Temperature variation of the refractive indices of yttrium lithiumfluoride

Norman P. Barnes and Donald J. GettemyUniversity of California, Los Alamos Scientific Laboratory, P.O. Box 1663, Los Alamos, New Mexico 87545

(Received 5 November 1979; revised 12 June 1980)The temperature variations of the refractive indices of LiYF4, YLF, have been measured in the

visible region of the spectrum. The temperature variation at 0.546 yum are -0.67x10e-'/C and- 2.30X 10-6 /'C for the ordinary and extraordinary refractive indices, respectively. A Sellmeier

equation was fit to more extensive room-temperature refractive-index data measured elsewhere.

INTRODUCTION

The temperature variations of the refractive indices of yt-trium lithium fluoride, LiYF 4, have been measured in thevisible region of the spectrum. The temperature variationof the refractive indices is responsible, in part, for the thermallensing that occurs when solid-state laser rods are pumped.The thermal lensing, in turn, can cause a degradation in thebeam quality by reducing the TEMoo mode radius and thusallowing higher-order modes to oscillate. Compensationtechniques can negate the effects of thermal lensing; however,

to accomplish this, the magnitude of the thermal lensing mustbe known. Toward this end, the variations of the refractiveindices with temperatures have been measured for yttriumlithium fluoride, hereafter referred to as YLF.

A Sellmeier equation has been fit to refractive index dataof YLF. The refractive index data, both ordinary and ex-traordinary, has been measured at room temperature else-where.' These data have been fit to a Sellmeier equation tofacilitate refractive-index calculations and provide a cdm-parison for room-temperature refractive-index data reportedhere.

1244 J. Opt. Soc. Am., Vol. 70, No. 10, October 1980 0030-3941/80/101244-04$00.50 () 1980 Optical Society of America 1244

INCIDENTRADIATIONEh %

116/ OPTIC"E, /AXIS'%

FIG. 1. Orientation of YLF crystal.

1. EXPERIMENTAL PROCEDURE

EMERGINGRADIATION

Eh

The YLF crystal prism was cut so that the temperaturevariation of both the ordinary and extraordinary refractiveindices could be measured with a single crystal prism. Thecrystal was fabricated from undoped YLF. Undoped YLFwas used for these measurements since the normal multiplydoped YLF, that is, Ho:Tm:Er:YLF, has many strong ab-sorption features which would tend to absorb at severalwavelengths of interest. The YLF was fabricated into a prismas shown in Fig. 1. The prism apex angle a was 450 55'20".The apex angle was measured by using an autocollimator anda rotary table. A larger, and more useful, apex angle could notbe used. A larger apex angle would result in a larger angle ofminimum deviation which would not allow the beam to passthrough the Dewar windows. The optic axis of the crystal wasin the plane of the prism ends. The ends of the prism wereoptically polished. This allowed the orientation of the opticaxis to be determined by rotating the prism around an axisperpendicular to the ends between a pair of crossed polar-izers.

The YLF prism was mounted in a vacuum Dewar whichallowed the prism to be cooled to liquid-nitrogen tempera-tures. The prism was mounted on a copper coldfinger thatcould be put in intimate contact with a reservoir of liquid ni-trogen. Silicon grease was used between the prism and thecoldfinger to enhance thermal conductivity. A copper plate,secured lightly by fine steel screws, held the crystal againstthe coldfinger. Steel screws were used to approximate thethermal expansion coefficient of YLF and to minimize thethermal conductivity. Silicon grease was also used betweenthe prism and the plate to enhance thermal conductivity.Thermocouples in both the coldfinger and the plate monitoredthe temperature. Since the heat conduction through thescrews is small compared with the heat conduction throughthe prism, the temperature of the prism is assumed to be theaverage of the temperatures of the coldfinger and the plate.In thermal equilibrium, the temperature difference betweenthe coldfinger and the plate was only a few centigrade degrees.Plain Pyrex flats were used as windows on the Dewar. TheDewar windows are plane and parallel to within 20". Theprism is placed in the Dewar so that the bisector of the prismapex angle is approximately parallel to the windows. Thesmall wedge in these flats was taken into account by measuringthe angle of deviation of the Dewar with the windows butwithout the prism.

