Water ice is of great interest to climate and remotesensing studies because it is found in type II polarstratospheric clouds ~PSCs!, cirrus clouds, contrails,and surface snow. Type II PSCs form at ;188 K inthe polar stratospheres during the winter and play animportant role in polar ozone depletion. Cirrusclouds occur primarily in the upper troposphere and
Bhavani Rajaram, David L. Glandorf, Daniel B. Curtis, andMargaret A. Tolbert [email protected]! are with the De-partment of Chemistry and Biochemistry and Cooperative Insti-tute for Research in Environmental Sciences, University ofColorado, Boulder, Colorado 80309. O. B. Toon is with the Lab-oratory for Atmospheric and Space Physics and the Department ofAtmospheric and Oceanic Sciences, University of Colorado, Boul-der, Colorado 80309. N. Ockman [email protected]! can beeached at 9 Monte Alto Road, Santa Fe, New Mexico 87505.
are found at all latitudes and seasons. They have aglobal coverage as high as 30–40%.1 Cirrus cloudsform at temperatures between 190 and ;250 K andre composed almost exclusively of ice crystals.ontrails are formed from aircraft emissions mainlyt temperatures below a threshold of the order of 233.2Cirrus clouds and contrails have been identified as
two of the most uncertain components in regulatingthe Earth’s climate and variability.3,4 Cirrus cloudsreflect solar radiation back to space but also contrib-ute to the greenhouse effect by absorbing and radi-ating terrestrial infrared. Whether these cloudstend to cool or heat the Earth depends in part on theirradiative properties, which in turn depend on theirmicrophysical properties ~for example, ice particleshape, size distribution, ice-water content, and opti-cal thickness!.
Satellite observations provide the opportunity toderive these microphysical properties on scales re-quired by general circulation models, but accurateoptical constants are needed for proper retrieval.
1 September 2001 y Vol. 40, No. 25 y APPLIED OPTICS 4449
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Solar radiation reaching the top of the atmospherehas approximately 49% of its energy at wavelengthsbelow 0.7 mm and ;51% in the longer-wavelengthregion up to 5.0 mm ~near-infrared region!,5 makinghe latter frequency region an important one. Sincehe optical constants of ice vary appreciably acrosshe near infrared, ice crystals of sizes found in cirrusredominantly scatter light at some wavelengths,hereas they absorb appreciable amounts of light at
ther wavelengths. Figure 1 shows some of thehannels used by satellite instruments in the near-nfrared region along with a typical absorbance spec-rum of water ice for a film whose thickness, 100 mm,s typical of cirrus particle sizes. Also summarizedre some primary applications of these instruments.his figure clearly demonstrates the need for accu-ate optical constants for water ice in the near-nfrared region, particularly near 6250 cm21, whereeveral satellites are making observations. The cur-ently available optical constants disagree with onenother at this wave number by at least 65%.Almost all the calculations of the single-scattering
roperties of cirrus clouds6–10 and microphysicalproperty retrievals of cirrus clouds or contrails11–15 atnear-infrared wavelengths used the refractive indicesof water ice from Warren’s compilation16 based on
easurements made during the 1950s and 1970s17–19
or the refractive indices from the research of Kou etal.20 and Gosse et al.21 Warren’s compilation of therefractive indices of ice was made at 266 K and isrecommended by him for use at temperatures be-tween 213 and 273 K.
For the regions of the band maxima ~5291–4739
Fig. 1. Near-infrared absorption spectrum of a ;100-mm-thickfilm of water ice along with some satellite instruments that havechannels in this region ~CERES, Clouds and the Earth’s RadiantEnergy System; MODIS, Moderate-Resolution Imaging Spectrora-diometer; ETM-plus, Enhanced Thematic Mapper Plus; ASTER,Advanced Spaceborne Thermal Emission and Reflection Radiom-eter; VIRS, Visible Infrared Scanner; and EOSP, Earth ObservingScanning Polarimeter!. Also summarized are the principal appli-cations of the instruments. More information about these instru-ments can be obtained from the following website: http:yyspsosun.gsfc.nasa.govycgi-binyeos-kshyeosiinstru.kshy.
450 APPLIED OPTICS y Vol. 40, No. 25 y 1 September 2001
and 6896–6211 cm ! Warren’s compilation used theimaginary indices calculated from Ockman’s17 trans-mission spectrum of a 100-mm sample of water ice at43 K. It was assumed that the optical constantsere independent of temperature between 243 and66 K. Ockman did record the transmission spec-rum of a 358-mm sample at 247 K but did not publisht either in his dissertation or in subsequent studies.
