- Email: [email protected]
THz generation in InAs
ARTICLE IN PRESS
Physica B 376–377 (2006) 618–621 www.elsevier.com/locate/physb
THz generation in InAs R.A. Lewis, M.L. Smith, R. Mendis, R.E.M. Vickers Institute for Superconducting and Electronic Materials, University of Wollongong, Wollongong NSW 2522, Australia
Abstract We have observed strong terahertz (THz) emission from the o1 0 04 face of p-type InAs when it is illuminated by ultrashort (o12 fs) pulses of near-infrared radiation. As the crystal is rotated about the surface normal, there are two maxima per rotation, suggesting optical rectification plays a role in the emission process. This holds whether the angle of incidence is 451, most convenient for technical application, or 751, the Brewster angle, where the THz output is strongest. The power of the THz radiation varies approximately quadratically with the pump power. The data are consistent with photocurrent surge being the main mechanism of THz emission. We have found that the p-type InAs produces about two orders of magnitude more power than a standard unbiased THz emitter, 1-mm thick ZnTe. r 2005 Elsevier B.V. All rights reserved. PACS: 72.80.Ey; 78.30.Fs Keywords: III–IV; Terahertz; Impurities; InAs
1. Introduction The terahertz (THz) part of the electromagnetic spectrum lies between radio waves and visible light. Many materials exhibit characteristic resonances at THz frequencies [1]. These include materials implicated in the war on terrorism such as the biological agent anthrax (surrogate, Bacillus subtilus) [2] and the chemical agent RDX (a plastic explosive), foundational biological entities such as DNA components (base, nucleoside, nucleotide) [3], and pharmaceutical materials both therapeutic (aspirin, lactose) and illicit (cocaine). Many packaging materials (plastic, paper, styrofoam) are transparent to THz radiation [4] allowing the examination of packaged powders for moisture content [5], or of packaged meat for spoilage. Glasses, oils and clothing all show distinctive THz responses. THz images of basal cell carcinoma (a skin cancer) reveal contrast between diseased and normal tissue [6] both in amplitude and phase. THz spectra have been used to examine burns on flesh. Unlike X-rays, ‘T-rays’ promise exquisite sensitivity in dermatological studies. Corresponding author. Tel.: +61 2 4221 3062; fax: +61 2 4221 5944.
E-mail address: [email protected] (R.A. Lewis). 0921-4526/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2005.12.156
The technology to transmit and receive THz radiation is much less developed than the electrical and optical methods so effective in the regions of the electromagnetic spectrum that bracket it. The single largest obstacle to the advance of THz science and technology is the lack of sufficiently powerful emitters. Generating coherent THz radiation typically involves three steps. Conversion from the visible to the near infrared (NIR) is relatively efficient. Conversion from the NIR to the THz is much less efficient. Ultrashort NIR pulses have been directed to many types of targets to produce coherent THz radiation. The targets include normal [7] and electrically-biased [8] air, the laser-plasma boundary [9] and Ta foil [10]. THz generation via (free) plasma has the disadvantages of requiring terawatt sources and not having demonstrated TDS and so this method will not be considered further. 2. Semiconductor targets Solid targets are classified as electro-optic (EO) or photoconductive (PC) depending on the mechanism by which the NIR fs pulse is converted to THz radiation. A widely used EO emitter is o1 1 04 ZnTe; a widely used PC
ARTICLE IN PRESS R.A. Lewis et al. / Physica B 376–377 (2006) 618–621
applied [18]. In spite of the recent activity in this area we believe the full parameter space has not been mapped, in particular the best growth and annealing conditions. Asgrown LT-GaAs has short carrier recombination times but low resistivity; annealing produces a large resistivity increase with little increase in the recombination time [19]. 3. p-InAs InAs is the most efficient emitter to date that is neither PC nor (conventional) EO. When lightly p doped, InAs becomes a powerful THz emitter, in contrast to the results of a systematic study of n-InAs. The exact mechanism of emission in p-InAs is still controversial. From a o1 1 14 face a 20-fold increase in emission is observed as the azimuthal angle is varied, interpreted as a base signal due to photocurrent surge on which is imposed a large signal due to optical rectification; the change in the direction of the E field detected at different azimuthal angles supports this interpretation as does, possibly, the weak magnetic field dependence. Measurements to date do not distinguish whether the optical rectification is bulk or surface-field induced, although the doping dependence suggests the
