NONLINEAR MAGNETO-OPTICAL EFFECTS IN DIELECTRICS EMBEDDED WITH FERROMAGNETIC NANOPARTICLES

Transcription

1 Development, 5-8 July, 9, Abuja, Nigeria NONLINEAR MAGNETO-OPTICAL EFFECTS IN DIELECTRICS EMBEDDED WITH FERROMAGNETIC NANOPARTICLES Arlene.P. Maclin, Professor of Engineering & Director, Center for Academic Excellence Norfolk State University, Norfolk, Virginia & M. M. Noel,Assistant Professor of Engineering, Norfolk State University, Norfolk, Virginia 354; Abstract Magneto-optic effects in transparent dielectric materials embedded with ferromagnetic nanoparticles have been investigated through simulation of a nonlinear wave equation. The possibility of generation of harmonics due to magnetic saturation in ferromagnetic nanoparticles was studied. A simplified nonlinear spring-mass system model that accounts for magnetic saturation and harmonic generation is presented. The simplified model is analyzed using the finite element and finite difference methods and results are compared with data from simulation studies.. INTRODUCTION Nonlinear optical effects occur due to the nonlinearity of constitutive relationships [] in Maxwell s equations: D ( E) E and B ( H) H Nonlinear effects due to nonlinearity of the constitutive relation for the electric field have been demonstrated through harmonic generation experiments. In particular Franken et al [] demonstrated the generation of ultraviolet light by passing a African Journal of Physics Vol., pp.49-58, (9) ISSN: PRINT: CD ROM: ONLINE:

2 Development, 5-8 July, 9, Abuja, Nigeria ruby laser beam through a quartz crystal. However due to difficulties in manufacturing high frequency ferromagnetic materials nonlinear optical effects occurring due to the nonlinearity of the magnetic field, constitutive relationship have not been extensively studied. Recently the nonlinearity of ferromagnetic resonances was used to generate second and higher harmonics in the microwave region of the electromagnetic spectrum [3]. With the maturation of nanotechnology it has now become feasible to manufacture high frequency ferromagnetic materials. This paper considers the generation of second and higher harmonics due to nonlinear magnetic effects resulting from embedding ferromagnetic nanoparticles in a transparent dielectric matrix.. RESULTS The Maxwell s equations are for electromagnetic wave propagation in linear media. However the linear wave equations are only valid at low field values. At high field values, the relative permeability is not constant but decreases monotonically until the free space value is reached. The decrease in relative permeability with increasing magnetic field value is due to the alignment of magnetic domains. When all magnetic domains have been aligned the permeability cannot increase any further resulting in a constant permeability for very large field values. This effect is modeled by the nonlinear wave equation (). The nonlinear wave equation can be numerically solved using a finite difference approximation scheme. First the partial derivatives are approximated by finite difference approximations. These approximations are then substituted in the nonlinear wave equation resulting in a difference equation that expresses the field at any time t and location (x,y) in terms of the field values at previous times (4). B B t () r a b h cb r sec ( ) B ( a bsec h ( cb)) () 5 B t

3 Development, 5-8 July, 9, Abuja, Nigeria Consider u u u c( u) ( ) (3) t x y u u( x x, y, t) u( x, y, t) u( x x, y, t) x x u u( x, y y, t) u( x, y, t) u( x, y y, t) y y u u( x, y, t t) u( x, y, t) u( x, y, t t) t t ux, y, t cd ( ux, y, t ux, y, t ux, y, t ux, y, t 4 ux, y, t ) ux, y, t ux, y, t (4) ( cd for convergence) The results of the finite difference approximation simulation are shown below. 5

4 Development, 5-8 July, 9, Abuja, Nigeria Time= Color: u Height: u Figure : shows a normal mode for the linear wave equation. The solution was obtained by finite element method with triangular elements over a rectangular domain Figure : shows a contour plot 5 of linear mode shown in figure

5 Development, 5-8 July, 9, Abuja, Nigeria Figure 3: Shows variation of the magnetic field with position after iterations fothe the nonlinear wave equation. The solution was obtained using a finite difference approximation scheme. In order to obtain an analytic solution for model validation, we model the nonlinear saturation behavior in terms of a nonlinear mass-spring system model for the electron response. m x x kx q E cos( t) (5) e e m Equation (5) can be solved using a singular perturbation method approximation. To this end, we assume the electron displacement x(t) due to the electromagnetic wave to be of the form: x( t) x ( t) x ( t) x ( t) x ( t)... (6) 3 3 Equation (6) can be substituted in equation (5) and coefficients of the parameter are equated to obtain a sequence of approximations. shown below in Figures 4. and 5. 53

6 velocity velocity Proceedings of the Second International Seminar on Theoretical Physics & National Development, 5-8 July, 9, Abuja, Nigeria position Fig. 4: Phase space plot for linear response position 5 x -3 Fig. 5: Phase space plot 54 for nonlinear response.

7 Development, 5-8 July, 9, Abuja, Nigeria 3. CONCLUSION Based on these preliminary results, we believe that embedding ferromagnetic nanoparticles in a dielectric medium can produce nonlinear magnetic effects due to hysteresis and saturation effects. Nonlinear magnetic effects induced by addition of ferromagnetic nanoparticles can be used for harmonic generation. Due to the inverse relationship between wave velocity and permeability, the refractive index can be increased due to the addition of ferromagnetic nanoparticles [4-7]. Future experiments are needed to investigate the effects the addition of ferromagnetic nanoparticles can make on optical properties in dielectric media. ACKNOWLEDGEMENT Several technical discussions about this work were held with Dr. Vitaliy Lomakin from the University of San Diego These discussions are gratefully acknowledged. This research was supported by the NSF funded Engineering Research Center for Integrated Access Networks ( 87) with the University of Arizona as the lead institution. Dr. Maclin gratefully acknowledges the invitation by the organizers of the ISOTPAND9 to participate in this very important meeting for the dissemination of these research results to scientists and engineers in Nigeria. It was a real pleasure to meet so many emerging African scientists interested in pursuing graduate work in physics and engineering REFERENCES [] Y. R. Shen, Principles of nonlinear optics, New York, Wiley-Interscience, 984. [] P. A. Franken, A. E. Hill, C. W. Peters, and G. Weinreich, "Generation of optical harmonnics," Phys. Rev. Lett. 7, 8-9,

8 Development, 5-8 July, 9, Abuja, Nigeria [3] G. P. Rodrigue, "A generation of microwave ferrite devices," Proceedings of the IEEE, Volume 76, Issue, Feb 988 Page(s): [4] C A F Vaz et al, "Ferromagnetic nanorings," in the Journal of Condensed Matter Physics, 7. [5] E. M. Brunsman et al, "Magnetic properties of carbon-coated, ferromagnetic nanoparticles produced by a carbon-arc method," in the Journal of Applied Physics, May 994. [6] L. Berger et al, "Ferromagnetic nanoparticles with strong surface anisotropy: Spin structures and magnetization processes," in Physics Review, Volume 77, March 8. [7] J. Nogues, "Exchange bias in ferromagnetic nanoparticles embedded in an antiferromagnetic matrix," in the International Journal of Nanotechnology, Volume, April 5. 56