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Magnetooptic ellipsometry in multilayers at arbitrary magnetization

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Abstract

The Yeh’s 4×4 matrix formalism is applied to determine the electromagnetic wave response in multilayers with arbitrary magnetization. With restriction to magneto–optic (MO) effects linear in the off–diagonal permittivity tensor elements, a simplified characteristic matrix for a magnetic layer is obtained. For a magnetic film–substrate system analytical representations of the MO response expressed in terms of the Jones reflection matrix are provided. These are numerically evaluated for cases when the magnetization develops in three mutually perpendicular planes.

©2001 Optical Society of America

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Figures (5)

Fig. 1.
Fig. 1. The magnetization M displayed as a cartesian vector sum of polar, M P , longitudinal, M L , and transverse M T . In the spherical coordinates M is specified by its magnitude | M | and the angles θM and ϕM .
Fig. 2.
Fig. 2. The geometry used in the modelling. The magnetization vector evolves on cone shaped surfaces about polar (a), longitudinal (b), and transverse (c) axes.
Fig. 3.
Fig. 3. Magneto–optical Kerr effect for an interface between vacuum and Bi0.96Lu2.04Fe5O12 magnetic garnet expressed in terms of the real part of the ratio r ps/r ss at an angle of incidence of 50 deg: the effect of the rotation of the magnetization vector, M, about the normal to the interface specified by an angle ϕM (a), about the axis parallel to the interface and the plane of incidence specified by an angle ϕy (b), and normal to the plane of incidence specified by an angle ϕx (c). The initial position of M is given by an angle θM between M and the interface normal at a fixed azimuth ϕM =0 deg (a), ϕM =90 deg (b), and ϕM =0 deg (c). The incident radiation is s-polarized.
Fig. 4.
Fig. 4. Magneto–optical reflection characteristics at a film/substrate system consisting of a Bi0.96Lu2.04Fe5O12 magnetic garnet film 1.5 µm thick deposited on a Gd3Ga5O12 substrate expressed in terms of the real part of the ratio r ps/r ss at an angle of incidence of 50 deg: the effect of the rotation of the magnetization vector, M, about the normal to the interface specified by an angle ϕM (a), about the axis parallel to the interface and the plane of incidence specified by an angle ϕy (b), and normal to the plane of incidence specified by an angle ϕx (c). The initial position of M is given by an angle θM between M and the interface normal at a fixed azimuth ϕM =0 deg (a), ϕM =90 deg (b), and ϕM =0 deg (c). The incident radiation is s-polarized. Note that the MO effect values are of two orders in magnitude higher than in the case of a single vacuum/BiLuIG interface.
Fig. 5.
Fig. 5. The effect on ℜ(r ps/r ss) of the angle of incidence, θ 0, ranging from -90 deg to +90 deg at an interface between vacuum and Bi0.96Lu2.04Fe5O12 magnetic garnet (a) and in a film/substrate system consisting of a Bi0.96Lu2.04Fe5O12 magnetic garnet film 1.5 µm thick deposited on a Gd3Ga5O12 substrate (b). The magnetization vector M is restricted to the plane of incidence. Its orientation is specified by an angle θM between M and interface normal. The incident radiation is s-polarized.

Tables (1)

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Table 1. The permittivity tensor elements of the materials used in modelling.

Equations (62)