arren16 obtained this spectrum from Ockman in1983, but, unfortunately, not early enough to makefull use of it in his compilation. Reding’s18 transmis-sion spectrum of a 250-mm film ~195 K! was the bestvailable at the time of Warren’s compilation.ence, for the valleys ~4739–3817- and 6211–5291-
m21 regions! between the bands mentioned at thebeginning of this paragraph, Warren’s compilationused imaginary indices calculated from the spectrumof Reding.18 These imaginary indices correspondingto 195 K were shifted to shorter wavelength to makethem appropriate for use at 266 K. Further, al-though both Ockman17 and Reding18 measured thetransmission of their water ice samples sandwichedbetween windows, the reflection losses at the air–window and the window–ice interfaces were nottaken into account in the calculation of the imaginaryindices. As shown in Appendix A, this neglect canresult in errors as high as 28%. Ockman’s17 trans-mission spectra were considered by Warren to bemore reliable than Reding’s18 since the sample thick-nesses were better known and also because there wassome ambiguity in Reding’s data that was due to hismislabeling the ordinate on his spectra. Warren,16
however, believed that because of the possibility ofdiffuse scattering from the imperfect ice surfaces inthis frequency region of small absorption, it is alwaysbetter to choose the lowest values available for theimaginary indices. Since the imaginary indicesfrom Reding’s data18 were lower in the valleys, theywere used in his compilation.16
The optical constant measurements of Kou et al.20
and Gosse et al. 21 correspond to temperatures close to250 K. This temperature is at the higher end ofwhat is relevant for cirrus clouds and much too highfor PSCs. Recently, temperature-dependent absorp-tion coefficients of water ice from 20 to 270 K in thenear infrared were published by Grundy andSchmitt.22 The data presented in the paper byGrundy and Schmitt22 are the result of fitting theabsorption coefficient spectrum with a set of 17Gaussian functions. The Gaussian functions werechosen such that the same set, with frequencies,widths, and intensities smoothly varying with tem-perature, were able to fit the absorption coefficientspectrum at all the temperatures considered ~20–270K!. The fitted parameters corresponding to theGaussian functions were then used to parameterizethe temperature-dependent water ice absorption co-efficient spectrum. To our knowledge the results ofGrundy and Schmitt22 are the only data available todate that cover the entire temperature range rele-vant to PSCs and cirrus clouds. Table 1 summarizesthe various experimental measurements of the opti-
m
Table 1. Summary of Recent Experimental Measurements of the Optical Constants of Water Ice in the Near Infrareda
cal constants and absorption coefficients of water icein the near infrared from each of these groups.
Figure 2~a! compares the imaginary indices ofGrundy and Schmitt22 at 270 K and the values inWarren’s compilation16 at 266 K. Figure 2~b! showsthe percentage difference between the two data setsrelative to the values of Grundy and Schmitt.22 Itcan be seen that the agreement is within 20% aroundthe 5000-cm21 band and in the two valleys on eitherside of it, but the difference is more than 20% almosteverywhere else. As shown in the following para-graphs, the values in Warren’s compilation16 andthose of Grundy and Schmitt22 differ from all theother results in the valleys around the 5000-cm21
band by more than 35%.The data of Gosse et al.21 and of Kou et al.20 agree
well with each other but do not agree well with War-ren’s compilation. The former authors did accountfor reflection losses at the interfaces by using theratio of spectra corresponding to different pathlengths to derive optical constants. Although notexact, our data have confirmed that the error in this
AuthorTemperature
~K!Sample
Thickness
Current study 166 16 films,;255–956 mm
176 14 films,;267–1055 mm
186 19 films,;247–1164 mm
196 18 films,;264–1135 mm
Grundy andSchmitt, 1998b
20–270 100 mm–1.0 cm
Gosse et al.,1995c
251 2 films,,1 and 309mm
Toon et al., 1994d 166 0–30 mm
Kou et al., 1993e 248 116–269 mm
Ockman, 1957f 247 358 mm
aAll these experiments measured transmission.bRef. 22.cRef. 21.dRef. 28.eRef. 20fRefs. 17 and 23.
1
approach is less than 10%. Since Warren basedsome of his values on an incomplete analysis of Ock-man’s data, we felt it would be significant if Kou etal.20 and Gosse et al.21 also agreed with the results ofthe reanalysis of Ockman’s spectrum.23 Thus we ob-tained Ockman’s transmission spectrum of a 358-mmsample23 of water ice ~247 K! and extracted imagi-nary indices of refraction from it by properly takinginto account the reflections at the air–glass and theglass–ice interfaces predicted by the theory of theoptical properties of thin solid films ~see AppendixA!.24 An important reason for doing this is that boththe temperature and the sample thickness are com-parable with what Gosse et al.21 and Kou et al.20 used.The model used to extract the imaginary indices isdescribed in detail in Appendix A.
