Intermode beat stabilizedlaser with frequency pulling

Shuko Yokoyama, Tsutomu Araki, and Norihito Suzuki

A frequency-stabilized two-mode He-Ne laser has been developed. The intermode beat frequency of theexperimental laser was approximately 600 MHz for a 25-cm cavity. The laser frequency in which themode stands is pulled to the center of the gain curve (frequency pulling). The degree of pulling dependson where the longitudinal modes stand in the broadened gain curve. Beat frequency is thereby changedperiodically of the order of hundreds of kilohertz with respect to cavity expansion. The frequency pullingwas effectively used for frequency stabilization of the laser. The standing position of the longitudinalmode lights was locked in the gain curve by controlling the change of intermode beat frequency. Amicrowave mixer was applied to extract the frequency change of the intermode beat. Excellentfrequency stability (1010 for the laser oscillation and 106 for the beat frequency) was attained. Thepolarization orthogonality of the proposed laser was superior to that of Zeeman lasers.

Key words: Intermode beat stabilized laser, frequency pulling, microwave mixer, heterodyne interfer-ometer.

IntroductionConsiderable need exists for a high-speed, high-resolution interferometer system. Among availablehigh-resolution interferometers, a heterodyne inter-ferometer in conjunction with a Zeeman laser iswidely used. However, present heterodyne interfer-ometers have limited target velocity because of thesmall beat frequency of the Zeeman laser. Themaximum beat frequency of the Zeeman laser is 3MHz (Hewlett-Packard catalog value), resulting invelocity limitation lower than 0.95 m/s. The Zee-man laser also exhibits complicated cross talk that iscaused by the magnetic field'; this lowers measure-ment accuracy. To increase the velocity limit andthe measurement accuracy, the application of anintermode beat of laser oscillation is attractive.Because the adjacent two longitudinal modes of inter-nal-mirror laser are orthogonally polarized, thesemodes are favorable for a heterodyne interferometer.

S. Yokoyama is with IDEC Izumi Company, Mikuni Honmachi,Osaka 532, Japan. T. Araki is with the Department of MechanicalEngineering, University of Tokushima, Jousanjima, Toukushima770, Japan. N. Suzuki is with the Department of PrecisionEngineering, Osaka Electro Communication University, Ney-agawa 572, Japan.

Received 11 January 1993; revised manuscript received 7 June1993.

0003-6935/94/030358-06$06.00/0.© 1994 Optical Society of America.

The beat frequency produced by intermode spectrallines is higher (frequency of 600-1000 MHz for 14-28cm cavity length) than for the Zeeman beat frequency(0.1-3 MHz). The use of the high-frequency beatsignal enables heterodyne interferometry that is freefrom velocity limitations.

Several methods of frequency stabilization of two-mode lasers, which use polarization properties ofinternal-mirror He-Ne lasers (polarization-stabilizedlaser), are used.- 5 The intermode beat of a polariza-tion-stabilized He-Ne laser has been effectively ap-plied for use in distance meters.6 However, thefrequency stability of the polarization-stabilized laseris not sufficient for highly precise measurements.Even though the intensity ratio of the two modes iskept constant, appreciable fluctuation in the inter-mode beat frequency of a polarization-stabilized laserexists, resulting in an ambiguous phase relationbetween the reference and objective light beams inthe heterodyne interferometer. A unique stabiliza-tion laser that can improve frequency stability by 1decade has been described.7 That laser, however,requires an additional interferometer optics for fre-quency stabilization.

The intermode beat frequency of the internal-mirror laser changes depending on where the longitu-dinal modes stand in the Doppler-broadened gaincurve. This phenomenon is explained by the conceptof frequency pulling. In the present paper, fre-quency pulling was used to stabilize the laser oscilla-tion. If intermode beat frequency is kept constant,

358 APPLIED OPTICS / Vol. 33, No. 3 / 20 January 1994

the standing position of the longitudinal mode lightcan be fixed in the gain curve. Typically, beat fre-quency varies of the order of hundreds of kilohertz forevery A of cavity expansion. This is because for amechanically fixed cavity length, optical cavity lengthchanges depending on polarization because of anisot-ropy of the cavity mirror. Such a cavity-lengthchange causes a deviation of mode-oscillation fre-quency in a few hundreds of kilohertz. The polariza-tions of adjacent modes are orthogonal to each otherbecause of mode competition. For these reasons, arepetitive oscillation line profile appears for every(X/2) x 2 of mechanical cavity expansion. Thusintermode beat frequency changes periodically for Aof cavity expanstion. (Note that because of polariza-tion flip,8 the beat frequency changed periodically forevery X/2 of cavity expansion in the present experi-ment.)