4
6
Photon frequency (THz) 8 10 12
14
1.0 TO LO PHONONS
0.8
InAs T = 295 K
0.6 Reflectivity
emitter is low-temperature grown (LT, LTG)-GaAs. The conventional comparison of these two classes of emitter is that the PC offers the better (by about 8 times) signal-tonoise ratio, but the EO has the greater spectral range (to about 30 THz compared to about 3 THz). A third category of targets will be classified as other mechanism (OM). This includes photocurrent emitters of several sorts. The most prominent OM target is p-InAs. Some targets exhibit a combination of EO, PC and OM operation. In EO emitters the fs pulse generates THz radiation due to a transient non-linear polarization of the material. This mechanism is described as optical rectification since the emitted THz radiation follows the envelope of the laser pulse. It is a bulk effect: increasing the thickness of the emitter will lead to greater power but at the cost of loss of phase matching and so bandwidth. In PC emitters the fs pulse generates THz radiation through creating photo-carriers by the above-bandgap illumination. These accelerate in an electrical field typically produced by applying a DC voltage between metallic electrodes. The resulting time-varying dipole emits a THz pulse. Under simplifying assumptions the THz power is directly proportional to the square of both the DC field and the laser power. PC emitters are more efficient than EO emitters at low pump power [11] but the higher pump powers and higher biases required to produce absolutely more power often favors EO emitters as the PC mechanism breaks down. In OM materials the mechanism is neither a pure PC or EO effect but may be a (photo)current surge, involving the separation of electrons and holes by the NIR illumination, followed by radiation from the resulting dipoles. The current surge may originate from (1) the surface electric field arising in the depletion layer or (2) the photo-Dember effect. These are not the only possible OM. Additional suggested mechanisms involve (3) electric-field-induced polarization or the inverse Franz–Keldysh effect [12], (4) magnetoplasma waves [13], and (5) coupled plasmaphonon modes [14]. ZnTe is regarded as the premier EO material below 5 THz but fundamental and higher-order phonon processes confuse the picture at higher frequencies, especially for thick crystals [15]. The choice of EO crystal depends on the wavelength of the fs excitation, as the coherence length is frequency dependent. Emission of 2-THz radiation from ZnTe peaks for laser pumping at 800 nm, while that from GaAs peaks for 1400 nm excitation [16]. LT-GaAs has emerged as being preferred to the more traditional semi-insulating (SI)-GaAs in view of the much shorter recombination times, o1 ps compared with 100 ps. Smaller recombination times lead to greater bandwidth, now extending from 0.3 to 7.5 THz [17]. LTGaAs has a smaller mobility than SI-GaAs leading to lower photocurrent, less photoheating and so less chance of thermal runaway. LT-GaAs produces similar amounts of THz radiation to SI-GaAs under the same pump power but with reduced photocurrent allowing a higher bias to be
619
0.4
0.2
0.0 100
200
300 Photon energy (cm-1)
400
Fig. 1. Room-temperature reflectivity of p-InAs.
500
ARTICLE IN PRESS R.A. Lewis et al. / Physica B 376–377 (2006) 618–621
620
latter. A narrow-gap semiconductor such as InAs has high electron mobility me . If this is coupled with a low hole mobility mh , a large transient current may arise (the Dember effect). Surprisingly, InSb, with me twice that of InAs, emits only 1% of the radiation [20]. This is accounted for by the presence of low-me conduction band valleys in InSb.
3.1. Doping p-InAs has an efficiency which, unexpectedly, depends strongly on the concentration of dopants and, critically, whether these are acceptors or donors [12]. Published and unpublished results indicate a strong dependence of THz power on the doping level, although the nature of the dependence varies between reports. The efficiency of the standard EO emitter ZnTe is also known to depend on the impurities, often crudely judged by the color and opacity of the crystal. We used room-temperature THz reflection spectroscopy of the distinctive reststrahlen feature (Fig. 1) to determine an upper limit for the carrier concentration. We deduce from these data that the sample is lightly doped. There is no evidence of the plasma above 100 cm 1, which corresponds to carrier concentration 6 1016 cm 3. We have sought transitions from the ground
state to the excited states of the Zn acceptor through lowtemperature (1.5 K) THz transmission measurements. So far we have not been able to detect these.