Figure 3~a! compares the new results from Ock-an’s 358-mm transmission spectrum23 with those of
Grundy and Schmitt22 at 250 K, Gosse et al.21 at 251K, and Kou et al.20 at 248 K. Figure 3~b! shows thepercentage difference between each of these threedata sets ~relative to Ockman’s data!, plotted as a
mple Preparation Optical Constant Determined
ensing water vapora cold silicon surface
Real ~n! and imaginary ~k! indexwith an iterative Kramers–Kroniganalysis
crystalline samplespared in closed cells
Absorption coefficient, a
ed cell arrangementced inside a cryostat
Imaginary index, k, from k 5 aly4p
ensing water vapora cold silicon surface
Real ~n! and imaginary ~k! indexusing an iterative Kramers–Kronig analysis
of water placed be-en windows in aperature-controlled
ostat
Imaginary index, k, from k 5 aly4p
e crystal grown inss cell
Absorption coefficient
Sa
Condon
Monopre
Wedgpla
Condon
Droptwetemcry
Singlgla
September 2001 y Vol. 40, No. 25 y APPLIED OPTICS 4451
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function of frequency. Figure 3~c! shows the fre-quency regions where the agreement with the newOckman result is within 10% and where it is within20%, overlaid on the new Ockman imaginary indexspectrum. On the whole the agreement between theimaginary indices of Gosse et al.,21 Kou et al.,20 andthe new results from Ockman’s spectrum23 are within10%. Although it is not too surprising that the re-sults of Gosse et al.21 and those of Kou et al.20 agreethis well with each other ~both measurements weredone in the same research group and in a similarmanner!, it is noteworthy that the results obtainedfrom Ockman’s spectrum recorded ;35 yr prior tothese results ~at around the same temperature and ofa sample of thickness comparable with the one usedby Gosse et al.21! agree as well as they do. The val-
es of Grundy and Schmitt22 agree to within ,5% inhe band maxima around 5000 and ;6700 cm21, butn the two valleys around 4450 and 5440 cm21 their
imaginary indices differ by as much as 35%.Given these appreciable differences between the
Fig. 2. ~a! Comparison of the frequency-dependent imaginary in-ices of refraction of Grundy and Schmitt22 at 270 K with those inhe compilation of Warren16 at 266 K ~1984!. ~b! Variation of the
percentage difference ~relative to the values of Grundy andSchmitt22! between the two data sets, plotted as a function ofrequency.
452 APPLIED OPTICS y Vol. 40, No. 25 y 1 September 2001
results of Grundy and Schmitt22 and those of theothers mentioned in the previous paragraph, we feltthat further data were needed at lower temperatures~especially around 186 K! since the only data in the3700–7000-cm21 region at such temperatures avail-ble to date are the absorption coefficients of Grundynd Schmitt.22 As pointed out by Arnott et al.25 it is
important to assess the accuracy of the currently
Fig. 3. ~a! Comparison of the frequency-dependent imaginary in-ices of refraction of Grundy and Schmitt22 at 250 K with those
obtained by Kou et al.,20 Gosse et al.,21 and the reanalyzed Ockmanesult.23 ~b! Variation of the percentage difference ~relative to Ock-
man! of the data sets mentioned in ~a!, plotted as a function offrequency. ~c! Frequency regions where the agreement with thenew Ockman result23 is within 10% ~black diamonds! and where itis within 20% ~gray diamonds!, overlaid on the new Ockman23
imaginary index spectrum ~solid curve!.
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available measurements of the refractive indices ofwater ice in the near-infrared region at various tem-peratures and under various growth conditions forproper interpretation of the observed radiation sig-natures of ice clouds.
2. Experiment
The apparatus used to measure infrared spectra hasbeen described in detail previously.26 Briefly, poly-crystalline ice films with total thickness of up to;1164 mm were formed by condensation of waterapor on a cold 10-mm-diameter silicon wafer ~trans-arent at these frequencies! mounted in a vacuumhamber and cooled by a liquid-nitrogen-cooled cryo-tat. Water vapor was introduced into the chamberhrough leak valves positioned to backfill vapors intohe chamber. With backfilled vapors the films grewlmost equally on both sides of the wafer. The tem-erature of the silicon wafer was measured with sev-ral type T thermocouples. The thermocoupleunctions were attached to a copper heating plate inlose thermal contact with the silicon. The siliconafer was maintained at constant temperatures of66, 176, 186, or 196 K to within 61 K. At these
temperatures the hexagonal form of ice is expected tobe most stable. In the near-infrared spectral region,however, the spectra of the hexagonal and the cubicforms of ice are identical. The measured tempera-tures were calibrated with the frost point of ice. Thetemperature-dependent vapor pressures of Marti andMauersberger27 were used for this.
The infrared absorbances of the films were mea-sured in transmission with a Nicolet Magna 550Fourier-transform infrared spectrometer equippedwith a liquid-nitrogen-cooled MCT-B ~mercury cad-mium telluride, type B! detector. The absorbance,
, is defined as the base 10 logarithm of the ratio,0yI, where I, in our experiment, refers to the light
intensity after passage through the vacuum–ice–substrate–ice–vacuum system and I0 refers to theight intensity after passage through the vacuum–ubstrate–vacuum system. Data were collected at-cm21 resolution with 12 scans added per spectrum.
In our experimental geometry, infrared light incidentnormal to the surface undergoes multiple reflectionswithin the substrate and within the films.28 Theheory used to extract the optical constants from theeasured transmission of infrared light through the
lm system has been described in detail previously28
and is discussed only briefly here. Determination ofthe optical constants is an iterative process with twosteps that are repeated until a converging solution isreached. In the first step the thicknesses of the filmsare calculated by minimization of the squared differ-ence between the calculated and the measured absor-bances of the infrared spectra at a given temperature.The first time through the iterative loop, previouslymeasured refractive indices28 are used to calculatehe thickness. In the subsequent iterations theewly calculated refractive indices are used. In theecond step the refractive indices are adjusted asollows: The imaginary index at each frequency is
1
determined to minimize the squared difference be-tween the calculated and the measured absorbancespectra for all the chosen films at that temperaturesimultaneously. Then the value of the real index ateach frequency is calculated with the Kramers-Kronig relationship.
