TECHNICAL NOTE

Optical-activity measurementswith bihelicoidal laser eigenstates

P. Lagoutte, Ph. Balcou, D. Jacob, F. Bretenaker, and A. Le Floch

A novel, doubly differential method for the measurement of optical activity that uses the so-calledbihelicoidal eigenstates of a laser is demonstrated. An experimental realization is presented, andprospects are discussed.Key words: Optical activity, helicoidal standing waves, laser intracavity measurement.

1. Introduction

Optical-activity measurements have routinely beenperformed by chemists and biologists for more than acentury,1,2 and they are still an active area of interestin chromatographic detection.3 In the classicalmethod, a linearly polarized light beam is injectedinto the compound, and the rotation angle u of theplane of polarization is directly measured at the exit.The measurement is made by adjustment of a crossedpolarizer and analyzer. The ultimate accuracy de-pends on the quality of the polarizers and is currentlyof the order of 1023 deg. Higher accuracy is instru-mental in the life sciences, in particular in the studyof racemization of chemicals. Different authors havetried to enhance the resolution by the use of multiplereflections in a resonator.4–7 These methods requirethe injection of an external light source into a passivecavity and therefore demand precise mode matching.The question arises then as to whether chiralitymeasurements can be performed inside an activecavity.Here we propose a novel method for the measure-

ment of small optical activities: a chiral medium isinserted directly into a laser cavity whose eigenstatesare bihelicoidal stationary waves. Helicoidal eigen-states, described for the first time, to our knowledge,by Evtuhov and Siegman,8 and independently by

The authors are with the Laboratoire d’Electronique Quantique,Physique des Lasers, Centre National de la Recherche Scientifique,Unite de Recherche Associee 1202, Universite de Rennes I, 35042Rennes Cedex, France. F. Bretenaker is also with the Societed’Applications Generales d’Electricite et de Mecanique, Boıte Post-ale 72, 95101Argenteuil Cedex, France.Received 20 June 1994; revised manuscript received 7 November

1994.0003-6935@95@030459-04$06.00@0.

r 1995 Optical Society of America.

Kastler,9 have the remarkable property of being eigen-states of chirality. Our goal is to show that bihelicoi-dal eigenstates,10 formed by segments of differenthandednesses, can be used to carry out differentialmeasurements that are free of systematic uncertain-ties. In Section 2 we explain the theoretical prin-ciple that underlies the technique. In Section 3 thescheme is applied to the specific case of a quartz rod.

2. Theoretical Principle

A chiral medium essentially rotates the plane ofpolarization of an incident light beam around thepropagation axis. A basic feature is that the sign ofrotation depends on the direction of propagation ofthe light. Let u be the rotation angle at the exit ofthe compound, at a given wavelength. We propose toprobe this medium with helicoidal waves, which aresuperpositions of pairs of counterpropagating circu-larly polarized waves, both of them right 1or left2circular. We now show that bihelicoidal eigenstatesof a linear laser, which consist of two segments ofhelicoidal waves of opposite handednesses, can beused to measure optical activity located inside thecavity differentially.Figure 1 presents the schematic diagram of the

laser, which consists of a two-mirror Fabry–Perotcavity that encloses a quarter-wave plate, QWP1, ahalf-wave plate, HWP, and a second quarter-waveplate, QWP2. a and b denote the angles between thex axis and the fast axis of the HWP and the QWP2,respectively. For simplicity we keep the fast axis ofQWP1 fixed at 45° of the x axis. Two eigenstatesexist in general, and these states have been studiedwith the spatial vectorial model, within the frame-work of the Jones matrix formalism.10 The polariza-tion is linear between M1 and QWP1 and betweenQWP2 and M2, helicoidal between QWP1 and HWP,