The frequency of the intermode beat is in subgiga-hertz. To date it has been difficult to control suchhigh frequency directly. In the present experiment,the intermode beat was heterodyned with a micro-wave mixer to extract the change of intermode beatfrequency. The result of these investigations is afrequency-stabilized two-mode laser.

Principle

Frequency PullingIn the laser medium, the propagation vector, k', iscomplex and is given by

k' =k[1 + 2n]2 | ki2n ) (1)

where X' and X" are the real and imaginary parts ofthe electric susceptibility, respectively; k is the propa-gation constant in the vacuum (note that k = 2r/X);and n is the refractive index of the laser medium at anonabsorbing frequency. The following relation-ship between X' and X" is termed the Kramers-Kronigequation:

X (v = A-V X- vxv)=AV

where c is the velocity of light, vm is the mthresonance frequency defined by Eq. (5), and y(v) isdefined by Eq. (6).

mcVm 2L

kX"(v)y(V) = -2n 2

(5)

(6)

Equation (4) indicates that the actual frequency (v) inwhich the mode stands is pulled to the center. Thedegree of pulling decreases as the longitudinal modefrequency gets closer to the center of the gain curve.This phenomenon is called frequency pulling. Therelation between the gain curve y(vm) and the amountof frequency pulling (v - vm) is shown in Fig. 1,where the shape of -y(vm) is assumed to be a Gaussiancurve. The curve shape of (v - v) versus vm corre-sponds to the inverse polarity of X'(Vm).

Power Spectra of Optical Beat

Confirmation of frequency pulling was achieved withan internal-mirror He-Ne laser (Siemens Model LGR-7621S, 25-cm cavity length, 2-mW output). Thislaser showed a periodical oscillation between twomodes and three modes as shown in Fig. 2(a). Theintermode beat is obtained by mixing the adjacentlongitudinal mode lights after passing through apolarizer with a polarization direction of 450 relativeto the mode polarization. When two modes (A, B orB, C) are superimposed in the gain curve, only onebeat of the spectral line is generated (fundamentalbeat frequency, Vb). When three modes (A, B, and C)exist, three beat spectra lines are generated from the

(2)

where v is the laser oscillation frequency and v andAv denote the frequencies at the center and thehalf-width of the gain curve, respectively. The phasecondition of laser oscillations is given by

kL[1 + 2n2 = mrr,

VmVO

VTVm(3)

where L is the length of the cavity. This equation istransformed with Eq. (2):

y(v)v = VM + (vo - V)

Vm + (Vo m) YITV

V0

A

eIVm

Fig. 1. Schematic shapes of the laser gain curve (y) and theamount of frequency pulling (v - v) with respect to v. The gaincurve is assumed to be Gaussian.

20 January 1994 / Vol. 33, No. 3 / APPLIED OPTICS 359

(a)gain curve

(b) (c)

5e!: Bi ' I-RN I I I l A -

8 _ T < _ 8 B_AXBI _K __ -

t ~~~9B-C

100kHz

600 MHz polarization directionFig. 2. Periodic movements of intermode beat spectral lines andlongitudinal modes corresponding with /2 cavity expansion.(situation of stage 10 is exactly the same as stage 1). (a) Longitudi-nal modes (A, B, C) standing in the gain curve, (b) movement ofadjacent mode beat spectral lines of approximately 600 MHz, and(c) movement of beat spectral line of approximately 1.2 GHz madefrom mode A and C.

For this experimental laser, the fundamental beatlines move by 150-250 kHz continuously except forstage 4-5, where the polarization flip8 was caused,resulting in a discontinuous movement of the lines(see polarization direction of longitudinal modes be-tween stages 4 and 5 in Figs. 2). The actual gaincurve is distorted because of the isotope effect.Because such distortion is evident at the wing of thegain curve, extraordinary movements of the beatlines were observed deviating from the theoreticalshifts in stages 5-7 (a smaller frequency between two600-MHz beat lines) and stages 2-4 (a larger fre-quency between the two 600-MHz lines and a 1200-MHz line).