Ice crystals, being uniaxial, are slightly birefrin-gent. They have been shown, however, to be isotro-pic within experimental uncertainties.29,30 Hencemeasurements can be made on bulk polycrystallinesamples as was done in this study. The variations inrefractive indices due to birefringence are expected tobe less than the uncertainty in our derived values.
3. Data Analysis
In the near infrared ~3700–7000-cm21 region! theabsorption by ice is weak compared with the absorp-tion in the 500–3700-cm21 region, and hence thickfilms ~at least 300 mm! are essential for accurate
etermination of the imaginary refractive index. Inhe case of such thick films the strong absorbance,owever, leaves only very narrow frequency windowsapproximately 4449 and 5436 cm21! where the ab-
sorbance is small enough for the detection of anyinterference fringes predicted by the model for thisfilm system. Hence a method of finding film thick-ness based on interference fringes cannot be success-fully used to determine an initial guess for the filmthickness. To overcome this difficulty, the absor-bances at the frequency n0 5 3960 cm21 @see Fig. 4~a!#measured relative to a baseline drawn through thepoints n1 5 3825 and n2 5 4438 cm21 @Fig. 4~a!# were
sed to determine the film thicknesses. In otherords, with reference to Fig. 4~a!, the absorbance
A~n0! 2 A9~n0!# was used as a measure of the filmthickness. The points n1 and n2 were chosen to co-incide with the two nearest local minima on eitherside of the chosen absorption feature @Fig. 4~a!#. Forthe purpose of calibration, imaginary indices, k, atthe various frequencies, n, were calculated from theabsorption coefficients, a, of Grundy and Schmitt22 byuse of the relationship k 5 ay4pn at each of theemperatures. Grundy and Schmitt22 calculated a
by dividing the natural logarithm of the “calculatedintrinsic transmission” ~p. 25812! by the thickness ofhe film. These k values were used along with theeal refractive indices from Toon et al. at 166 K,28 and
the film model to calculate the absorbance, @A~n0! 2A9~n0!#, at the chosen frequency, n0, for a set of inputice film thicknesses, d. The calculated absorbanceversus the film thickness is shown in Fig. 4~b!. Ashown in Fig. 4~b!, a plot of @A~n0! 2 A9~n0!# versus thelm thickness, d, is found to be linear. The result oflinear fit to these lines was used along with the
A~n0! 2 A9~n0!# values for the experimental spectra torrive at an initial estimate of the thicknesses of thehosen films at that temperature.
A second experimental difficulty had to do withiffuse light scattering from the imperfect ice sur-aces in frequency regions of relatively weak absorp-ion such as near 5500 cm21, resulting in spurious
baselines for the spectra. Typically, experiments
September 2001 y Vol. 40, No. 25 y APPLIED OPTICS 4453
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could be done at only one temperature on any givenday because of the time required for growing filmswith a broad range of thickness. The chamberwould then have to be completely pumped out beforeanother set of experiments could be performed.Since the amount of light scatter depends on factorssuch as the surface roughness of the films, which ishard to control during the course of an experiment,the losses due to light scattering varied from day today. This variability made it necessary to treat theeffects of light scattering in all the spectra recorded atthe various temperatures in a consistent manner be-fore any temperature-dependent optical constantscould be extracted.
Difficulties due to light scattering were also en-countered in the measurements of Grundy andSchmitt.22 They had to correct each spectrum tocompensate for the effects of light scattering, which,in some cases, were quite large. To do this, theydivided each spectrum by a “synthetic continuum.”
Fig. 4. ~a! Near-infrared absorption spectrum of water ice show-ing the criterion used to determine the thickness of the films. Thedifference in the absorbance, @A~n0! 2 A9~n0!#, at frequency n0 isused as a measure of the film thickness as explained in the text ~n0
is ;3960cm21, n1 is ;3825cm21, and n2 is ;4438cm21!. ~b! Icelm thickness plotted as a function of @A~n0! 2A9~n0!# for films atemperatures of 166, 176, 186, and 196 K.
454 APPLIED OPTICS y Vol. 40, No. 25 y 1 September 2001
This synthetic continuum was calculated based on“an examination of the least strongly absorbing wave-length regions of each spectrum, the requirementthat spectra of samples of different thicknesses yieldmutually consistent absorption coefficients, and theexpectation that the spectra at any temperaturewould be similar to spectra at adjacent temperatures”~Ref. 22, p. 25812!.
