5
Laser heterodyne spectrometer for helioseismology David A. Glenar, Drake Deming, Fred Espenak, Theodor Kostiuk, and Michael J. Mumma The technique of laser heterodyne spectroscopy has been applied to the measurement of solar oscillations. Coherent mixing of solar radiation with the output of a frequency-stabilized CO 2 laser permits the measure- ment of fully resolved profiles of solar absorption lines with high spectral purity and excellent frequency stability. We have used this technique to measure OH pure rotation lines in the infrared (11-/m) solar spectrum. Power spectra of these line frequency measurements show the well-known 5-min oscillations as well as significant velocity power at shorter periods. I. Introduction The well-known solar 5-min oscillations, first re- ported in 1962,1 were subsequently interpreted in terms of a global network of acoustic wave modes (p modes) trapped by the sharp density gradient at the sun's visible surface and by refractive bending in the solar interior. These and the complementary gravity (g) modes are important probes of solar interior dy- namics. Their complete characterization, from mea- surements of velocity power spectra in the photo- sphere, should lead to improved models of the solar interior. 2 Ideally measurements of power spectra from spec- tral line observations require (i) excellent long-term frequency stability corresponding to Doppler velocity shifts at or below 1 m/s over the duration of the velocity time series, (ii) significant spectral line shape informa- tion to insure that the desired parameter (e.g., position of line intensity minimum or line centroid) is being accurately measured in the presence of unrelated in- tensity fluctuations, (iii) imaging capability to sepa- rate power spectra of large scale and small scale mo- tions by spatial filtering, and (iv) operation over the entire range of velocities imposed by the relative mo- tions of the earth and sun. Laser heterodyne spectroscopy near 11-/,um using wide bandwidth multichannel rf receivers has unsur- passed ability for (i), (ii), and (iv). Coherent detection also provides diffraction-limited spatial resolution which may be used to isolate the power spectrum of high degree modes, up to the seeing limit. Single- element HgCdTe photomixers can be used for single- point observations on the sun or in the integrated light mode. Multielement arrays 3 can also be used for 1-D imaging in response to requirement (iii). We have constructed a frequency-stabilized CO 2 la- ser heterodyne spectrometer for observing solar veloci- ty fields. This paper discusses the capabilities out- lined above and describes the performance of the system, which is still being optimized. We also pre- sent sample velocity power spectra from observations of infrared solar OH absorption features. II. Basic Operation The performance of CO 2 laser heterodyne instru- ments and their uses for remote sensing are discussed in detail in several publications. 4 The system we use for solar velocity measurements is similar in design to one recently used for photospheric OH observations. 5 Figure 1 shows the instrument optics and rf processing electronics. In Fig. 1(a), signal radiation from either the sun or local blackbody is combined with that from a frequency-stabilized CO 2 laser local oscillator (LO) on a high-speed square-law detector. The detector, in our case one element of a 12-element HgCdTe photo- mixer array, 3 responds nonlinearly and the output sig- nal (So) contains both dc terms and information at the difference frequencies between local oscillator and sig- nal: So a PLO + Pg + 2VL0PSig CoS(WLO - LSig)t- (1) David Glenar is withColgate University, Department of Physics & Astronomy, Hamilton, New York 13346; the other authors are with NASA Goddard Space Flight Center, Planetary Systems Branch, Greenbelt, Maryland 20771. Received 19 August 1985. Only the intermediate frequency (IF) information is of interest in heterodyne spectroscopy. In Fig. 1(b) it is further amplified and passed through sixty-four ad- jacent 25-MHz rf filters where the rf power is detected and processed by an on-line microcomputer system. Thus, the source spectrum on either side of the fixed laser line position is preserved in shape and translated 58 APPLIED OPTICS / Vol. 25, No. 1 1 January 1986

Laser heterodyne spectrometer for helioseismology

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Page 1: Laser heterodyne spectrometer for helioseismology

Laser heterodyne spectrometer for helioseismology

David A. Glenar, Drake Deming, Fred Espenak, Theodor Kostiuk, and Michael J. Mumma

The technique of laser heterodyne spectroscopy has been applied to the measurement of solar oscillations.Coherent mixing of solar radiation with the output of a frequency-stabilized CO2 laser permits the measure-ment of fully resolved profiles of solar absorption lines with high spectral purity and excellent frequencystability. We have used this technique to measure OH pure rotation lines in the infrared (11-/m) solarspectrum. Power spectra of these line frequency measurements show the well-known 5-min oscillations aswell as significant velocity power at shorter periods.

