1018 J. Opt. Soc. Am./Vol. 73, No. 8/August 1983

Temporal and interference fringe analysis of TEMo1* lasermodes

J. M. Vaughan and D. V. Willetts

Royal Signals and Radar Establishment, St. Andrews Road, Great Malvern, Worcestershire WR14 3PS, UK

Received February 11, 1983

The properties of TEMo 1* doughnut modes have been examined by frequency analysis and by two-beam interfer-ence. Interpretation in terms of an evolving helix, made up to two orthogonal TEM 0 1 modes of different frequency,is supported by computer simulation of fringe patterns. These patterns are shown to correspond closely with pho-tographic recordings; the implications of the phenomena for the crossed-beam technique in laser velocimetry areoutlined. Finally, the possibility of developing a beam of pure helical cophasal surface is discussed and its interfer-ence patterns analyzed.

INTRODUCTION

In previous publications1"2 we have investigated two-beaminterference patterns that are due to TEMoo and TEM01 lasermodes of different relative power. It was suggested that theTEMo1* doughnut mode, composed of two TEM 01 modes inphase and space quadrature, may possess a cophasal surfaceof helical form. However, in order to explain the observedinterference patterns derived from several high-gain Ar+- andKr+-laser lines, it was necessary to suppose that the observeddoughnut mode was an evolving helix, which switched rapidlyback and forth between a left-hand and a right-hand charac-ter. In our earlier papers the interference theory of thesemodes was performed analytically in terms of phase contours;we now present new results on the temporal behavior of thesemodes that explain the previous assumptions of helicalhand-switching. These results are incorporated into com-puter calculations and graphical constructions of the fringeprofiles, which are found to be in good agreement with ex-perimental photographic recordings. The calculations havebeen carried out both for evolving helices of the TEMo1* modealone and for various admixtures of the TEMoo mode. In thefinal section we propose a simple means of producing a sta-tionary helical wave of either hand and draw attention tounusual interference properties that it should possess.

It may be remarked that interest in these modes is notconfined solely to the unusual character of the wave surfacesand their interference patterns. These modes also haveconsiderable practical importance in photon-correlationspectroscopy and in crossed-beam fringe velocimetry. In thefirst case considerable reduction in the strength of fluctuationsmay occur, leading to a reduced signal-to-noise ratio; thisimportant problem is discussed in detail, both experimentallyand theoretically, in Ref. 3. In velocimetry the reduced fringevisibility and changing phase may introduce considerabledistortion of experimental data; this is outlined in the presentpaper. These effects are of course particularly troublesomeif such modes are additionally present in a nominally TEMoolaser. In the past such effects were widely found, particularlywhen high-gain ion lasers were used, but they have not hith-erto been explained or evaluated in detail.

TEMPORAL MEASUREMENTS

The source employed in this work was a Kr+-ion laser oscil-lating at 476 nm, with an intracavity 6talon for longitudinal-mode selection. It must be emphasized, however, that thephenomena are quite general for high-gain laser lines. Similartwo-beam interference patterns have been seen in many linesfrom Ar+- and Kr+-ion lasers and have also been observed inthe interference patterns that are due to a doughnut modefrom a CO2 laser operating at 10.6 gim, viewed with a thermalimager. For the present work the laser output frequency wasanalyzed in the 10-500-MHz range with a 20-cm scanning,confocal, Fabry-Perot interferometer. If the whole modewere allowed to enter the interferometer, it was found toconsist of a single frequency when the laser was emitting aTEM 0o mode; by adjustment of Rtalon tilt and dischargecurrent a pure TEMo,* mode, with characteristic dark center,could be obtained and was found to consist of two frequenciesW2 and W3 separated by 128 MHz. The spatial distributionof these frequencies was investigated by isolating a smallfraction of the mode in the (p, 0) plane perpendicular to thepropagation direction. This was accomplished by scanninga small aperture in this plane and subjecting the light thatpassed through it to spectral analysis. Frequency W2 wasfound to be concentrated near the horizontal and W3 near thevertical axis, qualitatively consistent with spatial distributionsof the kind p 2 exp(-p 2/U2 )cos 2 0 and p2 exp(-p 2/a 2 )sin26 to