1245 J. Opt. Soc. Am., Vol. 70, No. 10, October 1980

A Hg spectral lamp was used to provide a source of illumi-nation. The strong emission lines at 0.4358, 0.5461, 0.5770,and 0.5790,4m were used for the measurements. The first twolines are reasonably well separated, so no monochrometer wasused to select a single line. The last two lines were unresolvedby the YLF prism. Consequently, the averaged angle ofminimum deviation was measured for these lines. This pro-cedure introduces little error since the refractive-index dif-ference of YLF between 0.5770 and 0.5790 Am is less that0.000 10. This is on the order of the uncertainty in the mea-surements. By using the average for the two spectral lines,the error should even be smaller. Polarized radiation wasobtained by placing a Glan prism between the spectral lampand the YLF prism. The polarization could be changed byrotating the Glan prism. The angle of minimum deviationcould be measured by using a prism-type spectrometer. Thediameter of the spectrometer table, upon which the angle-measuring vernier is inscribed, is 20 cm. With this arrange-ment, the angle of minimum deviation could be measured towithin 20".

II. MEASUREMENTS

The refractive index of YLF was measured at room tem-perature and liquid-nitrogen temperature by measuring theangle of minimum deviation. All measurements were donewith the YLF prism in a Dewar. The angle of deviation withonly the windows in the Dewar was subtracted from this toobtain the angle of minimum deviation, 3. The refractiveindex of the prism was then calculated using2

n = sin[(a + 5)/2]/sin(a/2).A small correction must be incorporated into the calculationto account for the nonunity refractive index of air. Takinginto account the integrated coefficients of linear expansionof YLF and the orientation of the prism, the correction for thenonequal coefficients of linear expansion was found to benegligible. The refractive index was measured for both thehorizontal and vertical polarizations. The vertical polariza-tion n, corresponds to the ordinary refractive index no. Thehorizontal refractive index nh in an uniaxial crystal with theorientation of the prism can be computed from2

2 = cos20/n2 + sin2O/n2,

where ne is the extraordinary refractive index and 0 is theangle the optic axis makes with the direction of propagationinside of the crystal. Rearranging this equation yields

ne = sin0/(1/n 2 - cos20/n2)1/2,where nv, has been substituted for no. The refractive indicesfor both the horizontal and vertical polarizations were mea-sured at both room temperature and liquid-nitrogen tem-perature. Using the measured refractive indices, the ordinaryand extraordinary refractive indices were calculated at bothtemperatures.

To decrease the uncertainty of the measurements, the de-termination of the temperature variation of the refractiveindices was repeated several times and the results averaged.The uncertainty in the measurement results from a combi-nation of the minimum resolvable angle set by diffraction andthe minimum measurable angle set by the spectrometer. The

Norman P. Barnes and Donald J. Gettemy 1245

TABLE I. Measured refractive indices and variation of refractive indiceswith temperature of YLF. Units of on/d Tare C-1. The root-mean-squaredeviations are in the same units as the quantity appearing directly abovethem.

Wavelength 0.435 8 0.5461 0.5780no 1.46136 1.45599 1.454 996no/lT -0.54 X 10-6 -0.67 X 10-6 -0.91 X 10-6C 0.30 X 10-6 -0.49 X 10-6 0.25 X 10-6no (calc) 1.45976 1.45493 1.45402Ano 0.00160 0.00106 0.000 97ne 1.483 89 1.478 26 1.477 05bne/bT -2.44 X 10-6 -2.30 X 10-6 -2.86 X 10-6a,, 0.48 X 10-6 0.70 X 10-6 0.95 X 10-6ne (calc) 1.48308 1.47775 1.47674Ane 0.00081 0.00051 0.00031