I. Introduction

The well-known solar 5-min oscillations, first re-ported in 1962,1 were subsequently interpreted interms of a global network of acoustic wave modes (pmodes) trapped by the sharp density gradient at thesun's visible surface and by refractive bending in thesolar interior. These and the complementary gravity(g) modes are important probes of solar interior dy-namics. Their complete characterization, from mea-surements of velocity power spectra in the photo-sphere, should lead to improved models of the solarinterior. 2

Ideally measurements of power spectra from spec-tral line observations require (i) excellent long-termfrequency stability corresponding to Doppler velocityshifts at or below 1 m/s over the duration of the velocitytime series, (ii) significant spectral line shape informa-tion to insure that the desired parameter (e.g., positionof line intensity minimum or line centroid) is beingaccurately measured in the presence of unrelated in-tensity fluctuations, (iii) imaging capability to sepa-rate power spectra of large scale and small scale mo-tions by spatial filtering, and (iv) operation over theentire range of velocities imposed by the relative mo-tions of the earth and sun.

Laser heterodyne spectroscopy near 11-/,um usingwide bandwidth multichannel rf receivers has unsur-passed ability for (i), (ii), and (iv). Coherent detectionalso provides diffraction-limited spatial resolution

which may be used to isolate the power spectrum ofhigh degree modes, up to the seeing limit. Single-element HgCdTe photomixers can be used for single-point observations on the sun or in the integrated lightmode. Multielement arrays 3 can also be used for 1-Dimaging in response to requirement (iii).

We have constructed a frequency-stabilized CO2 la-ser heterodyne spectrometer for observing solar veloci-ty fields. This paper discusses the capabilities out-lined above and describes the performance of thesystem, which is still being optimized. We also pre-sent sample velocity power spectra from observationsof infrared solar OH absorption features.

II. Basic Operation

The performance of CO2 laser heterodyne instru-ments and their uses for remote sensing are discussedin detail in several publications.4 The system we usefor solar velocity measurements is similar in design toone recently used for photospheric OH observations. 5

Figure 1 shows the instrument optics and rf processingelectronics. In Fig. 1(a), signal radiation from eitherthe sun or local blackbody is combined with that from afrequency-stabilized CO2 laser local oscillator (LO) ona high-speed square-law detector. The detector, inour case one element of a 12-element HgCdTe photo-mixer array,3 responds nonlinearly and the output sig-nal (So) contains both dc terms and information at thedifference frequencies between local oscillator and sig-nal:

So a PLO + Pg + 2VL0PSig CoS(WLO - LSig)t- (1)

David Glenar is withColgate University, Department of Physics &Astronomy, Hamilton, New York 13346; the other authors are withNASA Goddard Space Flight Center, Planetary Systems Branch,Greenbelt, Maryland 20771.

Received 19 August 1985.

Only the intermediate frequency (IF) information isof interest in heterodyne spectroscopy. In Fig. 1(b) itis further amplified and passed through sixty-four ad-jacent 25-MHz rf filters where the rf power is detectedand processed by an on-line microcomputer system.Thus, the source spectrum on either side of the fixedlaser line position is preserved in shape and translated

58 APPLIED OPTICS / Vol. 25, No. 1 1 January 1986

Page 2: Laser heterodyne spectrometer for helioseismology

SOLAR IR VISIBLE 1 2 TELESCOPEGUIDE SCREEN DICHROIC

L M~~IRROZ U \

1300K I ; t64 CONTIGUOLBLACKBODY 0-2 GH2 25 MH2 FILTER

OPTIONAL GAS CELL' "I KINEMATIC IF SPECTRUM1900K MIRRORS FROM PHOTO-MIXER

BLACKB0DY I PREAMPLIFIER RFH,~SOLAR R POWER

ALIGNMENT SANDPASS APIIR ~ 5LASER TELESCOPE FILTER AMPLIFER50 MH

3 9 07 ~~~~~~IR FOCUS ! Z , R OFF AX S X BEAMX-____ T / > ~BOW TIE''MXE" PAR'BOU BEASOMIRRORS I T1OR90°T CHOPPER

/ 9 BEAMSPLITTER FLATkS XS r ~~~~~~~~~~~M RROR_WIDE BANDWIDTH BEAM LASER PIEZOSTACK TUNABLEHg Cd Tel EXPANDER ATTENUATOR RPHOTOMIXER STABILIZED CO, LASER LOCAL OSCILLATOR ADJUSTABLE