3.2. Geometrical factors In some OM materials geometric considerations, including polarization measurements, establish that the emission mechanism is neither a pure PC nor EO effect [21]. For example, in one report THz emission from the o1 1 14 face of p-InAs dips to 2/5 of the maximum as the sample is rotated about the surface normal suggesting the signal is 60% due to an EO effect. Emission occurs in the absence of an antenna structure on the surface, let alone an applied bias, so is not due to PC. The suggested origin of the remaining 40% of the emission is in a (photo)current surge. Some of our measurements on the geometrical factors affecting emission from p-InAs are shown in Fig. 2. The sample was rotated by azimuthal angle y about the surface normal for angles of incidence 451 and 751. The THz power follows a cos 2y dependence superimposed on a substantial background. This suggests that both photocurrent surge and optical rectification are contributing, with the former dominating.
1.2
1.0
1.4
p-InAs
1.2
angle of incidence
THz power (relative to maximum power)
THz power (relative to average power)
45°
0.8 angle of incidence 45° 75°
0.6
0.4
75° 1.0
0.8
0.6
0.4
0.2
0.2
p-InAs
0.0
0.0
0
100 200 Azimuthal angle (degrees)
300
Fig. 2. Power dependence on azimuthal angle for p-InAs.
0
50 100 150 200 Pump power incident on crystal (mW) Fig. 3. THz power dependence on pump power.
250
ARTICLE IN PRESS R.A. Lewis et al. / Physica B 376–377 (2006) 618–621
3.3. Optical excitation
References
As shown in Fig. 3, the variation of THz emitted power with pump power is close to quadratic, in contrast to o1 1 14 p-InAs [12]. The quadratic dependence we observe is characteristic of both photocurrent surge at low fluences and of optical rectification [22].
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
3.4. Magnetic field While initial claims as to the enhancement of THz emission due to the application of magnetic field appear to have been overstated there is no doubt that a magnetic field may increase THz emission, if only through a better orientation of the radiating dipole [23]. A magnetic field parallel to the surface can increase THz emission one hundred times in undoped InAs. Such a strong effect has not been observed in lightly doped p-InAs [12]. We intend to explore this in future to determine if a variation due to the Lorentz-force rotation of the dipole, or another cause, is occurring. Acknowledgments This work was supported by the Australian Research Council and the University of Wollongong. We acknowledge the support of Prof. Chao Zhang.
[15] [16] [17] [18] [19] [20] [21] [22] [23]
621
B. Ferguson, X.-C. Zhang, Nat. Mater. 1 (2002) 26. T.J. Johnson, et al., Chem. Phys. Lett. 403 (2005) 152. J. Nishizawa, et al., Optics Commun. 246 (2005) 229. T.-I. Jeon, et al., Curr. Appl. Phys. 5 (2005) 289. V.B. Podobedov, et al., JQSRT 91 (2005) 287. J. Nishizawa, et al., Optics Commun. 244 (2005) 469. D.J. Cook, R.M. Hochstrasser, Opt. Lett. 25 (2000) 1210. T. Loffler, et al., Appl. Phys. Lett. 77 (2000) 453. W.P. Leemans, et al., Phys. Rev. Lett. 91 (2003) 74802. S. Nashima, et al., COMMAD Conference, December 2004. Y.C. Shen, et al., Appl. Phys. Lett. 85 (2004) 164. R. Adomavicius, et al., Appl. Phys. Lett. 85 (2004) 2463. J.N. Heyman, et al., Phys. Rev. B 64 (2001) 085202. M.P. Hasselbeck, et al., Phys. Rev. B 65 (2002) 233203; R. Huber, et al., Phys. Rev. Lett. 94 (2005) 027401. M. Schall, M. Walther, P. Uhd Jepsen, Phys. Rev. B 64 (2001) 094301. A.G. Stepanov, J. Hebling, J. Kuhl, Appl. Phys. Lett. 83 (2003) 3000. Y.C. Shen, et al., Phys. Rev. B 69 (2004) 235325. M.R. Stone, et al., IEEE Trans. Microwave Theory Technol. 52 (2004) 2420. I.S. Gregory, et al., Appl. Phys. Lett. 83 (2003) 4199. P. Gu, et al., J. Appl. Phys. 91 (2002) 5533. M. Reid, et al., Phys. Rev. B 71 (2005) 081306. A. Yariv, P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation, Wiley, New York, 1984. A. Corchia, et al., Phys. Rev. B 64 (2001) 205204.