Our procedure for compensating for the effects dueo light scattering is illustrated in Fig. 5. Essen-ially, we are assuming that the valleys in the 176 Kpectra are not affected by scattering losses and thathere is minimal temperature dependence at 4449,436, and 7000 cm21. As discussed in Section 4,
Grundy and Schmitt22 found little temperature de-pendence in the imaginary indices of refractionaround these three frequencies. Since water ice ab-sorbs more strongly at other frequencies, any errorsin the baseline correction would be significantly lessat those frequencies. In the analysis of our data itwas found that the spectra recorded at 176 K wereamong the ones that allowed maximum transmissionin the valleys around 4449 and 5436 cm21 ~and there-fore minimum effects due to scattering! for a giventhickness, and hence these spectra were used as thebasis for our correction for scattering losses. Theworst case of light scattering was encountered in ourdata at 206 K ~not included in this study!. The base-line of a given spectrum was defined by polynomial~order #2! fit through the two points corresponding tominimum absorption in the valleys ~4449 and 5436cm21! and a third point corresponding to the last datapoint around 7000 cm21 @Fig. 5~a!#. For the fre-quency region between 4449 and 3700 cm21 the base-line was defined to be the value of the absorbance at4449 cm21. The absorbance at each of these threefrequencies in the case of the 176 K spectra are foundto increase linearly with the ice film thickness @Fig.5~b!#. The results of a linear fit of the absorbance tothe film thickness @also shown in Fig. 5~b!# were then
sed to calculate the correct absorbances at thesehree frequencies for all the other spectra correspond-ng to different thicknesses at the other tempera-ures. These correct absorbances were then used toalculate the correct baselines for these spectra.hese correct baselines were then added on to theespective spectra after their original baselines hadeen subtracted out @Fig. 5~c!#. The resultingaseline-corrected spectra at each temperature werehen used as input for the calculation of the opticalonstants.
A total of at least 13 and as many as 19 films ofotal thickness varying between ;220 mm and ;1m were chosen for the calculation of the optical
onstants at each temperature. Of these, the thin-est films ~approximately 220–460 mm! with small-st measured transmission of ;3% ~maximumbsorbance between 1.0 and 1.5! were used to deter-ine the imaginary index of refraction ~k! in the re-
ion of strongest absorption ~4763–5214 cm21!. Thehickest films ~approximately 700 mm–1 mm! weresed to determine the imaginary index of refraction
p
ot
in the rest of the spectral regions between 3700 and7000 cm21. Films of intermediate thickness ~ap-
roximately 460–700 mm! were used to ensure con-tinuity and also served as a further check of theanalysis. Figure 6 shows typical fits between the
Fig. 5. ~a! Diamonds represent the points used to define the basepoints is obtained by fitting the points with a polynomial ~order #
riginal spectrum ~black trace! to yield the dark gray curve. ~b! Tho the three diamonds in ~a!, observed in the case of the spectra re
of scattering!. The squares represent the measured absorbancesa straight line. The correct absorbances at the three frequenciesuse of the equations for the straight lines shown, along with the tbaseline to be added on to the original-background-subtracted spsmooth curve through the three new points calculated as describedoptical constants is obtained by means of adding this new baselin
1
calculated and the measured absorbance at 166 Kthat we were able to achieve.
In the calculation of the real index of refraction ~n!with the Kramers–Kronig relation28 the anchor pointwas chosen to be the temperature-dependent real in-
f the spectrum. The thin black curve passing through the threeTo correct for scattering, this baseline is first subtracted from thess dependence of the absorbances at the frequencies correspondingd at 176 K ~chosen because these spectra showed the least effectsthe solid line is the result of fitting the measured absorbances toach of the spectra at all the other temperatures are calculated byess of the films. ~c! The thin black curve represents the [email protected]., gray curve in ~a! and ~c!#. We obtain it by again fitting a
b!. The final spectrum ~black trace! used in the extraction of thein black curve! to the gray curve.
line o2!.icknecorde, andfor ehicknectra
in ~e ~th
September 2001 y Vol. 40, No. 25 y APPLIED OPTICS 4455
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dex of refraction, nvis~T!, at the frequency nvis 515823 cm21.28 The temperature-dependent real in-
ex of refraction was calculated with the Lorentz–orenz equation,
R 5 @nvis~T!2 2 1#y$r~T!@nvis~T!2 1 2#%, (1)
where a value of R 5 0.2072 cm3 g21 was used for thespecific refraction.31 The temperature-dependentdensity of ice, r~T!, was calculated with the relation
r~T! 5 0.9167 2 1.75 3 1024T 2 5.0 3 1027T2, (2)
where T is in celsius and the density is in gramscm23.32
The Kramers–Kronig integration was carried outin the frequency interval of approximately 1.8–185,809 cm21. To extend the values of the imagi-nary index in the integrand outside the frequencyrange in which optical constant measurements weremade in this study ~3700–7000 cm21!, theemperature-dependent imaginary indices of Clapp etl.33 were used between ;3700 and 800 cm21 ~theseata were chosen since they covered the temperatureange of our study!. Between 800 and 500 cm21 the
imaginary indices of Toon et al. at 166 K28 were used.Beyond each end of the frequency range 500–7000cm21 the imaginary index is assumed to be a constantmatching the values at 500 and 7000 cm21. Thesessumptions for the imaginary index outside the00–7000-cm21 range have been shown to have an
insignificant effect on the real index of refraction be-tween 3700 and 7000 cm21.28
4. Results
The real and the imaginary indices of refraction ob-tained in this study at 166, 176, 186, and 196 K arepresented in Table 2.34 The real indices of refractionat the various temperatures are plotted as a functionof frequency in Fig. 7. It can be seen that there is nonoticeable temperature-dependence. The Lorentz–Lorenz equation @Eq. ~1!# and Eq. ~2! predict the realndices of refraction in the temperature range 166–96 K to change only by ;0.1%, consistent with ouresults. Figure 8~a! shows a comparison of the realndices from this study at 186 K with those of Toon et
Fig. 6. Typical fits between the measured ~black trace! and the calcorrespond to a temperature of 166 K.