be expected of TEM 01 modes. 4

These observations were confirmed by beat-frequencyanalysis in the region between dc and 1.8 GHz by using a smallsilicon avalanche photodiode coupled to a Tektronix 7L12spectrum analyzer. The magnitude of the beat-frequencysignal at I' '3 - W2 1 (equal to 128 MHz) maximized along theO = Jz45' directions and was zero along the vertical and hor-izontal axes, distributed qualitatively as p2 exp(-p2 /a 2)sin 26. We conclude from the spectral data that the TEMo,*mode does indeed result from the superposition of two TEM 0 1modes in space quadrature but that in our case they are non-degenerate in frequency, being adjacent longitudinal modesof the laser resonator whose length L is 117 cm (c/2L = 128MHz). There appears to be no fundamental reason why both

0030-3941/83/081018-04$01.00

J. M. Vaughan and D. V. Willetts

Vol. 73, No. 8/August 1983/J. Opt. Soc. Am. 1019

modes should not be degenerate; in the present case the de-generacy is almost certainly lifted by cavity asymmetry in-troduced by the Brewster windows, dispersive prism, etc. Thefact that the polarization direction, defined by the Brewsterwindows and the prism, is vertical and coincident with the axisof one of the TEM01 modes lends support to this notion.

A consequence of the different frequencies of the twoTEM01 modes is that their resultant evolves from a left- toright-handed helix and back again via intermediate TEM 01modes orientated along the bisectors of the w2 and w3 modes;the evolution repeats at 128 MHz. This fact explains theearlier assumption of an evolving helical mode of changinghand that was required to give the observed interferencepatterns. It is straightforward to show that a simple helix isequivalent to two TEM 0 1 modes of the same frequency inphase and space quadrature. More generally, two orthogonalTEM 01 modes of the same frequency form an equivalent basisset to two helical modes of opposite hand, also at the samefrequency.

FRINGE-PATTERN FORMULATION

Calculations of fringe intensity are carried out with thecoordinate basis shown in Fig. 1. The centers of the two in-terfering beams are at (+R, 0), and the two beams are propa-gating into the plane of the paper to intersect at an angle 2a.We consider the general case in which the beams are composedof both a Gaussian (TEMoo) mode at frequency w0 and adoughnut (TEMoi*) mode oscillating initially at a single fre-quency w1, i.e., operating as a pure helix. Then the expressionfor the electric field at any point in the plane is given by

E(p, 0, t) = C[exp(-p 12 /2U2 )sin(cvot - 01)

+ exp(-P 22 /2U2)sin(wOt + 02)]

+ (pi/oY)exp(-p12 /272 )sin(cvit ± 01 -01)

+ (p2/,f)exp(-p22 /2o-2)sin(wvt + 02 + 02), (1)

where the term within brackets represents the Gaussiancontribution and the remainder the helical wave; we have usedthe fact that the phase of a helical wave is equal to the azi-muthal angle 0. C is a constant representing the relativeTEMOO field strength, and o describes the mode radius. Inthe third term, the plus applies when the helices are unin-verted with respect to each other, the minus when they areinverted (Figs. lb and 1c). The phase angles Oi arise from the

'2 J

b

BY

��c4- -R- -+

C

x

A1 B AI(3QA 2

Fig. 1. (a) Coordinate system used for fringe intensity calculations.The angles Al and 02 are constrained within the limits -(7r/2) <01 $<(37r/2) and -(37r/2) <02 < (r/2) chosen to simplify the analysis. (b)Uninverted beams, (c) inverted beams, both shown displaced side-ways.

inclination a of the beams to the (p, 0) plane and are given by(2r/X)sin a X pi sin Oi (i = 1, 2). Since the beams are inclinedin opposite directions, the hi appear in Eq. (1) with oppositesigns. The intensity arising from the field of Eq. (1) is ob-tained by taking the square modulus and time averaging inthe usual way, with the result that

2I c 1/2 E (a,2 + bi2 ) + aia 2 cos(01 + 02)

i=1

+ bib2 cos(0 1 + 02 F 01 - 02), (2)

where

ai = C exp(-pi 2 /2U2 )

bi = (pi/o)exp(-pi 2 /20f2 )

i = 1,2,i = 1,2.