Physica B 376–377 (2006) 618–621 www.elsevier.com/locate/physb
THz generation in InAs R.A. Lewis, M.L. Smith, R. Mendis, R.E.M. Vickers Institute for Superconducting and Electronic Materials, University of Wollongong, Wollongong NSW 2522, Australia
Abstract We have observed strong terahertz (THz) emission from the o1 0 04 face of p-type InAs when it is illuminated by ultrashort (o12 fs) pulses of near-infrared radiation. As the crystal is rotated about the surface normal, there are two maxima per rotation, suggesting optical rectification plays a role in the emission process. This holds whether the angle of incidence is 451, most convenient for technical application, or 751, the Brewster angle, where the THz output is strongest. The power of the THz radiation varies approximately quadratically with the pump power. The data are consistent with photocurrent surge being the main mechanism of THz emission. We have found that the p-type InAs produces about two orders of magnitude more power than a standard unbiased THz emitter, 1-mm thick ZnTe. r 2005 Elsevier B.V. All rights reserved. PACS: 72.80.Ey; 78.30.Fs Keywords: III–IV; Terahertz; Impurities; InAs
1. Introduction The terahertz (THz) part of the electromagnetic spectrum lies between radio waves and visible light. Many materials exhibit characteristic resonances at THz frequencies [1]. These include materials implicated in the war on terrorism such as the biological agent anthrax (surrogate, Bacillus subtilus) [2] and the chemical agent RDX (a plastic explosive), foundational biological entities such as DNA components (base, nucleoside, nucleotide) [3], and pharmaceutical materials both therapeutic (aspirin, lactose) and illicit (cocaine). Many packaging materials (plastic, paper, styrofoam) are transparent to THz radiation [4] allowing the examination of packaged powders for moisture content [5], or of packaged meat for spoilage. Glasses, oils and clothing all show distinctive THz responses. THz images of basal cell carcinoma (a skin cancer) reveal contrast between diseased and normal tissue [6] both in amplitude and phase. THz spectra have been used to examine burns on flesh. Unlike X-rays, ‘T-rays’ promise exquisite sensitivity in dermatological studies. Corresponding author. Tel.: +61 2 4221 3062; fax: +61 2 4221 5944.
E-mail address: [email protected] (R.A. Lewis). 0921-4526/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.physb.2005.12.156
The technology to transmit and receive THz radiation is much less developed than the electrical and optical methods so effective in the regions of the electromagnetic spectrum that bracket it. The single largest obstacle to the advance of THz science and technology is the lack of sufficiently powerful emitters. Generating coherent THz radiation typically involves three steps. Conversion from the visible to the near infrared (NIR) is relatively efficient. Conversion from the NIR to the THz is much less efficient. Ultrashort NIR pulses have been directed to many types of targets to produce coherent THz radiation. The targets include normal [7] and electrically-biased [8] air, the laser-plasma boundary [9] and Ta foil [10]. THz generation via (free) plasma has the disadvantages of requiring terawatt sources and not having demonstrated TDS and so this method will not be considered further. 2. Semiconductor targets Solid targets are classified as electro-optic (EO) or photoconductive (PC) depending on the mechanism by which the NIR fs pulse is converted to THz radiation. A widely used EO emitter is o1 1 04 ZnTe; a widely used PC
ARTICLE IN PRESS R.A. Lewis et al. / Physica B 376–377 (2006) 618–621
applied [18]. In spite of the recent activity in this area we believe the full parameter space has not been mapped, in particular the best growth and annealing conditions. Asgrown LT-GaAs has short carrier recombination times but low resistivity; annealing produces a large resistivity increase with little increase in the recombination time [19]. 3. p-InAs InAs is the most efficient emitter to date that is neither PC nor (conventional) EO. When lightly p doped, InAs becomes a powerful THz emitter, in contrast to the results of a systematic study of n-InAs. The exact mechanism of emission in p-InAs is still controversial. From a o1 1 14 face a 20-fold increase in emission is observed as the azimuthal angle is varied, interpreted as a base signal due to photocurrent surge on which is imposed a large signal due to optical rectification; the change in the direction of the E field detected at different azimuthal angles supports this interpretation as does, possibly, the weak magnetic field dependence. Measurements to date do not distinguish whether the optical rectification is bulk or surface-field induced, although the doping dependence suggests the