456 APPLIED OPTICS y Vol. 40, No. 25 y 1 September 2001
l.28 at 166 K and Warren16 at 266 K. Figure 8~b!hows the variation of the percentage difference be-ween our results at 186 K and the values in Warren’sompilation adjusted with the Lorentz–Lorenz equa-ion for 186 K.16 From this figure it is seen that the
difference is between 10.12% and 20.22% in the3700–7000-cm21 region.
The imaginary indices of refraction, k, obtained inthis study at the various temperatures are shown inFig. 9~a!. Figure 9~b! shows the absolute change inthe imaginary indices at 196 K relative to the valuesat 166 K, as a function of frequency. Also shown isthe absolute change predicted by the 17-Gaussianmodel of Grundy and Schmitt.22 It is gratifying tosee that our observations in this narrow temperaturerange of 40 K agree well with the predictions that arebased on a study of a much wider temperature range,from 20 to 270 K. The differences in the absolutevalues between our observation and the predictionbased on the research of Grundy and Schmitt22 aremost likely due to the different methods used to cor-rect the baselines of the spectra in an attempt toaccount for the effects of diffuse light scattering at theice film surfaces. The temperature dependence ofthe feature around 6070 cm21, observed, for example,by Grundy and Schmitt22 and Ockman,17 is also ob-served by us. However, smooth trends in the peakmaxima or frequency shifts with temperature arehard to make out, since the predicted percentagechange for a change in temperature of 10 K variesbetween 4% and 7% across this frequency region,whereas the uncertainty in our results is expected tobe ;10%. The absolute difference between theimaginary indices at 250 K ~Gosse et al.21! and thoseat 196 K ~our data! are compared with that predictedby the model of Grundy and Schmitt,22 in Fig. 9~c!.
The model of Grundy and Schmitt22 predicts thevariation of k with temperature to be linear at mostfrequencies, but at other frequencies a second-orderpolynomial is required for fitting the predicted varia-tion of k with temperature. However, we have deter-mined that the Grundy and Schmitt22 variation of k
ith temperature along with our 196 K data yieldsalues of k that differ by a maximum value of 15%ompared with the results of using the linear interpo-
ed ~gray trace! absorption spectrum obtained. The spectra shown
culat
mubt
ma
Table 2. Optical Constants of Water Ice at 166, 176, 186, and 196 K in Table 2. (continued)
lation between 196 and 250 K, this maximum value ofthe difference occurring at 250 K. In other words, at250 K the difference between the results of Gosse etal.21 and those calculated by extrapolation of our dataat 196 K with the Grundy and Schmitt22 variation is at
ost 115%. To avoid any discontinuity in the k val-es around 196 or 250 K, and because the differencesetween the two methods is smaller than 15% at in-ermediate temperatures, we have chosen to recom-
the 3721–6981-cm21 Spectral Range
Fre-quency(cm21)
Temperature (K)
dkydT
166 K 176 K 186 K 196 K
n k n k n k n k
1
end a linear interpolation between our 196 K datand the 250 K data of Gosse et al.21 Table 2 thus lists
the slope, dkydT, that can be used to interpolate thevalues of the imaginary indices between our 196 Kdata and the 250 K data of Gosse et al.21 according to
k~T! 5 k~196 K! 1dkdT
~T 2 196!. (3)
Fre-quency(cm21)
Temperature (K)
dkydT
166 K 176 K 186 K 196 K
n k n k n k n k
September 2001 y Vol. 40, No. 25 y APPLIED OPTICS 4457
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u
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In Fig. 10~a! our results at 186 K are compared withthe results of Grundy and Schmitt22 at 250 K; thoseof Gosse et al.,21 also near 250 K ~since the data ofKou et al.20 and Gosse et al.21 agree well with eachother, only the former was chosen for the comparisonfor clarity!; the new results from the reanalysis of
ckman’s spectrum;23 and the data from Warren’scompilation. Figure 10~b! shows the variation of thepercentage difference calculated as
% difference 5 @k(other) 2 k~ours!#
3 100yk(ours) 2 (% difference due to DT!, (4)
where k~other! refers to results of Gosse et al.,21 War-en,16 the new results from Ockman’s spectrum,23
and Grundy and Schmitt.22 The term “% differencedue to DT” is the percentage difference in the imag-inary indices predicted by the 17-Gaussian model ofGrundy et al. for a change in temperature from 270 to186 K with reference to Warren’s compilation and fora change from 250 to 186 K with reference to theother data. Figure 10~c! shows the frequency re-gions where the agreement between the various otherdata and our results are within 10% and where theagreement is within 20%, overlaid on our k spectrumat 186 K.
It is seen from Figures 10~b! and 10~c! that after thedifferences due to temperature are taken into consid-eration
1. Our imaginary indices are within 10% of theresults of Gosse et al.,21 Kou et al.,20 and the newlyderived values from Ockman’s spectrum23 at almostall the frequencies except in the immediate vicinity ofthe two valleys around 4400 and 5400 cm21.
