Magneto-optical Effects in Transparent Media under Inhomogeneous Magnetization

The analysis of the Magneto-Optical (M. 0.) effects when the magnetization is inhomogeneous is separated in two cases : first, when no diffraction occurs inside the medium, the geometrical optics approximation is used to derive the phase of the optical fields and the ray paths. Second, with no negligible diffraction inside themedium, extensions of the previous theory are presented and applied to the particular case of diffraction by dipolar spinwaves. Experimental results are given and discussed for this last case. In most existing theories of the Magneto Optical (M. 0 . ) effects a spatially uniform magnetization is assumed. But this assumption may be removed for inhomogeneous magnetization, such as in media containing domains or propagating spin waves. Two cases may be distinguished from the analytical treatment : Negligible diffraction inside and not negligible diffraction. I. Negligible diffraction inside the medium. With negligible diffraction within the medium the geometrical optics approximation may be used to find the propagation characteristics. From the general theory of propagation in inhomogeneous media [I], we know that the light propagates with progressive refraction effect and that the phase of the optical field is no more a linear function of the run distance. This phase, say $, is now called the cr Eikonal )> and is determined by solving : (W)' = k i I n(r, t) 12 (1) where ko is the light wavenumber in the vacuum and n(r, t ) the (( local )) index of refraction, which may be deduced from the usual theory of M. 0. effects. Assuming for instance that only the linear M. 0. effects are relevant, gives easily : I (n&(r, t ) 1' = I ~r f AM],(r, t, 1' (2> where the + or sign refers to the positive or negative circular polarisation. E, is the relative permittivity, A the M. 0. constant, and Mll(r, t ) the magnetization component parallel to the light propagation. Solving eq. (I) for a beam passing through a magnetic slab for instance gives the phase configuration at the output, from which the diffraction pa.ttern may be inferred as a usual phase grating. For the other effect, that is the progressive refraction, the ray paths are determined from Fermat's principle or, equivalently, the lagrange's equations. The lagrangian may be written : where ds is an arc element of the ray path, u a parameter. Within the same assumptions that above, the ray paths may be computed. But, taking the possible curvature of the ray path into account, MI, must be now written : In figure 1 is presented a numerical result of this calculation assuming for M(r, t) the following variaFIG. 1. -Light Ray paths in a M. 0. material for various periodic inhomogeneities. K = 6.28 cm-1 for curve 1 = 62.8 cm-1 for curve 2 = 6.28.10+2 cm-1 for curve 3 ($1 This work is supported by La Direction des Recherches We have taken the following values for the paraet Moyens d'Essais, Paris, France. (**) This work is a part of a work thesis registered under meter : no A 04 850, CNRS, France. AMo = .67 ; E, = 2.2 $I = 4 4 . Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1971138 C 1 120 BY B. DESORMI~RE AND H. LE GALL This value of AM, is lo3 times that of the YIG at 1.15 pm. Using the true value of YIG for this constant gives a negligible refraction. 11. Diffraction inside the medium not negligible. The theory of Bragg diffraction by pure spinwaves was first reported by Auld and Wilson [2] ; Smith [3] has studied the Bragg diffraction by magnetoelastic waves ; diffraction by pure spinwaves was observed by Collins and Wilson [4]. We have extended the previous theory by including the following phenomena : the existence of an optical magnetization, [5] the influence of the quadratic M. 0. effects, [6] the influence of a more complicated variation for the magnetization than that associated with a plane spinwave. The calculations have been exte~lsively achieved for the particular case of diffraction by dipolar spinwaves propagating in a YIG bar when a turning point exists inside the bar. The chief theoretical results may be summarized as follows : The quadratic M. 0. effects associated with the static magnetization leads to different values for the incidence and diffraction angles under the phase matching condition. The observation of only one order of diffraction is tentatively attributed to the same quadratic M. 0. effects involving the dynamic component of the magnetization perpendicular to the light propagation. A large modification in the diffraction pattern was found when a large field gradient exists near the turning point, as evidenced in the figure 2. The RamanFIG. 2. Computed diffracted field vs. diffraction angle for various field gradients at the turning point. Normal incidence. Nath conditions are then more appropriated than the Bragg conditions to interpret the actual results. The guided character of the dipolar waves results in an off-axis propagation for the phase-vector of the spinwave, which may enhance or compensate the dissymetry in the light propagation directions produced by the quadratic M. 0. effects. The diffraction pattern may be easily found for each particular dipolar mode. Experiments were performed with a [loo] YIG bar using a 1.15 pm CW laser. Rf. power effects were observed as shown in figure 3. Along with this variaDiffraction Ang le 2 1 (degree) FIG. 3. Variation of the diffraction angle and the diffracted intensity vs. the rf-level. F = 2 GHz. tion, the applied field for the maximum diffraction was also found to decrease with an increasing rf. level. Experiments performed with a pulse modulation have revealed that peak power effects remain for the range of levels when the variation of the scattered intensity is nonlinear. Thus, the dipolar spinwave saturates from the high K-region at a relativeIy law oower level. In figure 4 is presented a comparison between the rf. absorption and the diffracted intensity versus the FIG. 4. Comparison between the rf-absorption and the diffracted intensity. applied field. The diffracted intensity was optimised for each field by moving the laser beam along the YIG bar. This result makes also in evidence the relation between the diffracted signal and the large dipolar resonances [7]. But the diffraction allows the position of these resonances to be measured : the numbers MAGNETO-OPTICAL EFFECTS IN TRANSPARENT MEDIA UNDER INHOMOGENEOUS C 1 121 Di f f rac ted i n t ens i ty under 'the arrows give the laser beam position from the nearest end face where the maximum diffraction occurs(in mm). In figure 5 are given the results of the diffracted intensity variation when the laser beam is moved along the heigth of the YIG bar, using a square loop for the spinwave excitation. The curve 1 was obtained with the loop assumed to be centered. For the curve 2 the YIG bar was displaced upward for about. 1 mm. This result shows the great sensitivity of the spinwave ' VIG profile with respect to the centering of the excitation loop. , . Acknowledgments. R. Pauchard is greatly acknowledged for carrying out the numerical calculations ; the authors are greatly indebted to G. Voluet for his : i t \ technical assistance. Prof. Pauthenet is acknowledged for his constant interest to this work. FIG. 5. Diffracted intensity vs. the laser beam position Many helpful1 discussions, with Prof. Auld, Dr Addiwith a vertical displacement. son and Wilson during a visit the Microwave Research Curve 1 : with loop assumed to be centered. Laboratory of Stanford University are greatly acknowCurve 2 : with the YIG bar up-displaced by about. 1 mrn. ledged.