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Two-phonon hyper-Raman scattering and polariton spectra in rutile crystal
Volume 34, number 3
OPTICS COMMUNICATIONS
September 19110
TWO-PHONON HYPER-RAMAN SCATTERING AND POLARITON SPECTRA IN RUTILE CRYSTAL
V.N. DENISOV, B.N. MAVRIN, V.B. PODOBEDOV and Kh.E. STERIN Institute of Spectroscopy, Academy oj' $ciences of the USSR, USSR 142092, Troitzk, Moscow Region Received 10 June 1980
The first observation of botl two-phonon and polariton hyper-Raman scattering in rutile crystal is reported. The effect of two-phonon states on Au-polariton dispersion branch has been found.
1. lnt.oduetion
The hyper.Raman scattering e ae to the hyperpolarizability modulation by crystal-lattice vibrations may be observed in the frequency regi 9n of the second harmonic of the exciting light. A number of papers published lately shows the possibilities and future applications of the hyper-Raman tezhnique for the study of crystal vibrations. It was shown for the first time that hyper-Raman scattering may be observed in a noncentrosymmetric crystal [ 1], where the second harmonic (2toi)and hyper-Raman scattering (2to i co) intensities are of the same order (toi is the fiequeney of the exciting light, to is the crystal-lattice vibration frequency). It was also shown [2-5] that silent vibrations in solids may be observed in hyperRaman scattering. Using the hyper-Raman technique, we studied the quadrupole nonlinearity which gives rise to anomalous polarization of the 2coiJine in a centrosymmetrie TiO 2 crystal [5]. The first observation of the hyper-Raman scattering by Y I U i t a U U I I p U l a i i t ~ t t ~ I ~ a ~ . . . . . . . .7 SrTiO 3 crystal [6], will undoubtfully stimulate the polariton spectroscopy development. The hyperRaman polariton scattering as compared to the Raman one used up to now for bulk polariton studies makes possible the polariton observations in centrosymmetric crystals and the study of the upper polariton branch in both cubic and unaxial crystals. In the experiments described here, the hyper...........
Raman scattering by phonons and polaritons of A usymmetry in centrosymmetric TiO 2 (D4h) 14 rutile crystal has been studied. The second-order hyper.Raman spectra were observed for the first time, the obvious advantages of the hyper-Raman scattering, as compared to IR-absorption, for the study of dipole twophonon states in rutile were shown, the effect of twophonon states on polariton dispersion in rutile was found.
2. Experiments The hyper-Raman spectra were excited by a pulsed YAG laser (coi = 9396 cm -1) and wer:- recorded by the multichannel device described in our previous paper [ 1]. The hyper-Raman spectra of Au-phonons were recorded in x(zzz)y 90 ° scattering geometry, where three indices z mean the polarization of two exciting photons and of the scattered one. Only one dipole Au-vibration, splitting into TO- and LO-components is expected to be active in rutile [7,8]. In order to observe polariton spectra ~e focused z-polarized exciting beam (divergence inside the crystal is ~0.7 °) was directed along the crystal x-axis and the scattering was collected along the same direction (x(zzz)x-scattering geometry). The scattered light wa~s collected by a large-aperture lens, while the scattering angles were controlled by ring diaphragms [9]. The scattering angle 0 and its collection angle A0 were con357
OPTICS COMMUNICATIONS
Volume 34. number 3
A
B
\
} •I
I
!
~., ~
I
I
:4~,.~"
c~ -I
~ O f ~,
I ~ . 1. Spectra of the hyper-Ranlan scattering by the upper branch Au-polaritons of rutile crystal:a-O = 0° , b-0 = 3.7°.
trolled by the average circular radius and the ring gap width, respectively. Inside the crystal A0 was ~0.3 °. In .v(:::)x geometry the scattered light, on passing throu~ the ring diaphragm, contains both transverse and oblique components of Au-polaritons. The hyperRaman scattering associated with Au(TO)-transverse polaritons was located mainly in the Oxy-plane and its intensity was about 10 times higher than that of the oblique polaritons. Typical hyper-Raman Au(TO ) polariton spectra are shown in fig. 1.