2. Even in these valleys, our results are within20% of the results of Gosse et al.,21 Kou et al.,20 and
Fig. 7. Real indices of refraction of water ice at the temperaturesof 166, 176, 186, and 196 K, plotted as a function of frequency.
458 APPLIED OPTICS y Vol. 40, No. 25 y 1 September 2001
the newly derived values from Ockman’s spectrum,whereas those of Grundy and Schmitt22 and War-ren16 differ by much more than 20%.
3. The imaginary indices of Grundy andSchmitt22 are within 5% of our results in the bandmaxima but are lower by more than 40% in the twovalleys ~approximately 4360–4620 and 5300–6020cm21!.
4. With reference to the values in Warren’s16
compilation, the agreement around the 5000-cm21
band maximum is between 115% and 220%.However, in the approximately 6360–6700-cm21
region and in the shoulder at frequencies below;4300 cm21, the imaginary indices in Warren’scompilation16 are higher by approximately 20% and30%, respectively. In the two valleys ~approxi-mately 4400–4500 and 5360–5780 cm21! the val-
es in Warren’s compilation16 are lower by 30% and60%, respectively.
Fig. 8. ~a! Comparison of the real indices of refraction ~n! obtainedin this study ~186 K! with the values obtained by Toon et al.28 ~166K! and the values from Warren’s compilation16 at 266 K ~not tem-perature adjusted!. ~b! Variation of the percentage difference inthe real indices of refraction ~relative to the values in Warren’scompilation16! obtained in this study ~186 K! and those from War-ren’s compilation temperature adjusted with the Lorentz–Lorenzrealtion to 186 K.
lto
o
sdqdts
The spectra of Grundy and Schmitt,22 like ours,required corrections in order to account for light-scattering effects from the ice samples. In their pa-per Grundy and Schmitt22 did not describe the actualcalculation of the synthetic continuum to correct the
Fig. 9. ~a! Imaginary indices of refraction ~k! at the temperaturesf 166, 176, 186, and 196 K, plotted as a function of frequency. ~b!
Variation of the absolute change in the imaginary indices ~k! at 196K relative to the values at 166 K, as a function of frequency. Thedashed curve represents our experimental observation, and the solidcurve shows the absolute change predicted by the 17-Gaussianmodel of Grundy and Schmitt22 ~c! Variation of the absolute changein the imaginary indices ~k! at 250 K relative to the values at 196 K,as a function of frequency. The dashed curve is the difference be-tween the results of Gosse et al. at 250 K and our experimentalobservation at 196 K. The solid curve is the absolute change pre-dicted by the 17-Gaussian model of Grundy and Schmitt.22
1
individual spectra for these losses. The much lowervalues of the imaginary indices of Grundy andSchmitt22 in the valleys are most likely due to over-correction for scattering losses. Warren16 chose theowest values available in the valleys from spectrahat had been incompletely analyzed for light lossesn the cell walls ~values derived from Reding18! for
Fig. 10. ~a! Comparison of the frequency-dependent imaginaryindices of refraction ~k! obtained in this study with those of Grundyand Schmitt22 at 250 K, Gosse et al.21 at ;250 K, the new valuesfrom the reanalysis of Ockman’s spectrum23 at 247 K, and War-ren’s compilation16 at 266 K. Data shown here are at their re-pective temperatures. ~b! Temperature-adjusted percentageifference @defined in Eq. ~4!# between these sets of data. ~c! Fre-uency regions where the agreement between the various otherata and our results are within 10% ~black diamonds! and wherehe agreement is within 20% ~gray diamonds!, overlaid on our kpectrum at 186 K ~thick black curve!.
September 2001 y Vol. 40, No. 25 y APPLIED OPTICS 4459
s
b
sn
wl
Tmt
4
partial correction of the effects of light scattering inthis frequency region of low absorption by water ice.
5. Conclusion
The real and the imaginary indices of water ice havebeen determined in the 166–196 K temperature re-gion between approximately 3700 and 7000 cm21.The real indices of refraction have been found to beweakly temperature dependent, consistent with pre-dictions based on the Lorentz–Lorenz equation. Theobserved temperature dependence in the imaginaryindices of refraction as a function of frequency hasbeen compared with predictions based on the 17-Gaussian model of Grundy and Schmitt.22
The transmission spectrum of water ice ~358-mmample at 247 K! recorded by Ockman23 in 1957 was
reanalyzed to extract optical constants, taking intoaccount that the transmission was measured throughthe air–glass–ice–glass–air system. These new re-sults have been found to agree with the results ofGosse et al.21 and Kou et al.20 ~which correspond to asimilar temperature and sample thickness! to within10% at almost all the frequencies.
