It is theoretically investigated the generation of higher harmonics of two-dimensional and three-dimensional terahertz electromagnetic beams in nonlinear crystals. The attention is paid to crystalline paraelectrics like SrTiO3 under the temperatures 60 - 200 K, these crystals possess the cubic nonlinearity. The bias electric field is applied to provide the dominating quadratic nonlinearity. The initial focusing of the beams not only increases the efficiency of generation of higher harmonics, but alto makes possible to select maxima of different higher harmonics at some distances from the input. At lower temperatures the nonlinearity behaves at smaller input amplitudes, whereas at higher temperatures the harmonic generation can be observed at higher frequencies up to 1.5 THz. In three-dimensional beams the peak amplitudes of higher harmonics can be bigger than in two-dimensional beams, but the ratios of these peak values to the maximum values of the focused first harmonic are smaller than in two-dimensional beams.
References
[1]
Biswas, A., Banerjee, A., Acharyya, A., Inokawa, H. and Roy, J.N. (2020) Emerging Trends in Terahertz Solid-State Physics and Devices: Sources, Detectors, Advanced Materials, and Light-Matter Interactions. Springer, New York.
https://doi.org/10.1007/978-981-15-3235-1
[2]
Carpintero, G., Garcıa Muñoz, L.E., Hartnagel, H.L., Preu, S. and Räisänen, A.V. (2015) Semiconductor Terahertz Technology Devices and Systems at Room Temperature Operation. Wiley, New York. https://doi.org/10.1002/9781118920411
[3]
Bründermann, E., Hübers, H.-W. and Kimmitt, M.F. (2012) Terahertz Techniques. Springer, New York. https://doi.org/10.1007/978-3-642-02592-1
[4]
Song, H.-J. and Nagatsuma, T. (2015) Handbook of Terahertz Technologies. Devices and Applications. CRC Press, Taylor & Francis Group, Boca Raton.
[5]
Rieh, J.-S. (2021) Introduction to Terahertz Electronics. Springer, New York.
https://doi.org/10.1007/978-3-030-51842-4
[6]
Choudhury, B., Rakesh, A.M. and Jha, M. (2016) Active Terahertz Metamaterial for Biomedical Applications. Springer, New York.
https://doi.org/10.1007/978-981-287-793-2
[7]
Ganichev, S.D. and Prettl, W. (2006) Intense Terahertz Excitation of Semiconductors. Oxford University Press, Oxford.
https://doi.org/10.1093/acprof:oso/9780198528302.001.0001
[8]
Kolejáka, P., Postava, K., Miáica, M., Kužel, P., Kadlec, F. and Pištora, J. (2018) Experimental Gouy Phase Shift Compensation in Terahertz Time-Domain Spectroscopy. Photonics and Nanostructures—Fundamentals and Applications, 31, 129-133.
https://doi.org/10.1016/j.photonics.2018.06.011
O’Sullivan, C. and Murphy, J.A. (2012) Field Guide to Terahertz Sources, Detectors, and Optics. SPIE Press, Bellingham. https://doi.org/10.1117/3.952851
[11]
Perenzoni, M. and Paul, D.J. (2014) Physics and Applications of Terahertz Radiation. Springer, New York. https://doi.org/10.1007/978-94-007-3837-9
[12]
Siegel, P.H. (2002) Terahertz Technology. IEEE Transactions on Microwave Theory and Techniques, 50, 910-928. https://doi.org/10.1109/22.989974
[13]
Mittleman, D. (2003) Sensing with Terahertz Radiation. Springer, New York.
https://doi.org/10.1007/978-3-540-45601-8
[14]
Atakaramians, S., Shahraam Afshar, V., Monro, T.M. and Abbott, D. (2013) Terahertz Dielectric Waveguides. Advances in Optics and Photonics, 5, 169-215.
https://doi.org/10.1364/AOP.5.000169
[15]
Woolard, D.L., Loerop, W.R. and Shur, M.S. (2003) Terahertz Sensing Technology, 2 Vols. World Scientific Publ., Singapore. https://doi.org/10.1142/5244
[16]
Lee, Y.-S. (2009) Principles of Terahertz Science and Technology. Springer, New York.