3. The hyper-Raman scattering by phonons A strong peak of Au(TO)-phonon at 173 cm -1 , two peaks at 0 and 822 cm-1, one very broad band covering the range from 200 to 1200 cm -1 with the maximum at "~530 cm -1 are observed in the hyperRaman spectrum in x(zzz)), geometry (fig. 2). The
I
,
I
•
•
i
.t
|
,
!
•
r
CO,
C"
I:~. 2. ,Xu-phonon hyper-Raman spectrum of ruffle crystal in xlz-_z)y scattering geometry. The part of the spectrum reduced 10 times is shown by a dashed curve.
September 1980
broad band contains also some peak at 385,450, 590 and 670 cm -1. All these lines were also ,bserved in an anti-Stokes hyper.Raman spectrum. The peak at co = 0 is due to both quadrupole scattering at the 2coi second harmonic frequency [5] and the second harmonic scattering from the crystal surface. The 822 cm-l-line, which is very strong in the x(zzx)y geometry is expected to be active in our x(zzz)y scattering geometry due to both the crystal misorientation and the large collection angle (A0 = 0.13 sr). The 822 cm-l-line is associated with the Eu(LO).vibration [8]. But the appearance of the broad band with some peaks at 3 8 5 - 6 7 0 cm -.l cannot be explained by depolarization, because in x(zzx),v scattering geometry their intensities are at least two orders of magnitude less than the 822 cm - l line intensity. One of those peaks at "~590 era-1 corresponds probably to the very weak band observed in IR spectra at 592 cm -1 . This band is not associated with fundamental vibrations [8]. We assume the broad band with peaks at 385,450, 590 and 670 cm -1 to be associated with two-phonon states in a TiO 2 crystal. The Raman spectra in this frequency region are known to contain a lot of strong peaks associated with the second-order phonon process [10]. We believe that the hyper-Raman technique has the advantages as compared to IR absorption in this case. For example, only one of the above mentioned peaks observed in the hyper-Raman scattering was detected at the tail of the very strong IR-absorption band [8]. It is difficult to realize the hyper-Raman scattering geometry in which only the Au(LO)-vibration could be active. So, the Au(LO)-vibration frequency was found as follows. When directing the phonon wave vector k at if.angle with the crystal optical axis Oz, the oblique phonons are observed in the hyper-Raman spectrum. In z(xxz),v geometry ( i = 45°) Au(LO)" phonon in TiO 2 crystal is mixed with high frequency Eu(LO)-phonon at 822 cm - I giving rise to oblique one at 811 cm -1 . So the Au(LO)-phonon frequency corresponds to " 8 0 0 cm -1 which is close to that at 796 cm -1 obtained from IR spectra [8].
Volume 34, number 3
OPTICS COMMUNICATIONS
4. The hyper-Raman ,scattering by polaritons The comparison of Au(TO)-phonon line intensity with that of all-second order modes shows them to be comparable, e.g. the integral oscillator strength of the second-order modes must be large. So, a considerable contribution of the two-phonon states to the crystal dielectric function e(co) would be expected. Taking into account two-phonon states, the dielectric function e(co) can be written as follows [ 11 ]" -
=
+
(1)
where coLO and coTO are related to Au-phonon, function F(co') is proportional to both the density of twophonon states and their oscillator strength at frequenP cy co, e** is the high-frequency dielectric constant along the Z.axis. The polariton dispersion co(k) is determined completely by the dielectric function: k2/4rt2co2 = e(co).
(2)
Thus, neglecting the influbnce of two.phonon states in a ruffle crystal, a reasonable difference between the experimental curves and the curves calculated by eq. (2) may be expected. The polariton dispersion co(k) was found from measured co(0)values with the use of the relation which follows from momentum and energy conservation laws [61: k 2 = 41r2 ([2niw i _ cosns] 2 + 4wicosnins(1 - cos 0 ) ) ,
(3)
wl~ere i and s are subscripts refer to the exciting and scattered light, respectively (co s = 2co i - co); k , 6oi, cos and co are given in terms of cm -1; n i = 2.7354 [12]. We have taken into account the ns(cos) dependence in our calculations. In 5300-6300 A region it may be written as: n s = 2.9595 - 0.0000327 to.