Stabilization of Intermode BeatLaser frequencies can be locked at the certain valuesby controlling the change of the intermode beatfrequency. To extract the change of beat frequency,the beats are mixed with a local oscillator (approxi-mately 600 MHz) resulting in heterodyned beats,which are lower than the analyzed beat frequency(vb). These heterodyned beats can be observed oscil-loscopically as shown in Figs. 3. When the laser'soscillation is bimodal, the heterodyned beat forms afundamental sine wave as in Fig. 3(a). However, thebeat forms a complex waveform as in Fig. 3(b) becauseof the overlap of the two beat lines in a three-modeoscillation. The curve shown in Fig. 4 results fromthe frequency-voltage (F/V) conversion of this hetero-dyned beat. One period of this curve corresponds tox/2 expansion of cavity length. On this curve, thesection denoted the two-mode section corresponds tothe period of the two longitudinal modes' oscillation(i.e., stages 8-10 in Fig. 2). This section shows asmooth slope all the time. These results indicatethat exploitation of the two longitudinal mode sec-

interactions of modes A-B, B-C, and A-C. Thefrequency of the beat synthesized from interactionsbetween modes A and C is double that of Vb. Notethat Vb can be approximated to

cVb =L (7)

Based on this relation, a beat frequency of approxi-mately 600 MHz should result from the examinedlaser (L = 25 cm).

Intermode beat spectral lines move periodicallythrough thermal expansion of the cavity length bymeans of frequency pulling. Observed movement ofseveral spectral lines is shown in Figs. 2(b) and 2(c),where (b) shows the fundamental beat lines (madefrom A-B and B-C) and (c) shows the higher beatspectral line (made from A-C). Stages 1 through 10indicate one period movement corresponding withA/2 thermal expansion of the cavity length.

Fig. 3. Observed waveforms of heterodyned beat spectral line,when (a) one spectral line stands under two-mode laser oscillationand (b) three spectral lines stand under three-mode oscillation.The observed waveforms are the fundamental beat lines (the1.2-GHz line was eliminated by the high-cut filter in the diplexer.

360 APPLIED OPTICS / Vol. 33, No. 3 / 20 January 1994

170

I

40zw

Cawcc

10 aL

-10

(n+1) A n

'- CAVITY LENGTHFig. 4. Changes of the heterodyned beat frequency as a functionof cavity expansion. The frequency-voltage (F/V) conversionsignal was taken from the output of ITG1 in Fig. 6.

tions may be a favorable means to stabilize the laser'sfrequency.

When we observed the intermode beat signal intime, we found that its frequency decreased withcavity thermal expansion as shown in Fig. 5. Thisphenomenon can be explained by means of differentia-tion of Eq. (7).

cdLdvb cdL (8)2L2

Assuming that L = 25 cm, Vb decreased by 768 Hz foran increase of dL = X/2 (one period change). It was

- 170

- 80

-40 Zw

10 CCa.

0

shrink 4- expand

CAVITY LENGTHFig. 5. Drift of the heterodyned beat frequency with respect to thecavity shrinking. The F/V conversion signal was taken from theoutput of ITG1 in Fig. 6.

found that the actual value of dvb measured at theadjacent bottom points of Vb in Fig. 5 was slightlylarger than 768 Hz. This is because the amount ofone period change of Vb by frequency pulling variesdepending on the value of Vb itself.

Experimental and Results

Frequency Stabilization SystemThe block diagram of the intermode beat frequency-stabilization system is shown in Fig. 6. The backbeam (intensity = 70 pLW) of this laser is used bymeans of a feedback pathway to stabilize the laser'sfrequency. The back beam passes through a polar-izer (directed at 450 to each of the longitudinalmodes), resulting in generation of intermode opticalbeats (approximated frequency, Vb = 600 MHz and2vb = 1.2 GHz). Resultant optical beats are directedto the avalanche photodiode through an optical fiberguide. After amplification, these beat signals aremixed with a local oscillator signal (frequency, f) witha double-balanced mixer. It is necessary to selectthe frequency of the local oscillator as close to vb aspossible to facilitate signal processing of the interme-diate frequency output.

The intermediate frequency output of the double-balanced mixer contains the objective heterodynedbeat signal (frequency, fh = Vb - f) and an imagesignal (frequency, b + f). To eliminate the imagesignal, the intermediate frequency output is filteredby a diplexer. The sigmoidal waveform of the fil-tered beat signal is converted to a square waveform bythe comparator. The frequency difference detectionunit compares the heterodyned beat frequency (fh)

with a reference frequency (fr). The reference fre-

Dilxr M.HCmaao unit I

Fig. 6. Block diagram of the intermode beat stabilization system.PL, polarizer; APD, avalanche photodiode; DBM, double-balancedmixer; ITG1, integrator with small time constant; ITG2, integratorwith large time constant.

