154 IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 17, NO. 1, JANUARY/FEBRUARY 2011

Recovery of Polaritonic Stopband of PressedPolycrystalline Calcium Fluoride Powder

in Terahertz Frequency RegionYi Jiang, Member, IEEE, and Yujie J. Ding, Senior Member, IEEE

Abstract—By measuring and comparing the reflection spectraof single crystal and pressed polycrystalline powder of calciumfluoride (CaF2 ) in the terahertz range, we have demonstrated re-covery of polaritonic stopband in the powder. By using a simplifiedmodel to fit our data, we have deduced the longitudinal-opticaland transverse-optical phonon frequencies and damping parame-ter for the polariton resonance. While the two phonon frequenciesfor the powder sample differ from those deduced for the single-crystal sample by less than 2%, the value for the deduced dampingparameter is significantly increased. Due to small sizes of CaF2powder particles, we have attributed such a significant increase todamping of polaritonic waves at the particle surfaces.

Index Terms—CaF2 , infrared spectroscopy, polariton reso-nance, polaritonic stopband, reflection, terahertz (THz).

I. INTRODUCTION

IN CRYSTALLINE solids, such as polaritons originatingfrom strong coupling between electromagnetic waves and

transverse-optical (TO) phonons in a single crystal, describequanta of the coupled phonon–photon transverse wave field. Dueto such coupling, a stopband (i.e., forbidden band) is establishedin the range between TO and longitudinal-optical (LO) phononfrequencies in polar crystalline dielectric materials [1]–[4]. Con-sequently, reflectivities within such a stopband are shown to ap-proach 100%. However, it is unclear how the coupling affectsthe spectrum of reflectivity in polycrystals, especially as dimen-sions of crystallites making up polycrystals are dramaticallyreduced.

Generally speaking, polaritons are fundamentally importantsince they dictate linear and nonlinear-optical properties of crys-talline solids. To some extent, the characteristics of the polari-tons are quite similar to the behaviors of photonic crystals ormetallic materials. Indeed, away from the polariton resonance,the reflectivity of light can be dramatically reduced. In suchcase, if a crystal sample is sufficiently thin, the transmission canbe significantly improved. This is similar to surface-plasmon-enhanced transmission demonstrated previously [5]. In addition,luminescence efficiency can be improved using surface Plasmon

Manuscript received January 29, 2010; revised March 9, 2010; acceptedMarch 26, 2010. Date of publication May 18, 2010; date of current versionFebruary 4, 2011. This work was supported by the U.S. Office of NavalResearch.

The authors are with the Department of Electrical and Computer Engineer-ing, Lehigh University, Bethlehem, PA 18015 USA (e-mail: yij206@lehigh.edu;yud2@lehigh.edu).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/JSTQE.2010.2047248

Fig. 1. Pictures for CaF2 single crystal (2 mm thick, left) and powder (300 μmthick, right).

polaritons [6]. On the other hand, frequency conversion [7], [8]and energy localization [9] based on nonlinearities may greatlybenefit from the forbidden band of polaritons as well as photoniccrystals.

In this paper, for the first time to the best of our knowledge, wedemonstrate that after CaF2 powder is pressed under a pressurethe spectrum of reflectivity undergoes recovery to that for a sin-gle crystal. In addition, by fitting the spectrum using a simplifiedformula, we have deduced the TO and LO phonon frequencies,which are quite close to the values commonly accepted for CaF2single crystal. Furthermore, we have observed an evidence forthe increase in damping of polaritons. We believe that such anincrease is primarily caused by the termination of the polaritonwaves at the surfaces of the individual crystallites.

II. SAMPLE PREPARATION AND EXPERIMENT

Samples of single crystal and polycrystalline powder of CaF2were obtained by us commercially. We have prepared two single-crystal CaF2 samples with their thicknesses of 500 μm and2 mm, respectively. On the other hand, CaF2 powder consistsof many CaF2 crystallites with their sizes in the range of 1–5 μm. It is worth noting that these crystallites have randomorientations. The purity of CaF2 powder is 99.5%. Each CaF2powder sample was prepared by us after pressing the powderunder a pressure in the range of 9.16–11.5 atm. Each powdersample has a diameter of 7 mm. Fig. 1 shows the photographsof single crystal with thickness of 2 mm and diameter of 25 mm(left) and pressed polycrystalline powder mounted on a metal-lic holder (right) with thickness of 500 μm and diameter of7 mm, respectively. Upon the visual inspection, as illustrated byFig. 1, the CaF2 single-crystal sample is highly transparent in thevisible region. However, all the CaF2 powder samples have

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JIANG AND DING: RECOVERY OF POLARITONIC STOPBAND 155

white color in the visible region, which are not transparent. Thisis caused by the strong scattering of the incident visible light bythe crystallites with the sizes of 1–5 μm, which are comparablewith the wavelengths of the visible light [10]. We measured thereflection spectra of the three samples, using Fourier transformIR spectroscopy (FTIR) from Bruker Optics (IFS 66v/S). Eachsample was mounted inside the sample compartment under vac-uum. The incident angle for the reflection measurement is closeto 90◦. Each reflection spectrum was obtained after taking anaverage over 256 interferograms measured in a single-sided ac-quisition mode. The spectral resolution of FTIR was set by us to2 cm−1 . The light source used by us is a mercury lamp generat-ing broad radiation in the THz region. A gold mirror was usedas a reference in the reflection experiment.

