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JOURNAL OF APPLIED PHYSICS 105, 07E709 共2009兲 Differential destructive interference of the circular polarization eigenmodes of scattered soft x rays at the grazing incidence in magnetic thin films Dae-Eun Jeong and Sang-Koog Kima兲 Research Center for Spin Dynamics & Spin-Wave Devices and Nanospinics Laboratory, Department of Materials Science and Engineering, Seoul National University, Seoul 151-744, Republic of Korea 共Presented 12 November 2008; received 12 September 2008; accepted 20 November 2008; published online 11 March 2009兲 Recently, the authors found that an additional magneto-optical effect that linearly polarized soft x rays incident on a single magnetic layer on a nonmagnetic substrate can be converted to any states among the linearly s- and p- and circularly left- and right-handed polarizations by changing the grazing angle of incidence in specular reflection geometry. In this article, the authors report that the physical origin of such an effect is the differential interference of the circular polarization eigenmodes of scattered soft x rays at the grazing incidence. Totally destructive interference takes place selectively for one helicity but not for the other one at a certain grazing angle and in a specific energy region just below the absorption edges, thus leading to differential circular reflectivity. Numerical calculations using an iterative method of transmission, reflection, and propagation matrices allow us not only to verify the underlying mechanism but also to find the necessary specific conditions of photon energy and incidence angle where such a phenomenon can occur. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3072080兴 A variety of magneto-optical 共MO兲 phenomena allow us to sensitively probe magnetic properties in magnetic materials as well as to manipulate the polarization state of soft or hard x rays that interact with magnetic moments.1–6 In particular, at the absorption edges or near-edges of incident soft x rays, the MO effects are much enhanced compared with those in the visible light range or in the nonresonant region, owing to electronic transitions from spin-orbit split-occupied states to spin-polarized empty states.7 In principle, the MO effects can be classified into two main effects: circular birefringence and circular dichroism.8 It is well known that the former leads to the rotation of a polarization state of beams propagating through or reflecting from magnetized materials, and that the latter leads to the ellipticity of the polarization state.9 The circular birefringence and dichroism can also be described macroscopically by the differential real ␦R,L and imaginary R,L parts of the refractive indices nR,L = 1 − ␦R,L + iR,L, respectively, for the left 共L兲- and right 共R兲-handed circular polarization 共LCP and RCP兲 modes of the beam. At the absorption L3 and L2 edges of 3d transition metals, the circular-mode dependent refractive indices become deviated from the unit value and largely differ in magnitude near the absorption edges of a given material, that is, the differences ⌬␦ = ␦R − ␦L and ⌬ = R − L vary dramatically with photon energy h across the edges.10,11 Recently we found an additional MO phenomenon related to both effects that allows us to produce any polarization states among the linear s- and p-polarization, and LCP and RCP modes, as well as any elliptical polarizations in reflection geometry, using polarizing optical elements of a a兲 Author to whom correspondence should be addressed. Electronic mail: [email protected]. 