(4)
The polariton wave vector kmin values which can be practicable in the experiment are given by eq. (3) with 0 = 0: kmin = 4~.coi(ni - ns) + 2mlsco.
(5)
The k m i n magnitude for the lowest polariton branch in the rutile crystal was found to be very large
September 1980
(~25000 cm -1). Therefore the Au(TO ) polariton frequency shift was expected to be small in hyperRaman scattering. But the upper polariton branch may be observed in hyper-Raman scattering at the frequencies above co = 900 cm -1 (kmi n ~ 6000 cm-l). The polariton study was possible by the apertures employed at scattering angles 0 varying from 0 to 6 ° inside the crystal (up to ~17 ° in air). Under such conditions the polariton frequency shift from 908 to 3300 cm-1 was observed at the upper branch. Using the upper branch co(k) measurements, we calculated the high-frequency dielectric constant e**. It was calculated from eq. ( 1 ) - ( 2 ) and co(k) measurements in the 2500-3300 cm -1 region, where the contribution of two-phonon states to e(co) was negligible. The dielectric constant e** was found to be 7.25 -+ 0.1 and was close to the value 7.2 obtained from the surface polariton spectra [ 13]. Using the found value e.~ and taking coTO = 173 c m - l , coLO = 800 cm -1 , we calculated the co(k) dispersion branch by eq. (2), where the two-phonon contributions were neglected. In the 1200-3300 cm-1 region the experimental values are close to those obtained from co(k) calculated curve. But below 1200 cm -1 the observed polariton frequencies are higher than the calculated ones. Tltis part of the dispersion branch is represented in fig. 3. One can see that below 1200 cm -1 the polariton at frequency ,~ has a smaller wave vector than that predicted by calculation. In accordance with eq. (2) the observed discrepancy shows smaller e(co)values as compared to those expected from the classical oscillator model, e.g. taking no account of the integral in eq. (1). We consider the above-mentioned discrepancy to be caused by the effect of the two-photon states observed independently in the hyper-Raman spectrum (fig. 2). The integral that denotes the two-phtonon state contributions to e(co) in eq. (1) with maximum at ~'530 cm -1 shows that, first, the integr:d value is negative in the region of interest above 900 cm-1, and, second, it decreases as the frequency co moves away from the edge of the two-photon state zone. The expected character of discrepancy at the upper polariton branch due to the interaction between polaritons and two-phonon states was found to be in qualitative agreement with the data obtained (fig. 3 ) Undoubtedly, the great two-photon state contribution to e(co) affects not only the behaviour of the 359
Xo|~mlc 34, ,mmber 3
OPTICS COMMUNICATIONS
September 1980
•
•0
•
6
....
i
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Ca), _ 1
!(, •
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J,
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;:~,<,C
C ~ -t
I
i,
, ,
.
¢~-[ ,
3:i,
y_
Fig. 4, Extraordinary refractive index in ruffle. ! i~. 3. Calculated Au-polariton dispersion of the upper branch in futile and experimental data points at different ~-'a:~"~ir~, angles inside the crystal. Solid curve was calculated wi~h E~, = 7.25, t.TO = 173 cm -l , ~LO = 800 cm -I taking no account of the two-phonon state contributions.
polariton dispersion branch, but also the Au(LO )phonon frequency. At the LO-phonon frequency e(Wl_o) = O. The Au(LO)-phonon frequency must increa~ due to negative two-phonon state contributk,,,s to e ( ~ ) at this frequency. So, in fact, the devia,~on ,ff ~he dispersion branch from that predicted by ~,he classical oscillator model may be even more than nt is shown in fig. 3 (solid curve was calculated with ~ t o = 800 c m - 1). Finally, the study of the upper polariton branch permits the determination of the crystal refractive index at IR. UsLng the experimental ~ ( k ) values and the relation k = 2mleW ,
(6)
we found the c ~ s t a l extraordinary refractive index n c in the 9 0 8 - 3 3 0 0 cm -1 region (fig. 4). It should be noted that n e measurements by means of the conventional method at those frequencies are impractical
[]21.