[17]
Robertson, W.M. (1995) Optoelectronic Techniques for Microwave and Millimeter-Wave Engineering. Artech Publ., Norwood.
[18]
Gevorgian, S. (2009) Ferroelectrics in Microwave Devices, Circuits and Systems. Springer, New York. https://doi.org/10.1007/978-1-84882-507-9
[19]
Vendik, O.G. (1979) Ferroelectrics in Microwave Technology. Sov. Radio, Moscow. (In Russian)
[20]
Vendik, O.G. and Zubko, S.P. (1997) Phenomenological Description of the Permittivity of Strontium Titanate as a Function of Applied Electric Field and Temperature. Technical Physics, 42, 278-281. https://doi.org/10.1134/1.1258678
[21]
Blinc, R. and Žekš, B. (1974) Soft Modes in Ferroelectrics and Antiferroelectrics. North-Holland, Amsterdam.
[22]
Lines, M.E. and Glass, A.M. (1977) Principles and Applications of Ferroelectric and Related Materials. Clarendon Press, Oxford.
[23]
Rabe, K.M., Ahn, C.H. and Triscone, J.-M. (2007) Physics of Ferroelectrics. A Modern Perspective. Springer, New York.
[24]
Strukov, B.A. and Levanyuk, A.P. (1998) Ferroelectric Phenomena in Crystals. Springer, New York. https://doi.org/10.1007/978-3-642-60293-1
[25]
Maugin, G.A. (1988) Continuum Mechanics of Electromagnetic Solids. North-Holland, Amsterdam.
Rez, I.S. and Poplavko, Yu.M. (1989) Dielectrics. Basic Properties and Applications in Electronics. Radio and Svyaz’, Moscow. (In Russian)
[28]
Poplavko, Yu.M., Pereverzeva, L.P., Voronov, S.O. and Yakimenko, Yu.I. (2007) Physical Material Science. Vol. 2. Dielectrics. KPI Publ., Kiev. (In Ukrainian)
[29]
Gassanov, L.G., Koshevaya, S.V., Narytnik, T.N. and Omel’yanenko, M.Yu. (1978) Parametric and Nonlinear Interaction of Electromagnetic Waves in Paraelectrics. Izv VUZ Radioelektronika, 21, 56-63. (English Transl. Radioelectronics and Communications Systems)
[30]
Gassanov, L.G., Koshevaya, S.V. and Omel’yanenko, M.Yu. (1980) On Frequency Multiplication in Paraelectrics. Radiotekhnika i Elektronika, 25, 1238-1243. (English Transl. Radio Engineering and Electronic Physics)
[31]
Koshevaya, S.V., Kononov, M.V. and Omel’yanenko, M.Yu. (1985) An Influence of Dispersion in Waveguiding Systems with Cubically Nonlinear Dielectrics. Izv VUZ Radioelektronika, 28, 53-56. (English Transl. Radioelectronics and Communications Systems)
[32]
Koshevaya, S.V., Grimalsky, V.V., Kotsarenko, Y.N. and Tecpoyotl-T, M. (2016) Modulation Instability of Transversely Limited Electromagnetic Waves of Terahertz Range in Strontium Titanate Paraelectric. Radioelectronics and Communications Systems, 59, 489-495. https://doi.org/10.3103/S0735272716110029
[33]
Grimalsky, V., Koshevaya, S., Escobedo-Alatorre, J. and Tecpoyotl-Torres, M. (2016) Nonlinear Terahertz Electromagnetic Waves in SrTiO3 Crystals under Focusing. Journal of Electromagnetic Analysis and Applications (JEMAA), 8, 226-239.