20 January 1994 / Vol. 33, No. 3 / APPLIED OPTICS 361

two modes section

quency is synthesized by a quartz oscillator. Thisfrequency is subsequently divided by a transistor-transistor logic counter to be variable. The fre-quency difference detection unit generates up anddown pulses that are proportional to the positive(fr - fh > ) and negative (fr - fh < 0) frequencydifferences of the two inputs, respectively. The uppulse is generated as positive polarity, whereas thedown pulse is generated as negative polarity, asdepicted in Fig. 6. The up-down pulse series areintegrated successively by an integrator (ITG1) with adischarge resistance (time constant = 3 ms). Theoutput of ITG1 corresponds to a F/V conversion ofthe frequency difference between fh and f. (Thecurves shown in Figs. 4 and 5 resulted from theoutput of ITG1). ITG1's output is amplified to con-trol the current of the heater, which is wound aroundthe laser tube. Thus the signal loop by way of ITG1forms a servocontrol loop (main control system).

Because the cavity length is determined by thethermal balance between input and output energy ofthe discharge tube, the heater current value is afunction of both the surrounding temperature andthe cavity length. If the servocontrol consists ofonly the main control system, the resultant controlsystem has one-to-one correspondence between theheater current and the value of (fh - f), that is, thecavity length. Thus both the value of (fh - f) andthe heater current should be changed when thesurrounding temperature changes. The value of(fh - f) is thereby dependent on the surroundingtemperature.

To stabilize the value of (fh - f) independent ofsurrounding temperature, such one-to-one correspon-dence must be broken. For this reason, anotherintegrator (ITG2) with an input of up-down pulsetrains is interpolated parallel to ITG1. The resultantservocontrol system has one-to-one correspondencebetween the total output of ITG1 and ITG2 and theheater current value.

ITG2 has no discharge resistance (actual time con-stant 2 min) and functions such that its outputlevel moves monotonically unless ( fh - fr ) = 0 aftercompletion of the main control. If the balancedvalue of (fh - fr) that is completed by the maincontrol system deviates from 0 Hz, ITG2 shifts boththe heater current and ITG1's output level such thatthe value of( fh -fr) becomes exactly 0 Hz. Once thevalue of (fh - fr) becomes exactly 0 Hz, ITG2 shiftsthe heater current along to keep (fh - fr) =_ 0 (there-fore, ITG1's output level remains unchanged, evenwhen the surrounding temperature changes). Ac-cordingly, the adjacent mode beat, b, is locked at(fo + f) independently of surrounding temperature.

The frequency difference curve for free-runningand stabilized operation is shown in Fig. 7. Thisfigure indicates that the curve shifted immediatelyafter the heater control switch was turned on andthat the beat frequency was approximately stabilizedby the main control loop with ITG, in 2 min.Approximately 1 min. later, ITG2 started to function

ON

170NI

80

40Ow

10aocr

-10

*- TIME [ 1 min / division ]

Fig. 7. Changes of the heterodyned beat frequency before andafter the action of the stabilization circuit. The stabilizationcontrol was turned on after progressing the free-running operation.The F/V conversion signal was recorded at the output of ITG1 inFig. 6.

such that the frequency difference curve converged to0 Hz within 3 min.

In the actual stabilization system, the frequenciesof the oscillators and the consequent value of Vb areset as follows. Because of the narrow bandwidth ofITG1, the conversion of ITG1 does not functionlinearly at the input frequency of the up-down pulsesabove 200 KHz. Therefore it was necessary to re-duce in advance the frequency difference between fhand fr to be smaller than 200 kHz. The local oscilla-tor and the reference frequency were set as follows:On controllable tube temperature, f was set slightlylower than vb as approximated by Eq. (7). Because vbdecreased in proportion to cavity expansion [shown inEq. (8)], Ah correspondingly decreased in time as thethermal expansion of the laser cavity reached ad-equate values (i.e., smaller than 200 kHz). In thisexperiment, the local oscillator was fixed at 606.996MHz; this value was realized by multiplying thequartz oscillation (50.5830 MHz) by 12. To use thisstabilization circuit, one should operate the laserwithin the two-mode section shown in Fig. 4.Because the stabilization circuit functions to reducethe difference between fh and f, the reference fre-quency should be set such that this section of thecurve crosses 0 Hz for the control tube temperature.The reference frequency was thus set at 39 kHz.Consequently b was locked to 606.996 MHz + 39kHz.

To obtain excellent resettability, turn-on timing ofstabilization operation is determined in free-runningoperation as follows: As shown in Fig. 5, cavitylength can be determined with the bottom value of Vbin each period. Therefore the cavity length is alwaysfixed at a certain value by starting the stabilizationoperation at the period with a bottom b that isconsistent with a settled value.