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ε ( n ) = ( ε 0 ( n ) i ε 1 ( n ) cos θ M ( n ) i ε 1 ( n ) sin θ M ( n ) sin ϕ M ( n ) i ε 1 ( n ) cos θ M ( n ) ε 0 ( n ) i ε 1 ( n ) sin θ M ( n ) cos ϕ M ( n ) i ε 1 ( n ) sin θ M ( n ) sin ϕ M ( n ) i ε 1 ( n ) sin θ M ( n ) cos ϕ M ( n ) ε 0 ( n ) )
γ ( n ) 2 E 0 ( n ) γ ( n ) ( γ ( n ) · E 0 ( n ) ) = ω 2 c 2 ε ( n ) E 0 ( n )
ε 0 ( n ) N z ( n ) 4 ( 2 ε 0 ( n ) N z 0 ( n ) 2 ε 1 ( n ) 2 sin 2 θ M ( n ) ) N z ( n ) 2 2 ε 1 ( n ) 2 sin θ M ( n ) cos θ M ( n ) sin ϕ M ( n ) N y N z ( n )
+ ε 0 ( n ) ( N z 0 ( n ) 4 ε 1 ( n ) 2 ) + N y 2 ε 1 ( n ) 2 ( 1 sin 2 θ M ( n ) sin 2 ϕ M ( n ) ) = 0
N z 1,3 ( n ) = N z 0 ( n ) ( 1 ε 1 ( n ) 2 4 ε 0 ( n ) N z 0 ( n ) 2 ) ± ε 1 ( n ) 2 ε 0 ( n ) 1 / 2 N z 0 ( n ) ( N z 0 ( n ) cos θ M ( n ) + N y sin θ M ( n ) sin ϕ M ( n ) )
+ ε 1 ( n ) 2 8 ε 0 ( n ) N z 0 ( n ) 3 ( N z 0 ( n ) 2 cos 2 θ M ( n ) N y 2 sin θ M ( n ) 2 sin ϕ M ( n ) 2 )
N z 2,4 ( n ) = N z 0 ( n ) ( 1 ε 1 ( n ) 2 4 ε 0 ( n ) N z 0 ( n ) 2 ) ε 1 ( n ) 2 ε 0 ( n ) 1 / 2 N z 0 ( n ) ( N z 0 ( n ) cos θ M ( n ) N y sin θ M ( n ) sin ϕ M ( n ) )
ε 1 ( n ) 2 8 ε 0 ( n ) N z 0 ( n ) 3 ( N z 0 ( n ) 2 cos 2 θ M ( n ) N y 2 sin θ M ( n ) 2 sin ϕ M ( n ) 2 )
E 0 ( 0 ) = M E 0 ( 𝓝 + 1 )
M = [ D ( 0 ) ] 1 D ( 1 ) P ( 1 ) [ ( D ) 1 ] 1 D ( 𝓝 ) P ( 𝓝 ) [ D ( 𝓝 ) ] 1 [ D ( 𝓝 + 1 ) ]
D 1 j ( n ) = i ε 1 ( n ) N z 0 ( n ) 2 cos θ M ( n ) i ε 1 ( n ) N y N zj ( n ) sin θ M ( n ) sin ϕ M ( n )
ε 1 ( n ) 2 sin 2 θ M ( n ) cos ϕ M ( n ) sin ϕ M ( n )
D 2 j ( n ) = N zj ( n ) D 1 j ( n )
D 3 j ( n ) = N z 0 ( n ) 2 ( N z 0 ( n ) 2 N zj ( n ) 2 ) ε 1 ( n ) 2 sin 2 θ M ( n ) sin 2 ϕ M ( n )
D 4 j ( n ) = ( ε 0 ( n ) N zj ( n ) i ε 1 ( n ) N y sin θ M ( n ) cos ϕ M ( n ) ) ( N z 0 ( n ) 2 N zj ( n ) 2 )
+ ε 1 ( n ) 2 sin θ M ( n ) sin ϕ M ( n ) ( N zj ( n ) sin θ M ( n ) sin ϕ M ( n ) N y cos θ M ( n ) )