360
References
Ill V.N. Denisov, B.N. Mavrin, V.B. Podobedov and Kh.E. Sterin, Optics Comm. 26 (1978) 372; JETP 75 (1978) 684. [21 V.N. Denisov, B.N. Mavrin, V.B. Podobedov and Kh.E. Sterin, FTT (Physica Tverdogo Tela) 20 (1978) 3485. [31 tt. Vogt and G. Neumann, Phys. Stat. Sol. 92b (1979) 57 [41 B.G. Varshal, V.N. Denis0v, B.N. Mavrin, (;.A. I'avlova, V.B. Podobedov and Kh.E. Sterin, Opt. i. Spektr. 47 (1979) 619. 151 V.N. Denisov, B.N. Mavrin, V.B. Podobedov, Kh.E. Sterin and B.G. Varshal, Opt. i Spektr° 49 (1980)406. [61 V.N. Denisov, B.N. Mavrin, V.B. Podobedov and Kh.E. Sterin, Pisma JETP 31 (1980) 111. [71 S.P.S. Porto, P.A. Fleury and T.C. Damen, Phys. Rev. 154 (1967) 522. [81 F. Gervais and B. Piriou, Phys. Rev. 10B (1974) 1642. [91 R. Claus, Rev. Sci. Instrum. 42 (1971) 341. [lOl J.H. Nicola, C.A. Brunhavoto, M. Abramovich and C.E.T. Concalves da Silva, J. Ram. Spectr. 8 (1979) 32. 1111 B.N. Mavrin and Kh.E. Sterin, Physica Tverdogo Tcla ! 8 (1976) 3028. [121 W.L. Bond, J. App!. Phys. 36 (! 968) 1674. I131 V.V. Bryksin, D.N. Mirlin and 1.I. Reshina, Pisma .IETP 16 (1972) 231.
OPTICS COMMUNICATIONS
September 19110
TWO-PHONON HYPER-RAMAN SCATTERING AND POLARITON SPECTRA IN RUTILE CRYSTAL
V.N. DENISOV, B.N. MAVRIN, V.B. PODOBEDOV and Kh.E. STERIN Institute of Spectroscopy, Academy oj' $ciences of the USSR, USSR 142092, Troitzk, Moscow Region Received 10 June 1980
The first observation of botl two-phonon and polariton hyper-Raman scattering in rutile crystal is reported. The effect of two-phonon states on Au-polariton dispersion branch has been found.
1. lnt.oduetion
The hyper.Raman scattering e ae to the hyperpolarizability modulation by crystal-lattice vibrations may be observed in the frequency regi 9n of the second harmonic of the exciting light. A number of papers published lately shows the possibilities and future applications of the hyper-Raman tezhnique for the study of crystal vibrations. It was shown for the first time that hyper-Raman scattering may be observed in a noncentrosymmetric crystal [ 1], where the second harmonic (2toi)and hyper-Raman scattering (2to i co) intensities are of the same order (toi is the fiequeney of the exciting light, to is the crystal-lattice vibration frequency). It was also shown [2-5] that silent vibrations in solids may be observed in hyperRaman scattering. Using the hyper-Raman technique, we studied the quadrupole nonlinearity which gives rise to anomalous polarization of the 2coiJine in a centrosymmetrie TiO 2 crystal [5]. The first observation of the hyper-Raman scattering by Y I U i t a U U I I p U l a i i t ~ t t ~ I ~ a ~ . . . . . . . .7 SrTiO 3 crystal [6], will undoubtfully stimulate the polariton spectroscopy development. The hyperRaman polariton scattering as compared to the Raman one used up to now for bulk polariton studies makes possible the polariton observations in centrosymmetric crystals and the study of the upper polariton branch in both cubic and unaxial crystals. In the experiments described here, the hyper...........