largest prism that could be fabricated from the YLF boule hadsides that measure 0.9-cm wide and 1.0-cm high. At the angleof minimum deviation, the minimum resolvable angle2 froma prism of this size is about 17 arc sec for visible radiation. Onthe other hand, the 20-cm-diameter spectrometer table hada vernier that allowed the angle of minimum deviation to bemeasured within 20 arc sec. Since these angles are compa-rable, both angles would have to be decreased to substantiallydecrease the uncertainity in the measurement. If the angleof minimum deviation could be measured to within the min-imum resolvable angle, the uncertainty in the refractive indexwould be about 0.000 09. The change in the refractive indices,even for a 2000C temperature difference, is only on the orderof 0.00014 for the ordinary refractive index and 0.000 51 forthe extraordinary refractive index. Since this change is sosmall, even for this large temperature difference, no attemptwas made to measure the refractive indices at intermediatetemperatures as the even smaller change in the refractiveindex would probably be smaller than the measurement un-certainty. The measured horizontal and vertical refractiveindices were used to calculate the ordinary and extraordinaryrefractive indices for a given wavelength and temperature.Then, for a given wavelength, the difference in either the or-dinary or the extraordinary refractive indices at the twotemperatures, Ani, was divided by the temperature differenceAT. Typical temperature differences were about 212'C. Toobtain the variation of the refractive indices with temperature,it is assumed that onil/T, can be approximated by Ani/AT.The measurements and calculations were repeated four timesand the results averaged to decrease the uncertainty. Theresults of this process appear in Table I. Thus anil/T wasdetermined several times and the results were averaged. Theroot-mean-square deviation of the measurements, aj, is givenbelow c)ni/)T in the table. Results show that both refractiveindices decrease slowly as the temperature increases. Thevariation of the extraordinary refractive index is about 3.7times that of the ordinary refractive index.

The variation of the refractive indices with temperature ofYLF is significantly lower than the variation of the refractiveindices with temperature of most window materials. A tab-ulation of some of the more common window materials alongwith the associated On/lT is listed in Table II. The bn/bTfor the ordinary refractive index of YLF is more than an orderof magnitude lower than any of the listed materials. This

1246 J. Opt. Soc. Am., Vol. 70, No. 10, October 1980

makes YLF of interest for a window material as well as a laserrod material.

The fractional root-mean-square deviation of these mea-surements, that is ai/(oni/lT), is quite large, primarily be-cause the variation of the refractive indices with temperatureis quite small. The uncertainty in the measurement of cn/)Tis not given for all materials in Table II. However, from thenumber of significant figures quoted, it is inferred that theuncertainty in the measurements is on the order of the un-certainty in these measurements in YLF. Certainly, the re-ported uncertainities are on the order of, or larger than, theuncertainities in the YLF measurements. Since bnil/T isso small for YLF, the fractional root-mean-square deviationappears large, while a similar quantity for other materialsappears small, even though the uncertainties may be com-parable. Even for YLF the average fractional root-mean-square deviation of the extraordinary refractive index is 0.28which is smaller than the average fractional root-mean-squaredeviation of the ordinary refractive index, 0.52. This resultsprimarily from the fact that onil/T is larger for the extraor-dinary refractive index. The fractional root-mean-squaredeviation of bnil/T is on the order of the fractional root-mean-square deviation of the difference in the angle of min-imum deviation. This is what would be expected becausemni/?T is approximately linearly related to the variation in

the angle of minimum deviation. In retrospect, the apparatusused to measure the variation of the refractive index withtemperatures appears not to have sufficient resolution todecrease the fractional root-mean-square deviation in themeasurement. Perhaps a better method would be an inter-ferometric measurements'7 of this variation.