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and helicoidal of opposite handedness between HWPandQWP2. The helicities are reversed for the secondeigenstate. When the frequency difference betweenthe two eigenstates is large, the coupling betweenthem can be made smaller than 1 when the activemedium is placed in a part of the cavity in which thepolarization is linear. The two modes can then oscil-late simultaneously. The chiral medium can be lo-cated in two positions, labeled 1 and 2. Helicoidalwaves are eigenstates of optical activity, that is, theirpolarization remains invariant when the waves aregoing through a chiral medium. Each eigenstatesimply undergoes a phase shift. The total round-tripphase shift F 1F 5 62u2, whose sign depends on boththe direction of propagation and the wave handed-ness, alters the two eigenfrequencies.Let us insert the compound in position 1. x and y

refer to the bihelicoidal eigenstates whose polariza-tions are along x and y, respectively, between M1 andQWP1. The spatial vectorial model yields the eigen-frequencies as eigenvalues of the round-trip Jonesmatrix for modes x and y as

n1x, y 5

c

2L 3q 6 12a 2 b

p11

42

F

2p24 , 112

where q is a positive integer number, L is the opticallength of the laser, and c is the speed of light invacuum. Note that the eigenfrequencies depend onthe retardation imposed by the different waveplatesand therefore on their relative angles a and b.10 Thebeat frequency Dn1 between these two eigenstates is

Dn1 5 n1x 2 n1

y 5c

L 12a 2 b

p11

42

F

2p2 . 122

Let us now insert the chiral medium in position 2.The frequencies associated with the two eigenstates

Fig. 1. Schematic diagram of the Fabry–Perot cavity, whichconsists of a plane mirror, M1, and a curved mirror, M2. Twoquarter-wave plates 1QWP1 and QWP22 and a half-wave plate1HWP2 are inserted inside the cavity. The angles between the fastaxes of QWP1, HWP, and QWP2 and the x axis are equal to 45°, a,and b, respectively. The dashed rectangles represent the twopossible positions for the chiral medium in the bihelicoidal stand-ing wave. The eigenstate polarizations between the mirrors andthe QWP’s are linear and oriented at 45° from the fast axes of theplates.

460 APPLIED OPTICS @ Vol. 34, No. 3 @ 20 January 1995

in this case are

n2x, y 5

c

2L 3q 6 12a 2 b

p11

41

F

2p24 . 132

Beats between these two eigenfrequencies occur atfrequency Dn2, which is given by

Dn2 5 n2x 2 n2

y 5c

L 12a 2 b

p11

41

F

2p2 . 142

Taking the difference Dn2 2 Dn1 eliminates all theangles of the waveplates, which are not known to highaccuracy. As c@2L, Dn1, and Dn2 can be accuratelydetermined from the experiment, one directly obtainsu as

u° 5 45°3Dn2 2 Dn1

1 c2L2 4 . 152

This simple result can be obtained thanks to thedouble differential nature of the experiment. In-deed, the cavity generates the two opposite bihelicoi-dal eigenstates, each of them exhibiting two oppositehandednesses in positions 1 and 2.

3. Application to Optical-Activity Measurement of aQuartz Rod

As a demonstration, we apply this scheme to measurethe optical activity of a 10-mm-long right-handedquartz rod in the midinfrared. Our cavity is amonomode linear He–Ne laser that oscillates at 3.39µm. The experimental arrangement is presented inFig. 2. The 0.95-m-long hemispherical cavity con-sists of a plane mirror and a spherical mirror of 6-mradius of curvature. The elements are mounted onInvar rods to reduce thermal drift. A diaphragmprevents the high-order transverse modes from oscil-lating by induction of diffraction losses. The activemedium is a 370-mm-long discharge tube, which isfilled with a 7:1 3He–20Ne mixture at a 1 Torr total

Fig. 2. Experimental setup used to measure optical activity withbeat frequencies: D1, D2, D3, photodiode detectors; PZT, piezoelec-tric transducer; P, linear polarizer; M1, plane end mirror; M2,curved end mirror.

pressure and is closed with quasi-perpendicular silicawindows. The helicoidal eigenstates are establishedby placement of waveplates 1two QWP’s and one HWP2inside the cavity. The chiral medium is studiedsuccessively in position 1 1between QWP1 and HWP2and then in position 2 1between HWP and QWP22.Actually, the chiral medium is left fixed inside thecavity, and we switch between the two positions bymoving the HWP on either side of it. The twomethods are totally equivalent in theory, but thelatter is easier in practice.Figure 3 shows the total output intensity as a