0021-8979/2009/105共7兲/07E709/3/$25.00 single magnetic thin film.6 The underlying physics of that phenomenon is still unknown, although its understanding is crucially important to polarizer 共or analyzer兲 implementation, enabling the simple and low-cost optical production 共or determination兲 of any of the variable polarization states of soft x rays. In this article, we report the underlying mechanism of this additional MO effect, the origin of which is differential circular interference 共or asymmetric circular interference兲 between the RCP and LCP modes at specific optical conditions, as confirmed by numerical calculations of the amplitude and phase of the electric field of each of the directly and multiply scattered soft x rays, using an iterative matrix method in the framework of the orthogonal RCP-andLCP-mode basis. We also reveal the necessary criteria for asymmetric circular interference. To verify the physical origin of the observed differential circular reflectivity, we consider a case of two interfaces, for which a magnetic thin film 共B兲 is sandwiched between vacuum 共A兲 and a nonmagnetic substrate 共C兲, as shown in Fig. 1. When a soft x-ray beam of an arbitrary polarization is incident on the magnetic medium, the RCP 共red rays兲 and LCP 共blue rays兲 components of the beam are partially reflected from the top 共A/B兲 interface, partially refracted differently through the top interface, and then separately propagated at a different propagation direction and magnitude. The paths of the individual RCP and LCP components thus diverge when they propagate in the magnetic layer. The RCP and LCP components also experience multiple reflections from both the top and the bottom interfaces of the magnetic layer. Thus, the phase difference between a soft x-ray beam r,D 共Eជ aR共L兲 兲 reflected directly from the top interface and that r,M ជ 共E 兲 multiply reflected inside the magnetic medium can aR共L兲 be as large as out-of-phase at a specific grazing angle of 105, 07E709-1 © 2009 American Institute of Physics Downloaded 19 Mar 2009 to 147.46.233.61. Redistribution subject to AIP license or copyright; see http://jap.aip.org/jap/copyright.jsp 07E709-2 D.-E. Jeong and S.-K. Kim FIG. 1. 共Color online兲 Schematic illustration of the beam paths of the RCP and LCD components of beams directly reflected from the top 共A/B兲 interface, and waves multiply reflected at the top and bottom interfaces inside the magnetic medium 共b兲 at the grazing incidence in a vacuum 共a兲/magnetic ជ r,D共M兲 and r,D共M兲 represent material 共b兲/nonmagnetic substrate 共c兲 system. E ab ab the amplitude and phase, respectively, of the electric field of the reflected beams. The superscripts D and M indicate the beams directly reflected from the top interface and ones encountering multiple reflections inside the magnetic layer, respectively. The subscripts a and b indicate the polarization components of the incident and reflected beams, respectively, where b is the RCP or LCP component. In the magnetic layer, the magnetization is longitudinally oriented along either direction of my = ⫾ 1, as indicated by the arrows. incidence only for the longitudinal magnetizations, due to a ជ r,D significant circular birefringence. Finally, the beams of E aR共L兲 ជ r,M are superposed at the exit of the top interface. and E J. Appl. Phys. 105, 07E709 共2009兲 FIG. 2. 共Color online兲 共a兲 Numerical values of ␦R,L and R,L versus h across the Co L3 and L2 edges, measured and determined from a Co film 共Ref. 12兲. The calculations of ⌬rsR in 共b兲 and ⌬rsL in 共c兲 versus both and h for the case of my = + 1 in a model system of vacuum/Co共9 nm兲/Sisubstrate. The thick black-colored curves are drawn according to the positions of and h where ⌬rsR共L兲 = 0. aR共L兲 ជ r,D + Eជ r,M for the RCP component Accordingly, the sum of E aR aR ជ r,D + Eជ r,M for the LCP one yields a totally destructive or of E aL aL interference depending on the difference in the phase beជ r,D and Eជ r,M . tween comparable-amplitude of E aR共L兲 aR共L兲 In order that the totally destructive interference effect r,D ជ r,M 兩 and 兩r,D − r,M 兩 ⬇ con兩 ⬇ 兩E can occur, the 兩Eជ aR共L兲 aR共L兲 aR共L兲 aR共L兲 ditions must be simultaneously satisfied because together r ជ r,D + Eជ r,M ⬇ 0, that is, complete de=E they satisfy Eជ aR共L兲 aR共L兲 aR共L兲 structive interference of the RCP 共LCP兲 component. In most r,D ជ r,M 兩 due to 兩 is much weaker than 兩E energy regions, 兩Eជ aR共L兲 aR共L兲 significant absorptions during their propagation inside the magnetic medium near the absorption edges. However, in a certain energy range just below the absorption edges, the r,D ជ r,M 兩 condition can be allowed. This condition is 兩 ⬇ 兩E 兩Eជ aR共L兲 aR共L兲 top共bottom兲 top bottom ⬍ raR共L兲 , where rab is fulfilled only in the case of raR共L兲 the amplitude reflection coefficient at the top 共bottom兲 interface, and a共b兲 is the polarization state of incident 共reflected兲 top bottom ⬍ raR共L兲 yields larger reflecbeams. Only the case of raR共L兲 tivities at the bottom interface than at the top interface for beams propagating inside the magnetic medium; hence, multiple reflections inside the magnetic layer can lead to the ជ r,D 兩 ⬇ 兩Eជ r,M 兩 condition. 兩E aR共L兲 aR共L兲 top bottom Next, in order to find the raR共L兲 ⬍ raR共L兲 condition, we bottom top numerically calculated the values of ⌬rsR = rsR − rsR and bottom top ⌬rsL = rsL − rsL for a vacuum/Co共9 nm兲/Si-substrate model system, using the numerical Co values ␦R,L and R,L 关Fig. 2共a兲兴,12 as shown in Figs. 2共b兲 and 2共c兲. The common energy regions, where both ⌬rsR ⬎ 0 and ⌬rsL ⬎ 0 are fulfilled, are close to 0 ⬃ 13 eV below the L3 edge or 0 ⬃ 4 eV below the L2 edge, and are similar to the pre-edge regions just below the resonant edges where ␦R,L ⬍ 0, marked by the gray color in Fig. 2共a兲.10–13 In fact, ⌬rsR共L兲 is a function of both h and , thus the ⌬rsR共L兲 ⬎ 0 condition varies with as well as h, as noted by the thick black curves in Figs. 2共b兲 and 2共c兲. That ជ r,D 兩 ⬇ 兩Eជ r,M 兩 condition is fulfilled only in the speis, the 兩E aR共L兲 aR共L兲 cific ranges of h and that satisfies ⌬rsR共L兲 ⬎ 0. The other condition required for destructive interference r,D ជ r,M , that is an out-of-phase difference between Eជ aR共L兲 and E aR共L兲 r,D r,M is, 兩aR共L兲 − aR共L兲 兩 = . This condition should also be selective for either the RCP or LCP component, in order to provide a large asymmetric intensity. Since the phase difference r,D r,M − aR共L兲 兩 varies with and this phase difference suffi兩aR共L兲 ciently differs for the RCP and LCP components at the grazr,D r,M ing incidence, the 兩aR共L兲 − aR共L兲 兩 = condition could be satisfied selectively for either the RCP or LCP component and not for the remaining opposite one. Thus, the differential r,D circular reflectivity can be obtained for the case of 兩aR共L兲 r,M − aR共L兲 兩 = only for one of the circular components and not for the other, while satisfying the ⌬rsR共L兲 ⬎ 0 condition. To r,D r,M − aR共L兲 兩 = is satisfind the certain angle for which 兩aR共L兲 fied, we also conducted numerical calculations of the amplir,D ជ r,M for the RCP and E tude and phase of the beams of Eជ aR共L兲 aR共L兲 and LCP modes separately as a function of for the same vacuum/Co 共9 nm兲/Si substrate model system. The calculations were conducted using an iterative operation of the reflection 共R兲, propagation 共D兲, and transmission 共T兲 2 ⫻ 2 matrices in the framework of the orthogonal RCP-and-LCPmode basis.14,15 In those calculations, we applied Maxwell equations to the model structure and the boundary conditions of the top 共vacuum/Co兲 and bottom 共Co/Si兲 interfaces. The r,D ជ r,M for the RCP 共LCP兲 mode are electric fields Eជ aR共L兲 and E aR共L兲 Downloaded 19 Mar 2009 to 147.46.233.61. Redistribution subject to AIP license or copyright; see http://jap.aip.org/jap/copyright.jsp 07E709-3 J. Appl. Phys. 105, 07E709 共2009兲 D.-E. Jeong and S.-K. Kim FIG. 3. 