4
6
Photon frequency (THz) 8 10 12
14
1.0 TO LO PHONONS
0.8
InAs T = 295 K
0.6 Reflectivity
emitter is low-temperature grown (LT, LTG)-GaAs. The conventional comparison of these two classes of emitter is that the PC offers the better (by about 8 times) signal-tonoise ratio, but the EO has the greater spectral range (to about 30 THz compared to about 3 THz). A third category of targets will be classified as other mechanism (OM). This includes photocurrent emitters of several sorts. The most prominent OM target is p-InAs. Some targets exhibit a combination of EO, PC and OM operation. In EO emitters the fs pulse generates THz radiation due to a transient non-linear polarization of the material. This mechanism is described as optical rectification since the emitted THz radiation follows the envelope of the laser pulse. It is a bulk effect: increasing the thickness of the emitter will lead to greater power but at the cost of loss of phase matching and so bandwidth. In PC emitters the fs pulse generates THz radiation through creating photo-carriers by the above-bandgap illumination. These accelerate in an electrical field typically produced by applying a DC voltage between metallic electrodes. The resulting time-varying dipole emits a THz pulse. Under simplifying assumptions the THz power is directly proportional to the square of both the DC field and the laser power. PC emitters are more efficient than EO emitters at low pump power [11] but the higher pump powers and higher biases required to produce absolutely more power often favors EO emitters as the PC mechanism breaks down. In OM materials the mechanism is neither a pure PC or EO effect but may be a (photo)current surge, involving the separation of electrons and holes by the NIR illumination, followed by radiation from the resulting dipoles. The current surge may originate from (1) the surface electric field arising in the depletion layer or (2) the photo-Dember effect. These are not the only possible OM. Additional suggested mechanisms involve (3) electric-field-induced polarization or the inverse Franz–Keldysh effect [12], (4) magnetoplasma waves [13], and (5) coupled plasmaphonon modes [14]. ZnTe is regarded as the premier EO material below 5 THz but fundamental and higher-order phonon processes confuse the picture at higher frequencies, especially for thick crystals [15]. The choice of EO crystal depends on the wavelength of the fs excitation, as the coherence length is frequency dependent. Emission of 2-THz radiation from ZnTe peaks for laser pumping at 800 nm, while that from GaAs peaks for 1400 nm excitation [16]. LT-GaAs has emerged as being preferred to the more traditional semi-insulating (SI)-GaAs in view of the much shorter recombination times, o1 ps compared with 100 ps. Smaller recombination times lead to greater bandwidth, now extending from 0.3 to 7.5 THz [17]. LTGaAs has a smaller mobility than SI-GaAs leading to lower photocurrent, less photoheating and so less chance of thermal runaway. LT-GaAs produces similar amounts of THz radiation to SI-GaAs under the same pump power but with reduced photocurrent allowing a higher bias to be
619
0.4
0.2
0.0 100
200
300 Photon energy (cm-1)
400
Fig. 1. Room-temperature reflectivity of p-InAs.
500
ARTICLE IN PRESS R.A. Lewis et al. / Physica B 376–377 (2006) 618–621
620
latter. A narrow-gap semiconductor such as InAs has high electron mobility me . If this is coupled with a low hole mobility mh , a large transient current may arise (the Dember effect). Surprisingly, InSb, with me twice that of InAs, emits only 1% of the radiation [20]. This is accounted for by the presence of low-me conduction band valleys in InSb.
3.1. Doping p-InAs has an efficiency which, unexpectedly, depends strongly on the concentration of dopants and, critically, whether these are acceptors or donors [12]. Published and unpublished results indicate a strong dependence of THz power on the doping level, although the nature of the dependence varies between reports. The efficiency of the standard EO emitter ZnTe is also known to depend on the impurities, often crudely judged by the color and opacity of the crystal. We used room-temperature THz reflection spectroscopy of the distinctive reststrahlen feature (Fig. 1) to determine an upper limit for the carrier concentration. We deduce from these data that the sample is lightly doped. There is no evidence of the plasma above 100 cm 1, which corresponds to carrier concentration 6 1016 cm 3. We have sought transitions from the ground
state to the excited states of the Zn acceptor through lowtemperature (1.5 K) THz transmission measurements. So far we have not been able to detect these.