This is the result for an unchanging helix. Our previous ob-servations show that the helix switches hand at the intermodefrequency 1I2 - '31 of 128 MHz. Writing the complete ex-pression for this case shows that the third term in expression(2) should be replaced with

1/2bib 2[cos(01 + 02 =F 02 - 01) + cos(01 + 02 i 01 + 02)]-

Under these circumstances the intensity becomes

2I 1/2 (ai2 + bi2)

+ [ala 2 + bib 2 cos(02 ± 0i)Icos(01 + 02)- (3)

COMPUTATIONAL PROCEDURE

A Cartesian basis set was chosen that is related to the polarcoordinates of Fig. 1 by

x = (-1)i+lR - pi sin Oi, i = 1, 2,

Y = Pi cos Oi, i = 1, 2.

A program to display I(x, y) was written in BASIC by using theHewlett-Packard 9800 Series calculator and plotter. Forfixed y = yo, I(x, y) was computed at a series of points in xdiffering by Ax, and these points were plotted to give a verticalsection through the surface I(x, y) at y = yo; yo was then in-cremented to yo + Ay, and the computation was repeated.The results were plotted with a vertical shift on the chart ofK X Ay (K < 1) and a horizontal stagger s. The process wasrepeated until the whole pattern was built up from a series ofvertical sections to give the illusion of an isometric projectionof the fringe pattern, as shown in Figs. 2a-2d. For small s, theapparent incidence angle of view is cos-1 K. The input datato the program were the limits in x and y and the respectiveincrements Ax and Ay, K and s, and the fringe data C, a, R,and (27r/X)sin a. In the examples shown, the centers of thebeams are marked if they are separated; a scale of a is alsoprovided. (27r/X)sin a was chosen to give 20 fringes across thediagram, and K was 0.8, so that the apparent incidence anglewas 37°.

The computations for various configurations may be com-pared with the experimentally obtained interferograms ofFigs. 3a-3f. These photographs were obtained by two-beaminterference, which provides a useful examination of spatialcoherence.5' 6 The inclusion of an intracavity 6talon in thelaser permitted longitudinal-mode selection and some degreeof tuning of the relative intensity of the two TEM01 modes.

J. M. Vaughan and D. V. Willetts

�-- -R-

1020 J. Opt. Soc. Am./Vol. 73, No. 8/August 1983

Fig. 2. Computer reconstruction of two-beam interferograms witha TEMoi* laser mode: (a) beams inverted (i.e., Fig. lc) but not dis-placed, (b) beams inverted and displaced horizontally, (c) beamsuninverted and displaced horizontally (i.e., Fig. lb); (d) like (c) butwith the addition of 14% TEMoo mode. Note the half-cycle dis-placements in the fringes in different regions; see text for details ofthe perspective. The centers of the separated beams are marked bycircles.

UUl

II'-

IUFig. 3. Two-beam interference fringes. The top row shows thebeams overlapped exactly without displacement: (a) GaussianTEMoo mode, (b) doughnut TEMo1* mode uninverted (i.e., Fig. lb),(c) doughnut TEMo1 * mode inverted (i.e., Fig. 1c). The lower rowshows the effect of lateral displacement of the beams: (d) doughnutTEMoi* mode uninverted and displaced horizontally, (e) doughnutTEMo 1* mode uninverted and displaced vertically, (f) doughnutTEMoi* mode inverted and displaced horizontally. Note in partic-ular the half-cycle changes on the vertical fringes and compare themwith Fig. 2.

