Download Differential destructive interference of the circular polarization

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
+ i␤R,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
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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
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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.
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