TO RF T POTENTIOMETIELECTRONICS _m

WIDE BAND I C CO2 CO2 LASER MIRROR-CHOPPERLOW NOISE L SPECTRUM LOCK-LOOP SYNC SIGNAL MICROCOMPPRE AMP LO I ANALYZER STABILZER S SI - --- … - SYSTEMPy EECTR CIRCUIT E AND DISPLAY

DETCO

a b

Fig. 1. C02 laser heterodyne spectrometer for balanced detection solar velocity measurements: (a) opticalelectronics.

to rf frequencies where a narrow slice of it can be storedand displayed as a double sideband spectrum. Weobserved two rotational transitions of OH in the solarspectrum in close coincidence with 13CO2 and 14CO2laser transitions near 11.1 and 11.3 ,im. In both cases,the center of the OH features were located at differ-ence frequencies accessible to our photomixer and pre-amplifiers. When operating near 11 m, 25-MHz widefilters provide sub-Doppler spectral resolving power(v/AV) of 1.1 X 106. The total spectral bandpass of1.6 GHz can show the entire line shape, which can befitted to a Voigt profile to extract a desired line shapeparameter such as line center position. This bandpassalso provides a Doppler range of about ±20 km/sec,which is more than adequate for the range of earth-sunrelative velocities. If the relative velocity of the in-strument and the sun are accurately known in advance,a frequency-tunable rf oscillator and mixer can be usedto keep the filter bank centered on the absorption line.

Since this is a coherent detection technique, theheterodyne angular field of view on the source (Q), thecollecting aperture (A), and the wavelength (X) arerelated by the so-called antenna theorem,6

X = AQ. (2)

This means that an instrument operating at 11.1 gumusing the NASA Goddard 1.2-m telescope will sampleabout a 2-sec of arc diam portion of the solar image atthe instrument-telescope infrared focus, neglectingtelescope drift. For ground-based applications, theimage will also be blurred by seeing, but this effect isless in the infrared than in the visible.7 As with anycoherent detection scheme, the aperture-solid angleproduct is conserved, which also permits integratedlight (whole sun) measurements using very small tele-scope apertures without loss of sensitivity.

Ill. Sensitivity Considerations

Even from a strong infrared source like the sun, thedominant instrumental noise source is local oscillatorshot noise. Signal detection is achieved by synchro-nous detection using a rotating blade mirror-chopper

ANALOGisS

front end; (b) rf processing

to alternately place the source image and a referenceimage at the focus. During a single chopping cycle, amicrocomputer digitizes and coadds the source (S) andreference (R) signal levels in separate 64-channel regis-ters synchronously with the chopper at a repetitionrate of -27 Hz. A single integration produces a plot of(S - R)/R, but each S and R average is also separatelystored to further examine and process the data. Twolocal blackbody reference sources and kinematic mir-rors permit a variety of operating modes, allowing anycombination of sun and blackbody to be placed in thesignal and reference beams. Absorption features froma reference gas cell in front of the blackbody can beused to periodically check the laser stability.

Consider a laser shot noise limited heterodyne sys-tem with an infrared source in one beam having uni-form power per unit frequency interval P8 (W/Hz) andno source in the other beam. For an integration time rin seconds and IF filter width B (Hz) the channel-to-channel signal-to-noise ratio (SNR) is

SNR = (Br) 12. (3)

The term A in the denominator of Eq. (3) representsthe product of certain unavoidable losses which de-grade the instrument's performance. For an unpolar-ized source like the sun, a factor of 2 loss occurs sinceonly the polarization component parallel to that of thelaser contributes to heterodyne signal. An additionalfactor of 2 is lost since the source beam is chopped witha 50% duty cycle. Other instrument degradation oc-curs from imperfect heterodyne quantum efficiency ofthe photomixer, mismatch of signal and local oscillatorfields at the detector and various optics losses. Totaldegradation varies with the frequency response of thephotomixer-preamplifier, and increases by a factor of2-3 at small IF frequencies for the present system. Anoverall A of 10 at small IF frequencies can be routinelyachieved by careful instrument alignment and choiceof optics.