Frequency StabilityBecause the laser is developed to use the intermodebeat, two-mode oscillation and stability of intermodebeat frequency are the focal points. Analysis of the

362 APPLIED OPTICS / Vol. 33, No. 3 / 20 January 1994

40-

100 0w

.40

TIME [2 min/division]Fig. 8. Changes of intermode beat frequency obtained from thepolarization stabilized laser. The F/V conversion signal wastaken from the output of ITG, in Fig. 6.

beat spectrum confirmed the dual-mode laser output.Resultant fluctuations of the heterodyned intermodebeat frequency, fh, were within 0.5 KHz, correspond-ing to a frequency stability of 106. This stability wasdependent on the quartz oscillator's stability (betterthan 106).

To evaluate laser system performance, the inter-mode beats generated from a typical polarization-stabilized laser (oscillating in two modes) was exam-ined. Beat fluctuations were recorded as describedin Fig. 7. A resultant fluctuation curve is shown inFig. 8. Comparison of the curves in Figs. 7 and 8confirmed that the stability of the proposed laser was2 decades superior to that of the conventional polariza-tion-stabilized laser.

Frequency stability of the 632.8-nm line of theresultant laser was also examined by mixing this laseroutput with an I2-stabilized laser (provided by OsakaElectro-Communication University, Neyagawa, Ja-pan). The square root of the Allan variance of thislaser was 4.0 x 10-10 (10 s), 4.5 x 10-10 (100 s), and4.5 x 10-1 (1000 s). [Note that the value ofthe polarization-stabilized laser was 1.0 x 10-9 (10 s),5.3 x 10-9 (100 s) and 9.6 x 10-8 (1000 s).]

To examine the frequency resettability, the fre-quency of the blue side mode was measured repeti-tively with an appropriate time interval. For thisreason, power switch turn-on actions of the examinedlaser were repeated 12 times during 2 weeks. Theresultant frequency was within 473 612 443.2 ± 0.20MHz.

Conclusion

A new laser system with a frequency that was highlystabilized in two modes was developed. A periodical

change in intermode beat frequency with cavity expan-sion was caused by frequency pulling. The change inbeat frequency was used to control the laser's fre-quency. For this reason, heterodyne technique devel-oped for processing of microwave signals was applied.The frequency change of the intermode optical beatwas effectively extracted with this technique. Theresultant beat frequency was 600 MHz. Stabiliza-tion values attained were 1010 for the laser frequencyand 106 for the beat frequency. (Note that forpolarization-stabilized lasers that use a conventionalapproach a beat frequency stability of 104 and a laserfrequency stability of 107 result.) The intermodebeat can be obtained without application of magneticfields, which are required for the generation of theZeeman beat. The magnet-free characteristic as-sures polarization orthogonality of the mode lines.

Therefore even undesirable mode cross talk can becompensated by optoelectronics devices. These fea-tures are satisfactory for the light source for high-speed and high-resolution heterodyne interferometers.The proposed electric heterodyne technique for laserstabilization can be further applied to the signalprocessing of a heterodyne interferometer.

We thank David M. Coleman, Department of Chem-istry, Wayne State University, for helpful discus-sions.

References1. Y. Xie and Y. Wu, "Elliptical polarization and nonorthogonality

of stabilized Zeeman laser output," Appl. Opt. 28, 2043-2046(1989).

2. R. Balhorn, H. Kunzmann, and F. Lebowsky, "Frequencystabilization of internal-mirror helium neon lasers," Appl. Opt.11, 742-744 (1972).

3. S. J. Bennett, R. E. Ward, and D. C. Wilson, "Commentson: Frequency stabilization of internal mirror He-Ne lasers,"Appl. Opt. 12, 1406 (1973).

4. A. Sasaki and T. Hayashi, "Amplitude and frequency stabiliza-tion of an internal-mirror He-Ne Laser," Jpn. J. Appl. Phys.21, 1455-1460 (1982).

5. T. Yoshino, "Frequency stabilization of internal-mirror He-Ne(X = 633 nm) lasers using the polarization properties," Jpn. J.Appl. Phys. 19, 2181-2185 (1980).

6. K. Seta and T. O'ishi, "Distance meter utilizing the interbeat ofa He-Ne laser," Appl. Opt. 29, 354-359 (1990).

7. T. Araki, Y. Nakajima, and N. Suzuki, "Frequency and intensitystabilization of a high output power, internal-mirror He-Nelaser using interferometry," Appl. Opt. 28, 1525-1528 (1989).

8. E. K. Hasle, "Polarization properties of He-Ne lasers," Opt.Commun. 31,206-210 (1979).

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