term,leading again to a zero beat at finite values of themagnetic field. For values of (k - 1c) and X verymuch less than A there is a region of excitation abovethreshold defined by

(1 - NT/N)Z(A , , <

for which the field value at which the zero beat occursdecreases as the error setting (k - kl)/IA is increasedin contrast to the behavior shown in Fig. 2. This hasbeen verified experimentally. 8

January 1966 / Vol. 5, No. 1 / APPLIED OPTICS 129

I I I I I I

05

xl. 20 ao

280-

n60-

C~40

20

00 1.0 2.0

Magnetic Field (Arbitrary Units)

Fig. 2. Beat frequency as a function of magnetic field computedfrom Eq. (9) for given values of the error setting, x = (k-kl)/A,

using = 0.1.

Culshaw and Kannelaud' also observed a rotation ofthe plane of polarization of the laser output for magneticfields less than 0.4 G which they ascribe to the level-crossing in zero field (Hanle effect'0). An independentexperimental investigation of the polarization effects inweak axial fields" shows that for fields less than 0.8 Gthere is only one elliptically polarized mode oscillating.This mode is practically linearly polarized at zerofield and shows decreasing ellipticity together with a

rotation of the major axis of the ellipse as the field isincreased, due to the changing Faraday rotation ofthe medium. Finally, a threshold field is reached atwhich a second mode starts to oscillate and the twomodes are almost oppositely circularly polarized.Therefore, it seems that the effect observed by Cul-shaw and Kannelaud was not a Hanle effect.

The author wishes to acknowledge helpful discus-sions with P. L. Bender and G. W. Series.

The research reported in this paper was supportedby the Advanced Research Projects Agency (ProjectDEFENDER), and monitored by the U.S. ArmyResearch Office (Durham) under a contract.

References1. W. Culshaw and J. Kannelaud, Phys. Rev. 136, A1209

(1964).2. D. F. Hotz, Appl. Phys. Letters 6, 130 (1965).3. P. Franken, Phys. Rev. 121, 508 (1961).4. J. N. Dodd, G. W. Series, and M. J. Taylor, Proc. Roy. Soc.

(London) A273, 41 (1963).5. A. Corney, B. P. Kibble, and G. W. Series, to be published.6. B. D. Fried and S. D. Conte, The Plasma Dispersion Func-

tion (Academic, New York, 1961).7. J. P. Barrat, J. Phys. Radium 30, 541, 633, 657 (1959).8. W. Culshaw, private communication.9. W. E. Lamb, Phys. Rev. 134, A1429 (1964).

10. A. C. G. Mitchell and M. W. Zemansky, Resonance Radi-ation and Excited Atoms (Cambridge Univ. Press, Cam-bridge, England, 1934).

11. H. de Lang and G. Bouwhuis, Phys. Letters 9, 237 (1964).

Meeting Reports continued from page 126

was located in the H. C. Orsted Institute simultaneously withan instrument and book exhibition.

The Congress covered a fairly wide range of topics, includinghigh-resolution infrared, Raman, visible, and ultraviolet spectra,electron-spin resonance, nuclear magnetic resonance, microwavespectra, intra- and intermolecular interactions, spectra of thecrystalline state, hydrogen bonding, spectral theory, and so on.The plenary lectures were delivered by R. Daudel, France, Th.Fbrster and R. L. M6ssbauer, GFR, and B. Stoicheff, Canada,on the application of wave-mechanical methods, polarizationspectra, recoilless absorption of gamma radiation, and on opticalmasers, respectively. Due to the wide range of subjects someone thousand participants heard about four-hundred lecturesorganized into ten sections. Thus, each participant attendedforty lectures at most if he was present throughout the meeting.

The scientific sessions were accompanied by successful socialevents. The reception by the Organizing Committee took placein the Old University Ceremonial Hall on 15 August, followed bythe Tivoli Anniversary Fireworks display. The reception by theCity of Copenhagen was given the next day in the CopenhagenTown Hall. On 17 August, the President of the OrganizingCommittee, B. Bak, received some fifty persons in the LangeliniePavillon. On 18 August the participants chose between NorthZealand, South Zealand, or Sweden excursions, and on 20 Augustthe Congress ended with the Congress Dinner in the HotelMercur.

From a scientific point of view the Congress is commonlybelieved to have been fully successful. The fact, however, thatthe time allotted for each lecture and subsequent discussion wasstrictly limited was reflected not only by the lectures being suc-cinct but by the discussions being short and in most of the cases

nonexistent. Discussions in depth went on outside the lecturehalls, in the hotels, at receptions, in the corridors of the OrstedInstitute, and in other places. Such informal discussions haveever been the most valuable feature of a congress, since it is throughthese opportunities that a wide and lively exchange of experiencestakes place, scientific contacts are made, and interchange ar-ranged. The lectures raised a number of new problems thathave not been discussed at any previous Molecular Congress, andin this respect the Danish meeting proved a definite advance tothat branch of science. Further, it was made possible for theleading experts and research workers from all parts of the worldand for scholars of that particular branch of science to meet andto become acquainted with each others' results and methods and,thereby, further to advance molecular spectroscopy.

Concurrent with the Congress, during the lunch break on 19August, the IUPAP Triple Commission of Spectroscopy held itsannual meeting.

The participants of the Congress would wish this reporter toexpress their full satisfaction with the success of the Congress andtheir thanks to the Organizing Committee and its head, ProfessorBak, for his excellent organization and kind hospitality.

Fourth National Meeting, Society for AppliedSpectroscopy, Denver, 30 August-3 September 1965

Reported by William L. Baun, AF Materials Laboratory (MAYA),

WPAFB, Ohio

The Fourth National Meeting of the Society for Applied Spec-troscopy was held in Denver, Colorado, from 30 August to 3September 1965.

Although the writer had not attended this meeting in previouscontinued on page 14

130 APPLIED OPTICS / Vol. 5, No. 1 / January 1966