3.2. Geometrical factors In some OM materials geometric considerations, including polarization measurements, establish that the emission mechanism is neither a pure PC nor EO effect [21]. For example, in one report THz emission from the o1 1 14 face of p-InAs dips to 2/5 of the maximum as the sample is rotated about the surface normal suggesting the signal is 60% due to an EO effect. Emission occurs in the absence of an antenna structure on the surface, let alone an applied bias, so is not due to PC. The suggested origin of the remaining 40% of the emission is in a (photo)current surge. Some of our measurements on the geometrical factors affecting emission from p-InAs are shown in Fig. 2. The sample was rotated by azimuthal angle y about the surface normal for angles of incidence 451 and 751. The THz power follows a cos 2y dependence superimposed on a substantial background. This suggests that both photocurrent surge and optical rectification are contributing, with the former dominating.
1.2
1.0
1.4
p-InAs
1.2
angle of incidence
THz power (relative to maximum power)
THz power (relative to average power)
45°
0.8 angle of incidence 45° 75°
0.6
0.4
75° 1.0
0.8
0.6
0.4
0.2
0.2
p-InAs
0.0
0.0
0
100 200 Azimuthal angle (degrees)
300
Fig. 2. Power dependence on azimuthal angle for p-InAs.
0
50 100 150 200 Pump power incident on crystal (mW) Fig. 3. THz power dependence on pump power.
250
ARTICLE IN PRESS R.A. Lewis et al. / Physica B 376–377 (2006) 618–621
3.3. Optical excitation
References
As shown in Fig. 3, the variation of THz emitted power with pump power is close to quadratic, in contrast to o1 1 14 p-InAs [12]. The quadratic dependence we observe is characteristic of both photocurrent surge at low fluences and of optical rectification [22].
[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
3.4. Magnetic field While initial claims as to the enhancement of THz emission due to the application of magnetic field appear to have been overstated there is no doubt that a magnetic field may increase THz emission, if only through a better orientation of the radiating dipole [23]. A magnetic field parallel to the surface can increase THz emission one hundred times in undoped InAs. Such a strong effect has not been observed in lightly doped p-InAs [12]. We intend to explore this in future to determine if a variation due to the Lorentz-force rotation of the dipole, or another cause, is occurring. Acknowledgments This work was supported by the Australian Research Council and the University of Wollongong. We acknowledge the support of Prof. Chao Zhang.
[15] [16] [17] [18] [19] [20] [21] [22] [23]
621
B. Ferguson, X.-C. Zhang, Nat. Mater. 1 (2002) 26. T.J. Johnson, et al., Chem. Phys. Lett. 403 (2005) 152. J. Nishizawa, et al., Optics Commun. 246 (2005) 229. T.-I. Jeon, et al., Curr. Appl. Phys. 5 (2005) 289. V.B. Podobedov, et al., JQSRT 91 (2005) 287. J. Nishizawa, et al., Optics Commun. 244 (2005) 469. D.J. Cook, R.M. Hochstrasser, Opt. Lett. 25 (2000) 1210. T. Loffler, et al., Appl. Phys. Lett. 77 (2000) 453. W.P. Leemans, et al., Phys. Rev. Lett. 91 (2003) 74802. S. Nashima, et al., COMMAD Conference, December 2004. Y.C. Shen, et al., Appl. Phys. Lett. 85 (2004) 164. R. Adomavicius, et al., Appl. Phys. Lett. 85 (2004) 2463. J.N. Heyman, et al., Phys. Rev. B 64 (2001) 085202. M.P. Hasselbeck, et al., Phys. Rev. B 65 (2002) 233203; R. Huber, et al., Phys. Rev. Lett. 94 (2005) 027401. M. Schall, M. Walther, P. Uhd Jepsen, Phys. Rev. B 64 (2001) 094301. A.G. Stepanov, J. Hebling, J. Kuhl, Appl. Phys. Lett. 83 (2003) 3000. Y.C. Shen, et al., Phys. Rev. B 69 (2004) 235325. M.R. Stone, et al., IEEE Trans. Microwave Theory Technol. 52 (2004) 2420. I.S. Gregory, et al., Appl. Phys. Lett. 83 (2003) 4199. P. Gu, et al., J. Appl. Phys. 91 (2002) 5533. M. Reid, et al., Phys. Rev. B 71 (2005) 081306. A. Yariv, P. Yeh, Optical Waves in Crystals: Propagation and Control of Laser Radiation, Wiley, New York, 1984. A. Corchia, et al., Phys. Rev. B 64 (2001) 205204.