Raman scattering by phonons and polaritons of A usymmetry in centrosymmetric TiO 2 (D4h) 14 rutile crystal has been studied. The second-order hyper.Raman spectra were observed for the first time, the obvious advantages of the hyper-Raman scattering, as compared to IR-absorption, for the study of dipole twophonon states in rutile were shown, the effect of twophonon states on polariton dispersion in rutile was found.
2. Experiments The hyper-Raman spectra were excited by a pulsed YAG laser (coi = 9396 cm -1) and wer:- recorded by the multichannel device described in our previous paper [ 1]. The hyper-Raman spectra of Au-phonons were recorded in x(zzz)y 90 ° scattering geometry, where three indices z mean the polarization of two exciting photons and of the scattered one. Only one dipole Au-vibration, splitting into TO- and LO-components is expected to be active in rutile [7,8]. In order to observe polariton spectra ~e focused z-polarized exciting beam (divergence inside the crystal is ~0.7 °) was directed along the crystal x-axis and the scattering was collected along the same direction (x(zzz)x-scattering geometry). The scattered light wa~s collected by a large-aperture lens, while the scattering angles were controlled by ring diaphragms [9]. The scattering angle 0 and its collection angle A0 were con357
OPTICS COMMUNICATIONS
Volume 34. number 3
A
B
\
} •I
I
!
~., ~
I
I
:4~,.~"
c~ -I
~ O f ~,
I ~ . 1. Spectra of the hyper-Ranlan scattering by the upper branch Au-polaritons of rutile crystal:a-O = 0° , b-0 = 3.7°.
trolled by the average circular radius and the ring gap width, respectively. Inside the crystal A0 was ~0.3 °. In .v(:::)x geometry the scattered light, on passing throu~ the ring diaphragm, contains both transverse and oblique components of Au-polaritons. The hyperRaman scattering associated with Au(TO)-transverse polaritons was located mainly in the Oxy-plane and its intensity was about 10 times higher than that of the oblique polaritons. Typical hyper-Raman Au(TO ) polariton spectra are shown in fig. 1.
3. The hyper-Raman scattering by phonons A strong peak of Au(TO)-phonon at 173 cm -1 , two peaks at 0 and 822 cm-1, one very broad band covering the range from 200 to 1200 cm -1 with the maximum at "~530 cm -1 are observed in the hyperRaman spectrum in x(zzz)), geometry (fig. 2). The
I
,
I
•
•
i
.t
|
,
!
•
r
CO,
C"
I:~. 2. ,Xu-phonon hyper-Raman spectrum of ruffle crystal in xlz-_z)y scattering geometry. The part of the spectrum reduced 10 times is shown by a dashed curve.
September 1980
broad band contains also some peak at 385,450, 590 and 670 cm -1. All these lines were also ,bserved in an anti-Stokes hyper.Raman spectrum. The peak at co = 0 is due to both quadrupole scattering at the 2coi second harmonic frequency [5] and the second harmonic scattering from the crystal surface. The 822 cm-l-line, which is very strong in the x(zzx)y geometry is expected to be active in our x(zzz)y scattering geometry due to both the crystal misorientation and the large collection angle (A0 = 0.13 sr). The 822 cm-l-line is associated with the Eu(LO).vibration [8]. But the appearance of the broad band with some peaks at 3 8 5 - 6 7 0 cm -.l cannot be explained by depolarization, because in x(zzx),v scattering geometry their intensities are at least two orders of magnitude less than the 822 cm - l line intensity. One of those peaks at "~590 era-1 corresponds probably to the very weak band observed in IR spectra at 592 cm -1 . This band is not associated with fundamental vibrations [8]. We assume the broad band with peaks at 385,450, 590 and 670 cm -1 to be associated with two-phonon states in a TiO 2 crystal. The Raman spectra in this frequency region are known to contain a lot of strong peaks associated with the second-order phonon process [10]. We believe that the hyper-Raman technique has the advantages as compared to IR absorption in this case. For example, only one of the above mentioned peaks observed in the hyper-Raman scattering was detected at the tail of the very strong IR-absorption band [8]. It is difficult to realize the hyper-Raman scattering geometry in which only the Au(LO)-vibration could be active. So, the Au(LO)-vibration frequency was found as follows. When directing the phonon wave vector k at if.angle with the crystal optical axis Oz, the oblique phonons are observed in the hyper-Raman spectrum. In z(xxz),v geometry ( i = 45°) Au(LO)" phonon in TiO 2 crystal is mixed with high frequency Eu(LO)-phonon at 822 cm - I giving rise to oblique one at 811 cm -1 . So the Au(LO)-phonon frequency corresponds to " 8 0 0 cm -1 which is close to that at 796 cm -1 obtained from IR spectra [8].