Previous measurements of the refractive indices were fittedto a Sellmeier equation in order to compare them with themeasurements reported here. The ordinary and extraordi-nary refractive indices of YLF have been measured between0.225 and 2.6 Am.' This refractive-index data was fitted toa Sellmeier equation of the form

n2 = A + BX2/(X2 - C) + DX2/(X2 - E).The results of the curve-fitting procedure appears in Table

TABLE II. Temperature variationcommon window materials.

of the refractive indices of several

Method ofMaterial ljn/c)T (106/0C) measurement References

SiO2 10 prism 3A12 0 3 13.1 prism 4BaF 2 -18.6 prism 5

-16.3 i 0.2 interferometer 13CaF2 -16.0 prism 5

-11.5 0.2 interferometer 13LiF -17.0 0.2 interferometer 13KCI -36.5 i 0.2 interferometer 13

-33.2 0.2 interferometer 7NaCi -31 prism 8CdTe 93 prism 9Ge 396 40 prism 10Si 154 15 prism 10ZnSe 10.7 i 1.0 interferometer 6

Norman P. Barnes and Donald J. Gettemy 1246

TABLE IlIl. Sellmeier coefficients for YLF. The units of the C and E pa-rameters are micrometers.2 To use the Sellmeier equation, all wavelengthsshould be in micrometers.

no ne

A 1.38757 1.31021B 0.70757 0.84903C 0.00931 0.00876D 0.18849 0.53607E 50.99741 134.95660a 0.000099 0.000074

III. The root-mean-square deviation of the measurementsto the derived curve, a, also appears in Table III. The root-mean-square deviation is less than half of the uncertainty inthe measurements, +0.0002. Using the derived Sellmeierequation, the index of refraction at the three wavelengths usedhere were calculated and appear in Table I. Since the averageangle of minimum deviation of the 0.5770- and 0.5790-gtmspectral lines was measured, this is compared with the indexof refraction at 0.5780 Aim. The difference between the cal-culated index of refraction and the measured refractive indexis somewhat larger than the uncertainty in the previouslyreported measurements. This may be due to slight compos-itional or growth differences in the crystals measured. Thefact that all of the differences between the two measurementshave the same sign and the differences for both the ordinaryand extraordinary refractive indices monotonically decreasedas the wavelength increased tends to support this hypothesis.However, a small variation in the refractive index for differentcrystals makes little difference in the accuracy of the mea-surement of the variation of the refractive indices with tem-perature.

III. EFFECT ON THERMAL LENSING IN LASERRODS

A small negative variation of the refractive index withtemperature tends to minimize the thermal-focusing effectsin solid-state laser rods. The thermally induced focal lengthof a solid-state laser rod, such as YAG, is given by 1 '12

f = (ra 2 kh/Ph)[Can3 + (1/2)(6n/bT)]-f,where Ph is the heat generated within the laser rod of radiusa, k, is the thermal conductivity, and a is the coefficient oflinear expansion. The parameter C is a function of the pho-toelastic constants and the polarization of the radiation. Fora material like YAG, both terms are positive with the firstterm significantly larger than the second. The thermallyinduced focal length of a YLF solid-state laser rod is notknown since the photoelastic constants are not known.However, if the photoelastic term is positive, it will be can-celled, at least to some extent, by the negative (Onil/T) term.While (bnj/bT) is relatively small for YLF, n is also relatively

small. Thus the term in brackets is expected to be small.Assuming the product kPh is roughly comparable for YLFand YAG, YLF may have a longer thermal focal length which,in turn, implies a larger TEMoo mode volume.

IV. SUMMARY

The variation with temperature of both the ordinary andthe extraordinary refractive indices of YLF has been measuredin the visible region of the spectrum. The parameters (ani/ZT) for both the ordinary and extraordinary refractive indicesare small and negative; however, this parameter is about 3.7times larger for the extraordinary polarization. A Sellmeierequation was fit to more extensive room-temperature re-fractive-index data measured previously. The root-mean-square deviation of the curve-fitting process is about half ofthe uncertainty of the refractive-index data. The relativelysmall negative values for the (Oni/lT) should help minimizethermal-focusing effects in solid-state laser rods.

ACKNOWLEDGMENTSThe authors wish to thank Thomas Pollak of Sanders As-

sociates for the undoped YLF material from which the prismwas made and Vito Lazazzera for fabricating and orienting theprism used for the measurements. This work was performedunder the auspices of the U.S. Department of Energy.

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1247 J. Opt. Soc. Am., Vol. 70, No. 10, October 1980 Norman P. Barnes and Donald J. Gettemy 1247