For purposes of estimating the continuum SNR dur-ing solar observations, Ps in Eq. (3) can be replaced

1 January 1986 / Vol. 25, No. 1 / APPLIED OPTICS 59

Page 3: Laser heterodyne spectrometer for helioseismology

Z 102/

101 /~~~~~~~~~~A=1100 M= 17

10-12 4 6 8 10 12 14

WAVELENGTH (MICRONS)

Fig. 2. Plot of Eq. (5) showing expected heterodyne signal-to-noiseratio in the balanced detection mode of operation. AD is theDoppler half-width at half-maximum for a molecular weight M of 17.

with the source flux from a thermal blackbody.blackbody spectral radiance is

B 0(T = h [exp (ki)- ' ]L k kT )

SCAN 519.1.01

1.00

0.99

1.01

1.00

0,zw1I-_zw.

C.

0,

The Zz0

(4) IwU(.

enCombining this with the fixed aperture-solid angle

product [Eq. (2)1, the single-sideband SNR in Eq. (3)can be expressed in terms of the source temperatureand operating wavelength

SNR = 2 [exp (jkT) h

0.99

1.01

1.00

(5)

Figure 2 shows signal-to-noise ratios predicted byEq. (5). In these curves, the individual filter band-widths have been fixed at /4 of the Doppler width(halfwidth at half-maximum) of a species with a molec-ular weight of 17 (e.g., OH). This choice of filter widthadequately samples the spectral line without an exces-sive number of intermediate frequency filters (our 25-MHz system considerably oversamples solar OH lineshapes at 11 gim). It also has the advantage of allowingfor increasingly wide passband filters at shorter oper-ating wavelengths which help to maintain the signal-to-noise ratio to wavelengths as small as 2 gim.

To fully realize the SNR predicted by Eq. (5), it isnecessary to operate in the null-balanced mode, whereequally strong sources are placed in both signal andreference beams.5 Gain nonlinearities in the filterbank electronics limit the SNR to <1000 when a stronginfrared source like the sun is present in only onebeam. Either two points on the photosphere can beused or the solar signal can be attenuated and balancedwith the local blackbody, in which case the blackbodytemperature should be used in Eq. (5). Regardless ofthe spectral distribution of the solar radiation, a localhigh temperature blackbody approximates the solarinfrared continuum extremely well over such a narrowbandpass and the blackbody temperature can be fine-adjusted to cancel the solar continuum signal to <1%.Since gain fluctuations are proportional to the differ-ence in signal between the two beams, they are greatly

0.99

1.01

1.00

0.99

17 DEC-83 21:62:41

16 32 48

CHANNEL NUMBER64

1.00SCAN 519. 17-DEC-83 21:82:41

0.97 _

0.94

1.00

0.97

5 0.94

m 1.00

0 9U)

U 0.94

0,

100

Z

0.97

0.94

1.00

0.97

0.94

.2'

/

/

17-DEC.83 21:723

LINE CENTER. CHANNEL -

55.417

. . Il

16 32 48

CHANNEL NUMBER64

Fig. 3. Portion of a 46-min time series of dual-beam solar OHobservations spanning a 5-min period. The left-hand column showsdifference signals between two beams on the photosphere; the right-hand column shows the double-sideband reconstructed line shapes.The line center is seen to move through one 5-min oscillation be-

tween the top and bottom of the right-hand series.

reduced in the null-balanced spectrum of a weak ab-sorption feature.

IV. Sample Observations

The vertical sequence of line shapes in Fig. 3 showspart of a time series of observations of the OH R22(26.5f)line with a rest wave number of 882.864 cm-'. The datawere acquired using a null-balanced laser heterodynesystem at the McMath solar telescope of the NationalSolar Observatory located at Kitt Peak, operating in the11.3-gm band of 14 CO2. This OH feature, with 10%absorption at disk center, is formed in local thermody-

60 APPLIED OPTICS / Vol. 25, No. 1 / 1 January 1986

LINE CENTER- CHANNEL -

54.787

S 82. 1 .. . . .. I I . ,

SCAN 22. 17 DEC-83 21:53:53 SCAN 522. 7-DEC.83 215t-?LINE CENTERCHANNEL - i57.817

. .. ... .. . ...

SCAN 526. 17 DEC 83 21:5:30

LINE CENTERCHANNEL -59.140

17-DEC-83 21:6:29SCAN 528.