Volume 34, number 3
OPTICS COMMUNICATIONS
4. The hyper-Raman ,scattering by polaritons The comparison of Au(TO)-phonon line intensity with that of all-second order modes shows them to be comparable, e.g. the integral oscillator strength of the second-order modes must be large. So, a considerable contribution of the two-phonon states to the crystal dielectric function e(co) would be expected. Taking into account two-phonon states, the dielectric function e(co) can be written as follows [ 11 ]" -
=
+
(1)
where coLO and coTO are related to Au-phonon, function F(co') is proportional to both the density of twophonon states and their oscillator strength at frequenP cy co, e** is the high-frequency dielectric constant along the Z.axis. The polariton dispersion co(k) is determined completely by the dielectric function: k2/4rt2co2 = e(co).
(2)
Thus, neglecting the influbnce of two.phonon states in a ruffle crystal, a reasonable difference between the experimental curves and the curves calculated by eq. (2) may be expected. The polariton dispersion co(k) was found from measured co(0)values with the use of the relation which follows from momentum and energy conservation laws [61: k 2 = 41r2 ([2niw i _ cosns] 2 + 4wicosnins(1 - cos 0 ) ) ,
(3)
wl~ere i and s are subscripts refer to the exciting and scattered light, respectively (co s = 2co i - co); k , 6oi, cos and co are given in terms of cm -1; n i = 2.7354 [12]. We have taken into account the ns(cos) dependence in our calculations. In 5300-6300 A region it may be written as: n s = 2.9595 - 0.0000327 to.
(4)
The polariton wave vector kmin values which can be practicable in the experiment are given by eq. (3) with 0 = 0: kmin = 4~.coi(ni - ns) + 2mlsco.
(5)
The k m i n magnitude for the lowest polariton branch in the rutile crystal was found to be very large
September 1980
(~25000 cm -1). Therefore the Au(TO ) polariton frequency shift was expected to be small in hyperRaman scattering. But the upper polariton branch may be observed in hyper-Raman scattering at the frequencies above co = 900 cm -1 (kmi n ~ 6000 cm-l). The polariton study was possible by the apertures employed at scattering angles 0 varying from 0 to 6 ° inside the crystal (up to ~17 ° in air). Under such conditions the polariton frequency shift from 908 to 3300 cm-1 was observed at the upper branch. Using the upper branch co(k) measurements, we calculated the high-frequency dielectric constant e**. It was calculated from eq. ( 1 ) - ( 2 ) and co(k) measurements in the 2500-3300 cm -1 region, where the contribution of two-phonon states to e(co) was negligible. The dielectric constant e** was found to be 7.25 -+ 0.1 and was close to the value 7.2 obtained from the surface polariton spectra [ 13]. Using the found value e.~ and taking coTO = 173 c m - l , coLO = 800 cm -1 , we calculated the co(k) dispersion branch by eq. (2), where the two-phonon contributions were neglected. In the 1200-3300 cm-1 region the experimental values are close to those obtained from co(k) calculated curve. But below 1200 cm -1 the observed polariton frequencies are higher than the calculated ones. Tltis part of the dispersion branch is represented in fig. 3. One can see that below 1200 cm -1 the polariton at frequency ,~ has a smaller wave vector than that predicted by calculation. In accordance with eq. (2) the observed discrepancy shows smaller e(co)values as compared to those expected from the classical oscillator model, e.g. taking no account of the integral in eq. (1). We consider the above-mentioned discrepancy to be caused by the effect of the two-photon states observed independently in the hyper-Raman spectrum (fig. 2). The integral that denotes the two-phtonon state contributions to e(co) in eq. (1) with maximum at ~'530 cm -1 shows that, first, the integr:d value is negative in the region of interest above 900 cm-1, and, second, it decreases as the frequency co moves away from the edge of the two-photon state zone. The expected character of discrepancy at the upper polariton branch due to the interaction between polaritons and two-phonon states was found to be in qualitative agreement with the data obtained (fig. 3 ) Undoubtedly, the great two-photon state contribution to e(co) affects not only the behaviour of the 359
Xo|~mlc 34, ,mmber 3
OPTICS COMMUNICATIONS
September 1980
•
•0
•
6
....