Details of the procedure for photographing the fringe pat-terns have been described in Refs. 1 and 2. In essence, theoriginal laser beam was divided at a prism beam splitter, andthe two beams were enlarged and recombined on a translucentscreen for ease of examination. The angle between the beamsand the degree of overlap could be varied in both the invertedand the uninverted arrangements. The clearest fringe pat-terns, as might be expected, were obtained with the twoTEM 01 modes of nearly equal intensity. In these two figuresthe corresponding theoretical and experimental recordingsare those of Figs. 2a and 3c, 2c and 3d, and 2b and 3f, respec-tively. It is clear that there is good correspondence betweenthe appropriate pairs of diagrams; the fringe-smearing regionwhere the phase difference between helices approaches ±7r/2is clearly reproduced in Fig. 2 as an area of constant nonzerointensity separating regions in which the fringes are of oppo-site phase. Figure 2d illustrates that addition of TEMoo modecontracts the phase-changed circle of Fig. 2c; in contrast, forthe inverted case it pushes the limbs of the phase-change cross

in Fig. 2a apart vertically. If the helices are separated to theextent shown in Fig. 2b, the addition of the same amount ofTEMoo power (C = 0.4) used in these calculations gives riseto a pattern similar to that of Fig. 2a.

The precautions needed in using the crossed-beam tech-nique in laser velocimetry have been outlined in Refs. 1 and2 and are graphically illustrated in Figs. 2 and 3. The problemis readily appreciated by considering a small particle tra-versing the fringes shown in these figures. If the beams arerecombined in an uninverted manner, then at the region ofexact overlap (e.g., Fig. 3b), there is at least no change of phase,although the beam profile varies with the particular trajectoryand is far from Gaussian. However, even in this uninvertedcase, there are changes of phase away from exact overlap (Fig.3d); on the other hand, if the beams are inverted and recom-bined, phase changes occur throughout (Figs. 3c and 3f). Inpractice it is unlikely that such a pure doughnut mode wouldpass unnoticed. Much more insidious is the possibility of aresidual TEMoo mode's filling in the hole of the TEMo1* modeto give an apparently smooth Gaussian-like profile. This maybe illustrated by the example with as low as 14% of the powerin the TEMoo mode. At first sight it might be thought that,with 86% of the power in the TEMo1 * mode, a beam wouldappear clearly to be a doughnut; in fact, the central intensityis as much as 37% of the maximum, and, given the inherentnonlinearity of the eye (and photographic emulsion) partic-ularly at high intensity, this central minimum might well passunnoticed, with the beam being classed as Gaussian. Indeed,several workers have commented on the occasional suddenenlargement of a beam as it snaps into a larger, but apparentlyGaussian, mode. It can now be recognized that this is almostcertainly the growth, as the laser gain conditions become fa-vorable, of a TEMo1 * mode around the lowest-order mode.Techniques of laser velocimetry employing the Doppler dif-ference or fringes technique usually rely on accurate mea-surement of the time intervals between fringe peaks (forstrong scattering) or on accumulating an intensity-correlationfunction (most particularly for weak scattering). Bothmethods rely on accurate knowledge of the fringe geometry;in the latter case, for work of the highest accuracy, reliableinformation is needed on the beam profile, usually consideredto be Gaussian (indeed, this assumption, that all trajectoriesof particles through the beam give rise to Gaussian scatteringprofiles of equivalent form, is usually fundamental). Ob-viously the distortion of the fringe patterns introduced byhigher-order laser modes must introduce considerable loss ofprecision and speed. The conclusion must be that stringentprecautions should be taken when two-beam fringe veloci-metry is used to ensure that the laser beam is in the lowest-order TEMoo mode. The precautions should include, at leastfor high-gain lasers, adjustment of an internal cavity apertureand, most importantly, detailed optical examination of thefringes under the actual conditions of use.

GENERATION OF HELICAL WAVE OFCONSTANT HAND

In an earlier section of this paper it was shown that theTEMo1* mode described in our experiments evolved back andforth between left- and right-hand helices since the two con-stituent TEM01 modes were oscillating at different frequen-cies. In Fig. 4 we show an arrangement designed to produce

J. M. Vaughan and D. V. Willetts

Vol. 73, No. 8/August 1983/J. Opt. Soc. Am. 1021

-\-c

3.900rotator\

BS1

3-

X/2 plate-

BS2,.I IU 1

I'l

I

W - PJPZM - _C5(9 00

RH LHFig. 4. Optical arrangement to produce a beam of pure helical co-phasal form from a beam initially in a two-spot TEMo1 mode of singlefrequency. The two paths between the beam splitters BS1 and BS2should be arranged to be approximately equal. The axis of the two-spot mode lies in the plane of the paper when the modes are shownnormal to the beam and is normal to the plane of the paper when themode axis lies along the beam. The optical path and phase betweenthe two beams may be varied by the piezoadjustable mirror at PZM.As the phase is changed the beam varies between a left-hand and aright-hand helix, as shown.