III. RESULTS AND ANALYSIS

A. Theoretical Analysis

Consider a simplified theoretical model for expressing com-plex dielectric constants as a function of frequency.

ε (ν) = ε (∞) +ε (∞)

(ν2

LO − ν2TO

)(ν2

TO − ν2 − iνγ)(1)

where νTO and νLO are the TO and LO phonon frequencies,respectively, ε (∞) is the high-frequency limiting value of thedielectric constant, and γ is the damping parameter of the polari-ton. According to (1), the two phonon frequencies, the dampingparameter, and the high-frequency dielectric constant can fullydefine the polariton resonance. The role of the phonons is tobe strongly coupled to photons. It is important for us to notethat only TO phonons can be coupled to photons. If γ = 0,ε(ν) < 0 for νTO < ν < νLO . From (1), we can find out thereal and imaginary parts of ε(v) as follows:

εr (v) = ε(∞) +ε(∞)

(v2

LO − v2TO

) (v2

TO − v2)

(v2TO − v2)2 + v2γ2

(2)

εi(v) =ε(∞)

(v2

LO − v2TO

)vγ

(v2TO − v2)2 + v2γ2

. (3)

The reflectivity for an incident THz wave can be then deter-mined from the following expression:

R(v) =[n(v) − 1]2 + κ(v)2

[n(v) + 1]2 + κ(v)2(4)

where n(v) and κ(v) are the real and imaginary parts of re-fractive index, respectively, which can be determined from thefollowing expression:

ε(v) = εr (v) + iεi(v) = [n(v) + iκ(v)]2 (5)

From (5), one obtains

n =1√2

√√ε2r + ε2

i + εr (6)

κ =1√2

√√ε2r + ε2

i − εr . (7)

Substituting (6) and (7) into (4), we have obtained

R =1 −

√2√

ε2r + ε2

i + 2εr +√

ε2r + ε2

i

1 +√

2√

ε2r + ε2

i + 2εr +√

ε2r + ε2

i

. (8)

Based on the expression given earlier, one can see that thereflectivity is a function of the TO phonon and LO phonon fre-quencies and damping parameter. If the damping is negligible,i.e., γ ≈ 0, we have obtained εi ≈ 0 from (3). As a result, R ≈ 1for the frequencies in the range from νTO to νLO following (1).In the other words, the spectral shape of the reflection peakappears to be a perfect square defined from νTO to νLO . Thisimplies that a THz wave is almost completely reflected withinsuch a frequency band. Therefore, the region with its frequen-cies between νTO and νLO defines a polaritonic stopband. Asγ is increased, the rising and falling edges of the square becomemore and more broadened.

B. Results and Discussions

Using (8) whereby εr and εi , are given by (2) and (3),we have fitted our data obtained on the single-crystal sam-ple and presented in Fig. 2(a) based on the nonlinear-least-square (NLS) fit. As a result, we have obtained νTO ≈253.2 cm−1 (39.49 μm) and νLO ≈ 474.3 cm−1 (21.08 μm)(see Table I). These two frequencies differ from the well-accepted values by just 1.5% and 2.4%, respectively. In addition,we have deduced the damping parameter of the polariton to beγ ≈ 18.1 cm−1 . This value is about four times higher than pre-vious values [11]. Such a factor can be attributed to the twodifferent models used, which have affected the deduced valuesof γ.

On the other hand, Fig. 2(b) illustrates our result of the re-flectivities measured on the CaF2 powder sample. It is worthnoting that the reflectivities within the reflection peak are sig-nificantly reduced compared with that for the single-crystal sam-ple. Similarly, using (8), we have achieved the NLS fitting tothe data presented in Fig. 2(b). As a result, we have obtainedνTO ≈ 256.2 cm−1 and νLO ≈ 482.7 cm−1 . These values are1.2% and 1.8% higher than those deduced from our single-crystal sample, respectively. Besides these phonon frequencies,we have also obtained the damping parameter for the polaritonwave to be γ ≈ 38.7 cm−1 from the fitting. This value is abouttwice higher than that obtained by us for the single-crystal sam-ple. Such a significant difference indicates that for some reason,the polariton wave excited in the CaF2 powder experiences asignificantly higher loss. Based on Heisenberg uncertainty prin-ciple, we have estimated the lifetime for the polaritons insidethe single crystal to be on the order of 150 fs. On the other hand,for the CaF2 powder, the lifetime for the polaritons is reducedto 68.5 fs. Since the sizes for the CaF2 crystallites within thepowder sample are in the range of 1–5 μm, the group velocitiesof the polaritons required to reach the back surface of each CaF2particle are within 6.83 × 108 cm/s and 3.41 × 109 cm/s.