i
•O
Ca), _ 1
!(, •
I
I
•
•
|
'
*
J,
i
;:~,<,C
C ~ -t
I
i,
, ,
.
¢~-[ ,
3:i,
y_
Fig. 4, Extraordinary refractive index in ruffle. ! i~. 3. Calculated Au-polariton dispersion of the upper branch in futile and experimental data points at different ~-'a:~"~ir~, angles inside the crystal. Solid curve was calculated wi~h E~, = 7.25, t.TO = 173 cm -l , ~LO = 800 cm -I taking no account of the two-phonon state contributions.
polariton dispersion branch, but also the Au(LO )phonon frequency. At the LO-phonon frequency e(Wl_o) = O. The Au(LO)-phonon frequency must increa~ due to negative two-phonon state contributk,,,s to e ( ~ ) at this frequency. So, in fact, the devia,~on ,ff ~he dispersion branch from that predicted by ~,he classical oscillator model may be even more than nt is shown in fig. 3 (solid curve was calculated with ~ t o = 800 c m - 1). Finally, the study of the upper polariton branch permits the determination of the crystal refractive index at IR. UsLng the experimental ~ ( k ) values and the relation k = 2mleW ,
(6)
we found the c ~ s t a l extraordinary refractive index n c in the 9 0 8 - 3 3 0 0 cm -1 region (fig. 4). It should be noted that n e measurements by means of the conventional method at those frequencies are impractical
[]21.
360
References
Ill V.N. Denisov, B.N. Mavrin, V.B. Podobedov and Kh.E. Sterin, Optics Comm. 26 (1978) 372; JETP 75 (1978) 684. [21 V.N. Denisov, B.N. Mavrin, V.B. Podobedov and Kh.E. Sterin, FTT (Physica Tverdogo Tela) 20 (1978) 3485. [31 tt. Vogt and G. Neumann, Phys. Stat. Sol. 92b (1979) 57 [41 B.G. Varshal, V.N. Denis0v, B.N. Mavrin, (;.A. I'avlova, V.B. Podobedov and Kh.E. Sterin, Opt. i. Spektr. 47 (1979) 619. 151 V.N. Denisov, B.N. Mavrin, V.B. Podobedov, Kh.E. Sterin and B.G. Varshal, Opt. i Spektr° 49 (1980)406. [61 V.N. Denisov, B.N. Mavrin, V.B. Podobedov and Kh.E. Sterin, Pisma JETP 31 (1980) 111. [71 S.P.S. Porto, P.A. Fleury and T.C. Damen, Phys. Rev. 154 (1967) 522. [81 F. Gervais and B. Piriou, Phys. Rev. 10B (1974) 1642. [91 R. Claus, Rev. Sci. Instrum. 42 (1971) 341. [lOl J.H. Nicola, C.A. Brunhavoto, M. Abramovich and C.E.T. Concalves da Silva, J. Ram. Spectr. 8 (1979) 32. 1111 B.N. Mavrin and Kh.E. Sterin, Physica Tverdogo Tcla ! 8 (1976) 3028. [121 W.L. Bond, J. App!. Phys. 36 (! 968) 1674. I131 V.V. Bryksin, D.N. Mirlin and 1.I. Reshina, Pisma .IETP 16 (1972) 231.