LINE CENTER[ CHANNEL -

56.255

. . I . . . . . . . . . I

SCAN 530. 17 DEC-83 21:57:21

. . . . . . . . . . I . ... .

_ . .

--- __. . __ _ . . _

_ 8 - wr.... , , :M

I . . . . . . . . . . I . . .

a

SCAN 628. 17 ...S 1662

SCAN 53G.

Page 4: Laser heterodyne spectrometer for helioseismology

GSFC OPTICAL SITEJ+ 50 - 30 SEC VS. 26 CK84 Y

0 5

WU) 0 >0 In

s, -50 EARTH. ROTATION LABORATORY PH3 LINE

(LASER STABILITY)

0 2000 4000 6000 8000TIME (SEC)

Fig. 4. Time series of photospheric vertical velocities from re-trieved OH line center positions showing the dominant 300-s oscilla-

tions.

namic equilibrium near the temperature minimum,where the visible optical depth is -0.01.

Each result in the left-hand column shows the chan-nel-by-channel difference in intensity between twopoints on the photosphere near disk center, spaced-30 sec of arc apart. Any difference in radial velocitybetween the two points will produce an intensity dif-ference which is proportional to the derivative of in-tensity with respect to frequency in the line profile.Each of these dispersion line shapes is added to a singlerepresentative OH profile produced from a long inte-gration with the photosphere in one beam and localreference blackbody in the other. This produced thereconstructed line shape series shown at the right.Each line shape at right was then fitted to a Voigtprofile to extract the line center position and produce atime series of velocity differences.

The dispersion line shape sequence shows the domi-nant 5-min period of the solar velocity oscillations.The relative velocity between the two points on thephotosphere is a maximum in the top frame and pro-ceeds through a complete cycle in the following fourframes, which span a time interval of -5 min.

Figure 4 (top trace) shows a time series of photo-spheric vertical velocity variations from measure-ments of the OH R22(27.5f) transition at 903.769 cm-'using a '3CO2 laser. These measurements were madeat the Goddard Space Flight Center optical test siteover a 145-min time period. This series was also car-ried out in the balanced detection mode, but with onebeam on the photosphere and the other imaged on alocal blackbody. The velocity noise from the instru-ment itself was checked separately by performing thesame procedure on a weakly absorbing laboratory PH 3feature formed in a low-pressure gas cell (lower trace)and found to have a rms value of -10 m/s. This noisearises from the finite SNR on the gas cell feature,which affects the statistical quality of the fit, and noton the stability of the laser which was estimated to bestable to <10 m/sec over the duration of measurement.The dominant 300-s period can be seen, especially near4000 s, but a number of other frequency componentsalso appear.

Figure 5 shows velocity power spectra from the God-dard single-beam measurements (a) and from thedual-beam measurements at Kitt Peak (b) with a beamseparation of 30 sec of arc. Both results show the

I-

82

m

3:

0a-

I I 1

25 - 3.3 mHz

20 - I

1 -

1 : 1 I I I

0

12

10

8

6

4

2

0

SUN VS. BLACKBODYGSFC OPTICAL SITET=145 MINAT= 25 SEC

0 4 8 12 16 20FREQUENCY (mHz)

(b)

Fig. 5. Sun vs blackbody (a) and sun vs sun (b) velocity powerspectra from OH line position measurements. Trace (a) shows allspatial frequencies equally weighted. In trace (b) modes with verysmall horizontal wave numbers (h << 0.29 Mm- 1 ) are suppressed.

predominant 5-min band at frequencies near 3.3 mHz.The lower resolution in (b) is a consequence of theshorter duration for that series. The large dc compo-nent in the single-beam result (a) is due to the earth'srotation. This does not appear in (b) since two adja-cent beams record only velocity differences on thesolar disk. All spatial frequencies are equally repre-sented in (a), up to the limits imposed by seeing whichwas 2-3 sec of arc in this run. The 30-sec of arc beamseparation in (b) corresponds to -22 Mm on the solarsurface (1 Mm = 106 m). Thus, motions from horizon-tal wavelengths vh >> 22 Mm will be common to bothbeams and should be largely eliminated from the pow-er spectrum. Defining the horizontal spatial wavenumber by kh = 2r/Xh means that values of kh << 0.29Mm-1 will be suppressed in the dual-beam result.Figure 5(b) shows structure which is substantiallyabove the noise level at temporal frequencies >4 mHzand a pronounced peak near 7 mHz [the feature near 7mHz in (a) may also be real].