https://doi.org/10.4236/jemaa.2016.810021
[34]
Grimalsky, V., Koshevaya, S., Escobedo-Alatorre, J. and Jatirian-Foltides, E. (2017) Formation of Short Terahertz Electromagnetic Pulses in Nonlinear Paraelectrics. Proceedings 30th International Conference on Microelectronics MIEL-2017, Nis, 9-11 October 2017, 91-94. https://doi.org/10.1109/MIEL.2017.8190076
[35]
Grimalsky, V., Koshevaya, S., Escobedo-Alatorre, J. and Jatirian-Foltides, E. (2019) Stimulated Brillouin Scattering of Terahertz Electromagnetic Pulses in Paraelectrics. Applied Physics B, 125, 15-22. https://doi.org/10.1007/s00340-018-7125-4
[36]
Grimalsky, V.V., Rapoport, Y.G., Boardman, A.D. and Koshevaya, S.V. (2018) Nonlinear Focusing of Picosecond Baseband Pulses in Paraelectric Crystals. Optical and Quantum Electronics, 50, 102-114. https://doi.org/10.1007/s11082-018-1369-4
[37]
Grimalsky, V.V., Rapoport, Yu.G., Koshevaya, S.V., Escobedo-Alatorre, J. and Tecpoyotl-Torres, M. (2021) Nonlinear Focusing of Picosecond Baseband Pulses in Paraelectric Crystals in a Wide Temperature Range. Optical and Quantum Electronics, 53, Paper 484. https://doi.org/10.1007/s11082-021-03104-6
[38]
Rapoport, Yu.G., Grimalsky, V.V., Koshevaya, S.V. and Melendez-Isidoro, D.L. (2015) Modulation Instability of Terahertz Electromagnetic Pulses in SrTiO3 Paraelectric. Proceedings 2015 IEEE 35th International Conference on Electronics and Nanotechnology (ELNANO), Kyiv, 21-23 April 2015, 131-134.
https://doi.org/10.1109/ELNANO.2015.7146851
[39]
Zamudio-Lara, A., Koshevaya, S.V., Grimalsky, V.V. and Ya?ez-Cortes, F. (2015) Frequency Multiplication of Terahertz Radiation in the Crystals of Strontium Titanate Paraelectric. Radioelectronics and Communications Systems (Izvestiya VUZ, Radioelektronika), 58, 411-416. https://doi.org/10.3103/S0735272715090034
[40]
Kozina, M., Fechner, M., Marsik, P., et al. (2019) Terahertz-Driven Phonon Upconversion in SrTiO3. Nature Physics, 15, 387-392.
https://doi.org/10.1038/s41567-018-0408-1
[41]
Kamarás, K., Barth, K.L., Keilmann, F., Henn, R., Reedyk, M., Thomsen, C., Cardona, M., Kircher, J., Richards, P.L. and Stehle, J.L. (1995) The Low Temperature Infrared Optical Functions of SrTiO3 Determined by Refectance Spectroscopy and Spectroscopic Ellipsometry. Journal of Applied Physics, 78, 1235-1240.
https://doi.org/10.1063/1.360364
[42]
Yashchyshyn, Y., Godziszewski, K., Bajurko, P., Modelski, J., Szafran, M., Bobryk, E., Pawlikowska, E., Tarapata, G., Weremczuk, J. and Jachowicz, R. (2013) Tunable Ferroelectric Ceramic-Polymer Composites for Sub-THz Applications. Proceedings 43rd European Microwave Conference, Nuremberg, 7-10 October, 2013, 676-679.
https://ieeexplore.ieee.org/document/6686746
https://doi.org/10.1109/APMC.2013.6695095
[43]
Bloembergen, N. (1965) Nonlinear Optics. W. A. Benjamin Inc., New York.
[44]
Kivshar, Y.S. and Agrawal, G.P. (2003) Optical Solitons. From Fibers to Photonic Crystals. Academic Press, New York.
https://doi.org/10.1016/B978-012410590-4/50012-7
[45]
Weiland, J. and Wilhelmsson, H. (1977) Coherent Non-Linear Interaction of Waves in Plasmas. Pergamon Press, London.
https://doi.org/10.1007/978-1-4757-1571-2_29
[46]
Samarskii, A.A. (2001) The Theory of Difference Schemes. Marcel Dekker Inc., New York. https://doi.org/10.1201/9780203908518