P ( n ) = [ exp ( i ω c N z 1 ( n ) d ( n ) ) 0 0 0 0 exp ( i ω c N z 2 ( n ) d ( n ) ) 0 0 0 0 exp ( i ω c N z 3 ( n ) d ( n ) ) 0 0 0 0 exp ( i ω c N z 4 ( n ) d ( n ) ) ]
D ( n ) = [ 1 1 0 0 N z 0 ( n ) N z 0 ( n ) 0 0 0 0 N z 0 ( n ) ( ε 0 ( n ) ) 1 / 2 N z 0 ( n ) ( ε 0 ( n ) ) 1 / 2 0 0 ( ε 0 ( n ) ) 1 / 2 ( ε 0 ( n ) ) 1 / 2 ]
[ E 0 s ( r ) E 0 p ( r ) ] = [ r ss r sp r ps r pp ] [ E 0 s ( i ) E 0 p ( i ) ]
r ss = [ E 0 s ( r ) E 0 s ( i ) ] E 0 p ( i ) = 0 = M 21 M 33 M 23 M 31 M 11 M 33 M 13 M 31
r ps = [ E 0 p ( r ) E 0 s ( i ) ] E 0 p ( i ) = 0 = M 41 M 33 M 43 M 31 M 11 M 33 M 13 M 31
r sp = [ E 0 s ( r ) E 0 p ( i ) ] E 0 s ( i ) = 0 = M 11 M 23 M 13 M 21 M 11 M 33 M 13 M 31
r pp = [ E 0 p ( r ) E 0 p ( i ) ] E 0 s ( i ) = 0 = M 11 M 43 M 13 M 41 M 11 M 33 M 13 M 31
E 0 ( 0 ) = [ E 01 ( 0 ) E 02 ( 0 ) E 03 ( 0 ) E 04 ( 0 ) ] T = [ E 0 s ( i ) E 0 s ( r ) E 0 p ( i ) E 0 p ( r ) ] T
E 0 ( 𝓝 + 1 ) = [ E 01 ( 𝓝 + 1 ) E 02 ( 𝓝 + 1 ) E 03 ( 𝓝 + 1 ) E 04 ( 𝓝 + 1 ) ] T = [ E 0 s ( t ) 0 E 0 p ( t ) 0 ] T
S ( n ) = D ( n ) P ( n ) ( D ( n ) ) 1
S ( n ) = [ S 11 ( n ) S 12 ( n ) S 13 ( n ) S 14 ( n ) S 21 ( n ) S 11 ( n ) S 23 ( n ) S 24 ( n ) S 24 ( n ) S 14 ( n ) S 33 ( n ) S 34 ( n ) S 23 ( n ) S 13 ( n ) S 43 ( n ) S 44 ( n ) ]
S 11 ( n ) = cos β ( n )
S 12 ( n ) = i N z 0 ( n ) 1 sin β ( n )
S 21 ( n ) = i N z 0 ( n ) sin β ( n )
S 34 ( n ) = i N z 0 ( n ) ε 0 ( n ) 1 sin β ( n )
S 43 ( n ) = i N z 0 ( n ) 1 ε 0 ( n ) sin β ( n )
S 33 ( n ) = cos β ( n ) q ( n ) sin β ( n )
S 44 ( n ) = cos β ( n ) + q ( n ) sin β ( n )
S 13 ( n ) = N z 0 ( n ) 1 ε 0 ( n ) 1 / 2 ( l ( n ) sin β ( n ) + i a n )
S 14 ( n ) = ε 0 ( n ) 1 / 2 ( p ( n ) sin β ( n ) + i b n )
S 23 ( n ) = ε 0 ( n ) 1 / 2 ( p ( n ) sin β ( n ) i b n )
S 24 ( n ) = N z 0 ( n ) ε 0 ( n ) 1 / 2 ( l ( n ) sin β ( n ) i a n )
a n = i 2 ( e i β ( n ) Δ ( n ) + e i β ( n ) Δ ( n ) )
b n = i 2 ( e i β ( n ) Δ ( n ) + + e i β ( n ) Δ ( n ) )
Δ ( n ) ± = ω 2 c d ( n ) ε 1 ( n ) ε 0 ( n ) 1 / 2 N z 0 ( n ) 1 ( N z 0 ( n ) cos θ M ( n ) ± N y sin θ M ( n ) sin ϕ M ( n ) )