Using (1) whereby γ is set to 0, we have plotted the polaritonfrequency, ν, versus the polariton wavevector, k = 2π

√ε (ν)/c,

in Fig. 3(a). Due to the strong phonon–photon coupling, the

156 IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 17, NO. 1, JANUARY/FEBRUARY 2011

Fig. 2. Solid curves—measurements of reflectivity versus frequency for CaF2(a) single crystal (2 mm thick) and (b) powder (300 μm thick); dashed curves—NLS fits to data.

TABLE ILIST OF PARAMETERS FOR BULK CaF2 DEDUCED

BY FITTING DATA IN FIG. 2(a) AND (b)

dispersion of the polariton wave is dramatically modified fromthat for the photons (see Fig. 3(a)). It is worth noting thatthe obvious saturation for the wavevectors being larger than1 × 10−7 cm−1 in Fig. 3(a) is attributed to the polaritons.We have also plotted the normalized group velocity, given by(2π/c)dν/dk versus k in Fig. 3(b). The dramatic decrease ofthe group velocities is also caused by the polaritons. Accord-ing to Fig. 3(b), the group velocities in the range of 6.83 ×108 cm/s − 3.41 × 109 cm/s corresponds to the frequencies

Fig. 3. (a) Dispersion of polaritons and (b) velocity of polaritons versus fre-quency, based on a simplified model, i.e., (1) whereby γ = 0 and two phononfrequencies are deduced from our measurement made on the CaF2 powder.

for the polariton waves in the range of 199–238 cm−1 . If thedamping parameter of the polaritons is 4.63 cm−1 based on [11],the high-frequency end of the polaritons being damped by theback surface extends to 249 cm−1 , which is quite close to theTO frequency. Therefore, these estimates support our claim thatthe polaritons are severely damped by their termination at theback surfaces of the crystallites.

It is worth noting that the dispersion behavior illustrated byFig. 3(a) is similar to that for metallic gratings [12]. Therefore,one may expect that the THz wave in the vicinity of the polari-ton resonance can be significantly slowed down. Indeed, aftertaking into consideration the damping parameter with its valuegiven in [11], we have plotted the group velocity as a functionof the frequency (see Fig. 4). One can see from Fig. 4 that atthe frequency of 254 cm−1 , the group velocity is reduced by afactor of 240 relative to the speed of light in vacuum. Such a dra-matic reduction is attributed to the polaritons. Following [12],one should be able to fabricate gratings formed by periodicgrooves just beneath the CaF2 surface with graded depths. Onecan then separate different frequency components, using such aunique structure, as illustrated in [12]. However, the absorptionof the THz wave at the polariton resonance can be extremely

JIANG AND DING: RECOVERY OF POLARITONIC STOPBAND 157

Fig. 4. Group velocity is plotted versus frequency, using phonon frequenciesand damping parameter given by [2] and [11].

Fig. 5. Absorption coefficient is plotted versus frequency.

high (see Fig. 5). Indeed, at 254 cm−1 the absorption coefficientis 1.21 μm−1 . Such a dramatic increase in the absorption coef-ficient is the manifestation of the polaritons. As the frequencyis reduced, the absorption coefficient is reduced by up to twoorders of magnitude (see Fig. 5). Therefore, it may be advanta-geous for us to set the frequencies to those of being significantlylower than the TO frequency, see Fig. 5. At these frequencies,the reduction of the group velocity can be a factor of around 10.On the other hand, when the frequency is higher than a criticalvalue, i.e., 255.6758 cm−1 , the group velocity is either higherthan the speed of light in vacuum or negative, correspondingto the so-called superluminal regime [14]. However, since thecorresponding absorption is so high, it may be quite challengingfor researchers to observe such a phenomenon.

It is worth noting that within the frequency range of100–500 cm−1 , molecular species in the air may affect the re-flection spectrum of the pressed CaF2 powder. In Fig. 6, we havemade a comparison between the reflection spectra measured onthe CaF2 powder under vacuum and in air, respectively. One can

Fig. 6. Reflection spectra were measured on pressed CaF2 powder undervacuum and in air, respectively.

