2022   04   en   p.50-56 R.J. Kasumova1, Sh.Sh. Amirov1,2,3*,
On the theory of four-wave interaction in dissipative metamaterials


An analytical expression is obtained for the threshold pumping amplitude under phase-matching conditions. It is shown that the threshold pump amplitude increases with increasing losses, the input intensity of the idler wave, and with increasing nonlinear coupling coefficients for pump waves. The conditions under which losses can be compensated in the case of propagation of an backward signal wave under parametric four-wave interaction are analyzed. It is shown that, in the presence of a parametric coupling between forward and backward waves, compensation of signal wave losses by losses of direct waves will allow the parametric amplification and generation of the backward wave to be realized at the threshold pump amplitude.

Keywords: metamaterials, constant intensity approximation, four wave mixing, compensation of losses.
PACS: 78.67.Pt; 42.65-k; 42-70-a.


Received: 28.11.2022


1. Physics department, Baku State University, 23 Z. Khalilov str., Az-1148, Baku, Azerbaijan
2. Department of Medical and Biological Physics, Azerbaijan Medical University, A. Gasimzade str., 14, AZ 1022, Baku, Azerbaijan
3. Department of Physics and Electronics, Khazar University, 41 Mahsati str., Az 1096 Baku, Azerbaijan
Corresponding author e-mail: phys_med@mail.ru

[1]   S. Zhang, W. Fan., N.C. Panoiu, K.J. Malloy, R.M. Osgood, S.R.J. Brueck. Experimental demonstration of near-infrared negative-index metamaterials. Phys. Rev. Lett. 2005. V. 95. P. 137404- 8.
[2]   Xiao Shumin, V.P. Drachev, A.V. Kildishev, Ni Xingjie, U.K. Chettiar, Yuan Hsiao-Kuan, V.M. Shalaev. Loss-free and active optical negative-index metamaterials. Nature. 2010. V. 466. P. 735 - 742.
[3]   V.V. Klimov. Nanoplasmonics. UFN, 178, 875-880 (2008).
[4]   A.K. Popov, V.M. Shalaev. Negative-index metamaterials: second-harmonic generation, Manley-Rowe relations and parametric amplification. Appl. Phys. B. 2006. V. 84, P. 131-137.
[5]   J.F. Reinjes. Nonlinear optical parametric processes in liquids and gases. – Orlando: Wiley, 1984. – 510 p.
[6]   I.V. Shadrivov, A.A. Zharov, Y.S. Kivshar. Second-harmonic generation in nonlinear left-handed metamaterials, JOSA B23, № 3, 529-534 (2006)].
[7]   A.K. Popov, S.A. Myslivets, V.M. Shalaev. Coherent Nonlinear-optical Energy Transfer and Backward-wave Optical Parametric Generation in Negative-index Metamaterials. Physica B, 405, 2999 (2010).
[8]   A.K. Popov, S.A. Myslivets, T.F. Georg, V.M. Shalaev. V.M. Four –wave mixing, quantum control and compensating losses in doped negative-index photonic metamaterials. Optics Letters. 2007. V. 32, P. 3044 - 3046.
[9]   S. Palomba, Sh. Zhang, Y. Park, G. Bartal, X. Yin, X. Zhang. Optical negative refraction by four-wave mixing in thin metallic nanostructures. Nature materials. 2012. V. 11. P. 34 - 38.
[10]  Z.H. Tagiev and A.S. Chirkin. “Fixed intensity approximation in the theory of nonlinear waves,” Zh. Eksp. Teor. Fiz. 73, 1271–1282 1977.
[11]  Z.H. Tagiev, R.J. Kasumova, R.A. Salmanova, N.V. Kerimova. Constant- intensity approximation in a nonlinear wave theory. J. Opt. B: Quantum Semiclas. Opt. 2001. V. 3 – P. 84 -87.
[12]  V.G. Dmitriev, and L.V. Tarasov. Prikladnaya Nelineynaya Optika [Applied Nonlinear Optics] (Fizmatlit, Moscow, 2004).
[13]  Z.H. Tagıev, R.J. Kasumova, L.S. Gadjieva. Phase effects at second-harmonic generation in metamaterials. Journal of Russian Laser Research, 2011, v. 32, № 2, March, p.188-199.
[14]  R.J. Kasumova, G.A. Safarova, Sh.A. Shamilova, N.V. Kerimova. International Journal of Engineering and Technology IJET-IJENS, 15, № 06, 19, 2015.
[15]  R.J. Kasumova, Sh.Sh. Amirov, Sh.A. Shamilova. Parametric interaction of optical waves in metamaterials under low-frequency pumping. Quantum Electronics. 2017. V. 47(7). P. 655-660.
[16]  R.J. Kasumova, Z.H. Tagıev, Sh.Sh. Amirov, Sh.A. Shamilova, G.A. Safarova. Phase effects at parametrical interaction in metamaterials. Journal of Russian Laser Research. 2017. V. 38. P. 211 – 218.
[17]  Z.H. Tagiev, R.J. Kasumova, R.A. Salmanova. Theory of degenerate four-wave mixing in the prescribed intensity approximation. Optics and spectroscopy. – 1999. –V. 87(1). - P. 86 – 89.
[18]  V.G. Veselago. The electrodynamics of substances with simultaneously negative values of ε and μ. Sov. Phys. Usp., 1968. V. 10. P. 509 – 514.
[19]  D.R. Smith, W.J. Padilla, D.C. Vier, S.C. Nemat Nasser, S. Schultz. Composite medium with simultaneously negative permeability and permittivity. Phys. Rev. Lett. 2000. V. 84. P. 4184 - 7.
[20]  S.A. Akhmanov, V.A. Visloukh, A.S. Chirkin. UFN,1986, v.149, issue 3, p. 449-509.