b

Fig. 5. Phase-difference contour maps (thin lines) and fringe pat-terns (thick lines, continuous and dashed) for two-beam interferenceof a true helical TEMoi* laser mode: (a) inverted and separatedbeams, (b) uninverted and separated beams. The contours are la-beled in phase difference.

a helical TEMoI* mode of either hand. A single-mode linearlypolarized laser is forced to oscillate on a TEM 0 1 mode by in-clusion of a pin within its resonator. Its output is fed into aMach-Zehnder interferometer having in one arm an opticalbeam rotator, such as a Dove prism, so that the two beams arein space quadrature. Collinearity of output polarization isensured by an appropriately orientated half-wave plate, andfine length tuning of one of the interferometer arms (e.g., bypiezoelectric means) is provided so that the desired phaserelation between the two TEM0 1 beams may be produced. Asthis phase relationship is varied, it is readily seen that the

resultant mode moves through a series of patterns. Startingin phase to produce a pure right-hand helix, a 7r/2 changeproduces a two-lobe distribution (with its axis lying betweenthe axes of the two primary beams). A further 7r/2 changegives a left-hand helix, and another 7r/2 change produces atwo-lobe distribution with its axis at right angles to the pre-vious one. A final 7r/2 displacement, making 2i7r, brings themode back to the original right-hand helix.

The fringe patterns to be expected from such a single-handhelical mode may be analyzed by recourse to the phase-dif-ference diagram method developed in Ref. 1. In previouswork calculations were performed modulo 7r since hand-switching phenomena were in evidence; for a single-hand helixthe effects of the multivalued nature of the phase-differencesurface must be included. Singularities called branch points 7

arise at the centers of the helical modes, joined by a discon-tinuity in the surface, which is a branch line. Figure 5 showsthe phase-difference contour maps for both uninverted andinverted displaced TEMo1* modes with predicted fringepatterns overlaid. Branch lines are shown as double (wherethere is a cliff of altitude 27r), arranged to contract when aTEMOO mode is mixed in. The fringes are drawn continuousand dashed to distinguish between adjacent order numbers;the order number changes by one as a branch line is crossed,corresponding to the 2 7r phase change. A fringe terminateson each of the branch points, but, since these points are at zerointensity, the fringe will not terminate abruptly. Investiga-tions of single-hand interference patterns will be the subjectof a future publication.

ACKNOWLEDGMENTS

We are indebted for useful discussions to several colleaguesat the Royal Signals and Radar Establishment and in par-ticular to W. A. Cambridge, formerly of Trinity College,Cambridge, for pointing out the existence of branch lines andfor the computer programming.

REFERENCES

1. J. M. Vaughan and D. V. Willetts, "Interference properties of alight beam having a helical wave surface," Opt. Commun. 30,263-267 (1979).

2. D. V. Willetts and J. M. Vaughan, "Properties of a laser mode witha helical cophasal surface," in Laser Advances and Applications,B. S. Wherrett, ed. (Wiley, New York, 1980), pp. 51-56.

3. P. N. Pusey, J. M. Vaughan, and D. V. Willetts, "Effect of spatialincoherence of the laser in photon-correlation spectroscopy," J.Opt. Soc. Am. 73, 1012-1017 (1983).

4. H. Kogelnik and T. Li, "Laser beams and resonators," Appl. Opt.5, 1550-1566 (1966).

5. M. Bertolotti, B. Daino, F. Gori, and D. Sette, "Coherence prop-erties of a laser beam," Nuovo Cimento 38, 1505-1514 (1965).

6. M. Carnevale and B. Daino, "Spatial coherence analysis by inter-ferometric methods," Opt. Acta 24, 1099-1104 (1977).

7. P. M. Morse and H. Feshbach, Methods of Theoretical Physics(McGraw-Hill, New York, 1953), Part I, Sec. 4.4.

J. M. Vaughan and D. V. Willetts