Our imaginary indices have been compared withthose of Gosse et al.,21 Kou et al.,20 the new resultsfrom Ockman’s spectrum,23 Grundy and Schmitt,22
and Warren’s compilation.16 Our results agree withthose of Gosse et al.,21 Kou et al.,20 and the newresults from Ockman’s spectrum23 to within 10% inthe regions around the band maxima and to within20% in the two valleys between these bands when thedifferences expected because of temperature effectshave been taken into account. The agreement withGrundy and Schmitt22 is closer in the regions of theand maxima than in the two valleys.In view of the comparisons of the available data
ets, we recommend that, close to 250 K, the imagi-ary indices of Gosse et al.,21 Kou et al.,20 or the new
results from Ockman’s spectrum23 be used. Be-tween 166 and 196 K our imaginary indices of refrac-tion should be used. For temperatures between 196and 250 K we propose a linear interpolation scheme,and the slopes at the various frequencies required forthis have been provided.
Appendix A: Equations for the Calculation of the LightEnergy Transmitted through a Cell
The matrix formalism of Heavens24 has been used toderive the transmitted energy through the four-layersystem described in Fig. 11. The thickness of the icefilm, d2, is assumed to be much smaller than that ofthe windows. In accordance with Eq. 4 ~106b! in thebook by Heavens24 the transmitted energy is given by
T 5~t1t2t3t4!~t*1t*2t*3t*4!
aa*, (A1)
here tn is the Fresnel coefficient for transmission ofight across the ~n 2 1!un interface and t@n#
* is itscomplex conjugate. tn is complex and is in generalequal to 1 1 rn, where rn is the Fresnel coefficient forreflection at the ~n 2 1!un interface.
460 APPLIED OPTICS y Vol. 40, No. 25 y 1 September 2001
For our system and for normal incidence
r1 5~1 2 n1!
~1 1 n1!~here after referred to as g1!, (A2)
r2 5 g2 1 ih2, (A3)
where
g2 5n1
2 2 n22 2 k2
2
~n1 1 n2!2 1 k2
2, h2 52n1k2
~n1 1 n2!2 1 k2
2 . (A4)
Therefore
~t1t2t3t4!~t*1t*2t*3t*4! 5 l142 1 m14
2 , (A5)
where l14 and m14 are the real and the imaginaryparts of the above product and are equal to
l14 5 ~1 2 g12!~1 2 g2
2 1 h22!,
m14 5 22g2h2~1 2 g12!. (A6)
he derivation of the denominator in Eq. ~A1! is muchore complicated and is presented below following
he notation of Heavens24:
a 5 p14 1 iq14, (A7)
where
p14 5 p13 2 g1r13, q14 5 q13 2 g1s13. (A8)
p13, q13, r13, and s13 are given by
p13 5 p12p3 2 q12q3 1 r12t3 2 s12u3,
p13 5 p12q3 1 q12p3 1 r12u3 1 s12t3,
r13 5 p12r3 2 q12s3 1 r12v3 2 s12w3,
s13 5 p12s3 1 q12r3 1 r12w3 1 s12v3. (A9)
p12, q12, r12, and s12 are given by are given by
p12 5 1 1 g1g2,
q12 5 g1h2,
r12 5 g1 1 g2,
s12 5 h2. (A10)
Fig. 11. Schematic of the transmission of light through a cell
Fh
Table 3. Sample Calculation with the Cell Model
3
Defining
a2 52pk2d2
l, g2 5
2pn2d2
l, (A11)
we can now define p3, . . . , w3 as
p3 5 exp~a2!cos g2,
q3 5 exp~a2!sin g2,
r3 5 2g2 exp~a2!cos g2 1 h2 exp~a2!sin g2,
s3 5 2h2 exp~a2!cos g2 2 g2 exp~a2! sin g2,
t3 5 g2 exp~2a2!cos g2 2 h2 exp~2a2)sin g2,
u3 5 g2 exp~2a2!sin g2 2 h2 exp~2a2!cos g2,
v3 5 exp~2a2!cos g2,
w3 5 2exp~2a2!sin g2.
or the empty cell the following substitutions willave to be made
g2 5 2g1,
h2 5 0,
a2 5 0,
g2 5 2pd2yl.
Ockman’s 358-mm sample spectrum was recordedwith glass windows. The refractive indices of glassthat were used were taken from a handbook of opticalconstants.35 For the real indices of refraction of ice,Warren’s16 values were used. Warren’s16 imaginaryindices of refraction of water ice were used as aninitial guess in the fitting program. The final valuesof the imaginary indices of refraction are found byminimization of the squared difference between themeasured transmission and that calculated based onthe above cell model.
Table 3 gives sample inputs used in the program,along with the final values of the imaginary indicesobtained at those frequencies as described above.Also included in the table are the values of the imag-inary indices that would be obtained when the trans-mission through the cell is not taken into account,followed by the percentage error due to the neglect ofthe correct model. It can be seen that neglecting thereflections at the various interfaces can lead to sig-nificant errors in the determination of k. The fre-quencies shown in Table 3 were chosen to match thelocal maxima and minima in the absorption spectrumof water ice in this frequency region. It can be seenthat the errors range from 2% to 28%, the largesterrors occurring at the valleys near the 5000-cm21
band. These are the same regions were the discrep-ancies between the previously available values of theimaginary indices were the largest.
We thank W. M. Grundy for his help in accessinghis data. We are very thankful to S. G. Warren forsending us the Ockman data and for having saved itfor all these years. This research was supported by
1
NASA-EOS grant NAG5–8127 and by NSF-ATM9711969.
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