In the subphotospheric cavity, mode trapping oc-curs only for frequencies below 5 mHz,9 and reso-nances might therefore be unexpected at 7 mHz.However, higher frequency resonances have been gen-erated in 1-D nonlinear models of chromospheric oscil-lations.10 It is likely that the lower chromosphereforms an additional overlying cavity defined by thetemperature minimum and the steep temperature gra-dient in the transition region. This is also supportedby recent dual-beam submillimeter continuum mea-surements 1 1 which also reveal enhanced power at fre-quencies as high as 7 mHz. These observations showthe effects of adiabatic compression just above thetemperature minimum and may reveal some of thesame dynamics as our OH line measurements.

1 January 1986 / Vol. 25, No. 1 / APPLIED OPTICS 61

PERIOD ISECI

600 200150 100 80 60

K , A I A..I 1k.8).8..... A L-

(a)I lI I I I I I

SUN VS. SUN30 ARCSEC THROW -KlT PEAKT=46 MINAl= 21 SEC

Page 5: Laser heterodyne spectrometer for helioseismology

The authors wish to thank G. Halsey for helpfulsuggestions during this investigation. We also thankJ. Faris, H. Huffman, and J. Guthrie for instrumentalassistance and A. Hill for programming support. Sup-port for D. Glenar was provided by a NASA/ASEESummer Faculty Fellowship and by the NASA Astron-omy Program.

References1. R. B. Leighton, R. W. Noyes, and G. W. Simon, "Velocity Fields

in the Solar Atmosphere, I. Preliminary Report," Astrophys. J.135, 474 (1962).

2. Reviews of recent helioseismology research and future direc-tions can be found in F. Deubner, "Helioseismology: Oscilla-tions as a Diagnostic of the Solar Interior," Ann. Rev. Astron.Astrophys. 22, 593 (1984); R. W. Noyes and E. J. Rhodes, Jr.,"Probing the Depths of a Star: the Study of Solar Oscillationsfrom Space," Report of the NASA Science Working Group onthe Study of Solar Oscillations from Space (1984).

3. Wide bandwidth multielement photomixers for IR heterodyneapplications have been developed by D. Spears at MIT LincolnLabs., see D. L. Spears, "Planar HgCdTe Quadrantal Hetero-dyne Arrays with GHz Response at 10.6 um," Infrared Phys. 17,5 (1977).

4. See, for example, T. G. Blaney, "Signal-to-Noise Ratio andOther Characteristics of Heterodyne Radiation Receivers,"Space Sci. Rev. 17, 691 (1975); M. M. Abbas, M. J. Mumma, T.Kostiuk, and D. Buhl, "Sensitivity Limits of an Infrared Hetero-dyne Spectrometer for Astrophysical Applications," Appl. Opt.15,427 (1976); T. Kostiuk and M. J. Mumma, "Remote Sensingby IR Heterodyne Spectroscopy," Appl. Opt. 22, 2644 (1983).

5. D. Deming, J. J. Hillman, T. Kostiuk, and M. J. Mumma, "Ther-mal Bifurcation in the Upper Photosphere Inferred from Het-erodyne Spectroscopy of OH Rotational Lines," Sol. Phys. 94,57(1984).

6. A. E. Siegman, "The Antenna Properties of Optical HeterodyneReceivers," Proc. IEEE 54, 1350 (1966).

7. R. W. Boyd, "Wavelength Dependence of Seeing," J. Opt. Soc.Am. 68, 877 (1978).

8. The National Solar Observatory of the National Optical Astron-omy Observatories is operated by the Association of Universitiesfor Research in Astronomy under contract from the NationalScience Foundation.

9. J. W. Leibacher and R. F. Stein, "Oscillations and Pulsations,"in The Sun as a Star, NASA SP-450 (1981).

10. J. Leibacher, P. Gouttebroze, and R. F. Stein, "Solar Atmo-spheric Dynamics. II. Nonlinear Models of the Photosphericand Chromspheric Oscillations," Astrophys. J. 258, 393 (1982).

11. C. Lindsey and C. Kaminski, "Temporal Variations in the SolarSubmillimeter Continuum, Astrophys. J. 282, L103 (1984).

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continued on page 101

62 APPLIED OPTICS / Vol. 25, No. I / 1 January 1986

Meetings continued from page 471986April