Fig. 7. Reflectivity is plotted vs. frequency for UO2 for two different valuesof damping parameter of polariton, based on (8) in the text.

see that for the frequencies higher than 447 cm−1 , the absorptionof the incident radiation by the molecules present in the air has anegligible effect on the reflection spectrum. For the frequenciesin the range of 447–231 cm−1 , the molecular absorption signif-icantly affects the reflectivities. Despite of the presence of thesemolecular absorption peaks, one can still use the spectrum ofthe reflectivity to distinguish CaF2 from other crystalline solids.However, for the frequencies lower than 231 cm−1 , the reflectiv-ities are dramatically affected by the absorption of the incidentradiation by the molecules present in the air. It is worth notingthat when the absorption of the radiation due to the presenceof the molecules becomes too high, the power of the radiationafter propagating through air becomes too low. Therefore, thereflectivity determined by the ratio of the reflected and incidentpowers can be more than unity due to the nearly zero incidentpowers, as one can see from Fig. 6. Obviously, the huge spikesbelow 231 cm−1 , appearing in Fig. 6, are artifacts.

158 IEEE JOURNAL OF SELECTED TOPICS IN QUANTUM ELECTRONICS, VOL. 17, NO. 1, JANUARY/FEBRUARY 2011

IV. CONCLUSION

We have measured the reflection spectra of CaF2 single crys-tal and polycrystalline powder in the terahertz range. By fittingthe reflection curves, we have deduced the LO and TO frequen-cies from both of the single crystal and of the powder. Thecorresponding differences between them are within 2%. How-ever, the damping parameter for the CaF2 powder is more thantwice larger than that for the single crystal. Such a compari-son indicates that the polariton waves excited inside the CaF2particles were severely damped by their termination at the backsurfaces of the crystallites.

Our result presented here illustrates the feasibility that themeasurement of reflection spectrum across a polariton reso-nance can be developed into an effective technique for inves-tigating and identifying a crystalline solid UO2 and PuO2 bydeducing the TO and LO phonon frequencies. For example, TOand LO phonon frequencies of 448 cm−1 and 578 cm−1 canbe used to fingerprint and identify UO2 , which has exactly thesame lattice structure as CaF2 , by measuring the spectrum ofreflectivity. According to (8) above, the reflectivities of UO2reach the maximum values at the frequencies of 477 cm−1 and482 cm−1 , corresponding to two different values of the dampingparameter of the polariton, respectively, see Fig. 7. Moreover,the investigation of polariton resonances may lead to potentialapplications such as utilizing high-reflectivity mirrors in lasercavities, waveguide confinement of THz waves [13], slowingdown and perhaps stopping far-IR and THz waves, and realiz-ing nonlinear devices with enhanced performances.

ACKNOWLEDGMENT

The authors would like to thank Dr. R. Song for providing hisexpertise during the preparation of the powder samples.

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Yi Jiang (M’08) received the B.S. and Ph.D degrees,in 2001 and 2006, respectively, both from ShandongUniversity, Jinan, China.

In 2006, he was a Visiting Scientist at LehighUniversity, Bethlehem, PA, where he is currentlya Research Associate in the Department of Elec-trical and Computer Engineering. He is the au-thor or coauthor of more than 20 refereed jour-nal articles in optoelectronics, nonlinear optics,and Rutherford backscattering/channeling and ionimplantation. His research interests include ter-

ahertz generation, amplification and detection, nanostructures, and theirapplications.

Dr. Jiang is a member of the Optical Society of America.

Yujie J. Ding (M’04–SM’05) received the B.S. de-gree from Jilin University, Jilin, China, in 1984, theM.S.E.E. degree from Purdue University, Washing-ton, DC, in 1987, and the Ph.D. degree from JohnsHopkins University, Baltimore, MD, in 1990.

During 1990–1992, he was a Postdoctoral Fel-low, and then became an Associate Research Sci-entist at Johns Hopkins University. During 1992–1999, he was an Assistant Professor, and then be-came an Associate Professor of physics at BowlingGreen State University, Bowling Green, OH. Dur-

ing 1999–2002, he was an Associate Professor of physics at the University ofArkansas, Fayetteville. In 2002, he joined Lehigh University, Bethlehem, PA,where he is currently a Professor in the Department of Electrical and ComputerEngineering. He is the author or coauthor of more than 140 refereed journal arti-cles in optoelectronics, nonlinear optics, and quantum electronics. His researchinterests include terahertz generation, amplification and detection, nanostruc-tures and nanodevices, Raman scattering in nitride heterostructures, and theirapplications.

Prof. Ding is a Fellow of the Optical Society of America. He was the re-cipient of the Class of 1961 Professorship from Lehigh University in 2003 andthe Outstanding Young Scholar Award from Bowling Green State University in1996.