共Color online兲 Calculation results for the electric-field amplitudes and phases of the individual components of the RCP and LCP components for beams directly reflected at the vacuum/Co interface and for ones multiply reflected inside the Co layer versus with a linearly s-polarized incident beam of h = 770.1 eV and my = + 1. The model system is the vacuum/Co 共9 r,M r corresponds to 兩sR共L兲 nm兲/Si-substrate. The phase difference ⌬sR共L兲 r,D − sR共L兲兩. described simply as 共 EE 兲 = RD共 EE 兲 and 共 EE 兲 = RM共 EE 兲 with r,D aR r,D aL i R i L r,M aR r,M aL i R i L −1 −1 R M = 共TA/BDBRB/CDB−1TA/B DBRB/CDB−1TA/B −1 + TA/BDBRB/CDB−1共RB/ADBRB/CDB−1兲2TA/B + ¯兲, where Ri/j and Ti/j are the reflection and transmission matrices, respectively, for the interface between the i and j media, and Di is the propagation matrix from the top of the i medium to the bottom. The results for a linearly s-polarized incident beam at h = 770.1 eV and the longitudinal magnetization of my = + 1 are shown in Fig. 3. It is distinctly evident that both the ជ r,D兩 = 兩Eជ r,M 兩 and 兩r,D − r,M 兩 ⯝ conditions are satisfied at 兩E sR sR sR sR the specific angle of = 0.66° only for the RCP component. By contrast, the counterpart LCP component shows a dissimilar aspect in the phase variation versus angle, but shows similar behavior to that of the RCP component in the amplitude variation versus angle, as shown in Fig. 3共b兲. For the r,D r,M 兩 = 兩Eជ sL 兩 condition is satisfied at the LCP component, the 兩Eជ sL r,D r,M − sL 兩 angle of = 0.66°, but the phase difference 兩sL = 120° is not out-of-phase at that angle. These results clearly demonstrate that totally destructive interference occurs selectively for the RCP component only at the unique angle of = 0.66°, and that it does not occur for the LCP component 共vice versa for the opposite longitudinal magnetization, my = −1兲. For the LCP component, at another specific angle r,D r,M ជ r,D兩 is not equal to = 1.46°, 兩sL − sL 兩 is out-of-phase but 兩E sL r,M 兩; thus, totally destructive interference does not occur 兩Eជ sL for this component at that angle. From these calculations of ជ r,D and Eជ r,M in the framethe amplitudes and phases of E sR共L兲 sR共L兲 work of the RCP-and-LCP-mode basis, we can definitively provide the energy and grazing angle of incident beams satជ r,D 兩 = 兩Eជ r,M 兩 and 兩r,D − r,M 兩 = isfying the necessary 兩E sR,L sR,L sR共L兲 sR共L兲 conditions, which yield differential destructive interference and, in turn, differential circular reflectivity. In summary, the results of the present study facilitate a fundamental understanding of the optical conversion of a linear polarization state of incident soft x rays into any linear and circular polarization modes of soft x rays reflected from a magnetic thin film. The physical origin of that magnetooptical effect is ascribed to the differential destructive interference of the opposite photon helicities through the out-ofphase difference of directly and multiply reflected beams with their comparable amplitudes at the grazing incidence and in the pre-edge energies below the absorption edges. This work was supported by Creative Research Initiatives 共ReC-SDSW兲 of MOST/KOSEF. 1 G. Schütz, W. Wagner, W. Wilhelm, P. Kienle, R. Frahm, and G. Materlik, Phys. Rev. Lett. 58, 737 共1987兲. 2 C. T. Chen, F. Sette, and S. Modesti, Phys. Rev. B 42, 7262 共1990兲. 3 C. Kao, J. B. Hastings, E. D. Johnson, D. P. Siddons, G. C. Smith, and G. A. Prinz, Phys. Rev. Lett. 65, 373 共1990兲. 4 J. B. Kortright, D. D. Awschalom, J. Stöhr, S. D. Bader, Y. U. Idzerda, S. S. P. Parkin, Ivan K. Schuller, and H.-C. Siegmann, J. Magn. Magn. Mater. 207, 7 共1999兲. 5 G. Srajer, L. H. Lewis, S. D. Bader, A. J. Epstein, C. S. Fadley, E. E. Fullerton, A. Hoffmann, J. B. 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Downloaded 19 Mar 2009 to 147.46.233.61. Redistribution subject to AIP license or copyright; see http://jap.aip.org/jap/copyright.jsp