function of cavity frequency for the two eigenstatestogether 1a2 without any chiral medium in the cavity,1b2 with the quartz rod in position 1, and 1c2 with thequartz rod in position 2. The frequency scale is 20MHz@division. The frequency splitting between theeigenstates corresponds to the difference between thetwo maxima. This frequency splitting obviously var-ies depending on the amount of optical activity that ispresent in the cavity and the position of the chiralmedium. The difference between the splittingsshown in Figs. 31b2 and 31c2 is approximately 15 MHz,indicating a rotation u of approximately 4°. To ob-tain beats between the bihelicoidal eigenstates, wemust force the laser to operate in a simultaneityregime. To do this we first separate the intensityversus frequency profiles of the eigenstates, as pre-sented in Figs. 31a2–31c2, with polarizing beam split-ters. The resulting output-intensity profiles, de-tected with photodiodes D1 and D2 1Fig. 22, aresuperposed in Fig. 31d2. In the central frequencyrange, both modes oscillate together. The laser isservolocked to the frequency for which the eigenstate

Fig. 3. Profiles of output intensity versus frequency, obtainedwhen the cavity length was varied, 1a2 without any chiral mediuminside the cavity, 1b2 with the quartz rod in position 1, and 1c2 withthe quartz rod in position 2. The laser intensity is in arbitraryunits. The frequency scale is 20 MHz@division. 1d2 Superposedfrequency profiles of the eigenstates, obtained through a polarizingbeam splitter. The frequency scale is 10.4 MHz@division. Thearrow that points to curve 1d2 indicates the point at which thesystem is servolocked.

intensities are equal, in the middle of the simultane-ity range. The error signal is provided by an elec-tronic differential circuit that compares the two inten-sities, and the correction signal is sent on to thepiezoelectric transducer that carries mirror M1. Thelaser output goes through a polarizer to a photodiode,D3, to yield the beat signal. The beat frequency isthen obtained from a spectrum analyzer.Figure 41a2 presents the two beat-frequency spectra

for positions 1 and 2. The frequency difference be-tween the peaks is measured to be 14.540MHz. Thisfrequency difference depends only on the rotatorypower of quartz, the rod length, and the laser free-spectral range c@2L. The free-spectral range isreadily obtained from beat-frequency measurementsbetween longitudinal modes, with an accuracy of 61kHz. Figure 41b2 displays a detailed profile of thebeat-frequency peak. The full-width at half-maxi-mum is 1 kHz, so the resulting accuracy of themeasurement is du 5 3.1024 deg. As a whole, weobtain

Dn2 2 Dn1 5 14.540 6 1023 MHz,

with c@2L 5 157.429 6 1023 MHz.Using Eq. 152, we find that the value of u is 4.1562 6

3.1024 deg for a 10-mm-long rod. Our experimentalvalue for the rotatory power of quartz at 3.39 µm isthus

3u°43.39µm 5 0.41562 6 3.1025 deg mm21.

1a2

1b2

Fig. 4. 1a2 Intensity spectrum of beats between the eigenstates1linear scale2 for positions 1 and 2 of the chiral medium. Themeasured beat frequencies are Dn1 5 53.182 MHz and Dn2 5

67.722 MHz. 1b2Detailed profile of the beat peak in position 1.

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Extrapolation from the formula of Lowry and Ad-ams,11 which is valid up to 2.4 µm, yields a value of0.44 deg mm21, in reasonable agreement with ourmeasurement.

4. Conclusion

Anew method for the measurement of optical activitythat uses bihelicoidal eigenstates of a laser is pre-sented. It is experimentally demonstrated with aquartz rod in the midinfrared. This first experimen-tal realization has achieved accuracies of the order of3.1024 deg. More sophisticated setups, which arecapable of frequency stability of the order of 10 Hz12,131for instance, setups that use Zerodur cavities2, shouldresult in an accuracy of approximately 1026 deg. Thenew method, which is also applicable to the visiblespectral range, is thus capable of sensing minuteoptical activities, and so it is promising for numerousapplications.

This work was partially supported by the Directiondes Recherches, Etudes et Techniques and the Etab-lissement Public Regional de Bretagne. We thankthe Fichou S. A. society for the loan of the quartz rod.Discussions with J.-P. Tache, G. Ropars, M. Vallet,N. H. Tran, R. Le Naour, and J. C. Cotteverte aregratefully acknowledged.

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