β ( n ) = ω c d ( n ) N z 0 ( n )
p ( n ) = ε 1 ( n ) ( N z 0 ( n ) cos θ M ( n ) ) 2 ε 0 ( n ) 1 / 2 N z 0 ( n ) 2
l ( n ) = ε 1 ( n ) ( N y sin θ M ( n ) sin ϕ M ( n ) ) 2 ε 0 ( n ) 1 / 2 N z 0 ( n ) 2
q ( n ) = ε 1 ( n ) ( N y sin θ M ( n ) cos ϕ M ( n ) ) ε 0 ( n ) N z 0 ( n )
ω c d ( n ) N z 0 ( n ) 1
M = ( D ( 0 ) ) 1 S ( 1 ) D ( 2 )
r ss = r ss ( 01 ) + r ss ( 12 ) e 2 i β ( 1 ) 1 + r ss ( 01 ) r ss ( 12 ) e 2 i β ( 1 )
r ps , sp = t ss ( 01 ) t pp ( 10 ) { β 1 e 2 i β ( 1 ) [ p ( 1 ) ( r ss ( 12 ) + r pp ( 12 ) ) + l ( 1 ) ( r ss ( 12 ) r pp ( 12 ) ) ]
+ i 2 ( 1 e 2 i β ( 1 ) ) [ ± p ( 1 ) ( 1 + r ss ( 12 ) r pp ( 12 ) e 2 i β ( 1 ) ) l ( 1 ) ( 1 r ss ( 12 ) r pp ( 12 ) e 2 i β ( 1 ) ) ] }
× [ ( 1 + r ss ( 01 ) r ss ( 12 ) e 2 i β ( 1 ) ) ( 1 + r pp ( 01 ) r pp ( 12 ) e 2 i β ( i ) ) ] 1
r pp = r pp ( 01 ) r pp ( 12 ) e 2 i β ( 1 ) 1 + r pp ( 01 ) r pp ( 12 ) e 2 i β ( 1 ) + i 2 q ( 1 ) ( 1 e 2 i β ( 1 ) ) t pp ( 01 ) t pp ( 10 ) 1 r pp ( 12 ) 2 e 2 i β ( 1 ) ( 1 + r pp ( 01 ) r pp ( 12 ) e 2 i β ( 1 ) ) 2
r ss ( ij ) = N z 0 ( i ) N z 0 ( j ) N z 0 ( i ) + N z 0 ( j )
r pp ( ij ) = ε 0 ( i ) N z 0 ( j ) ε 0 ( j ) N z 0 ( i ) ε 0 ( i ) N z 0 ( j ) + ε 0 ( j ) N z 0 ( i )
t ss ( ij ) = 1 + r ss ( ij )
t pp ( ij ) = ( ε 0 ( i ) / ε 0 ( j ) ) 1 / 2 ( 1 r pp ( ij ) )
r ps ( 01 , pol ) cos θ M ( 1 ) = i ε 1 ( 1 ) N ( 0 ) cos θ ( 0 ) cos θ M ( 1 ) N z 0 ( 1 ) N z 0 ( 1 ) ( N ( 0 ) cos θ ( 0 ) + N z 0 ( 1 ) ) ( N ( 0 ) N z 0 ( 1 ) + ε 0 ( 1 ) cos θ ( 0 ) )
= i 2 p ( 1 ) t ss ( 01 ) t pp ( 10 )
r ps ( 01 , lon ) sin θ M ( 1 ) sin ϕ M ( 1 ) = i ε 1 ( 1 ) N ( 0 ) cos θ ( 0 ) sin θ M ( 1 ) sin ϕ M ( 1 ) N y N z 0 ( 1 ) ( N ( 0 ) cos θ ( 0 ) + N z 0 ( 1 ) ) ( N ( 0 ) N z 0 ( 1 ) + ε 0 ( 1 ) cos θ ( 0 ) )
= i 2 l ( 1 ) t ss ( 01 ) t pp ( 10 )
r ps , sp = 2 ω c d ( 1 ) ε 1 ( 1 ) N ( 0 ) cos θ ( 0 ) ( ε 0 ( 1 ) N z 0 ( 2 ) cos θ M ( 1 ) + ε 0 ( 2 ) N y sin θ M ( 1 ) sin ϕ M ( 1 ) ) ε 0 ( 1 ) ( N ( 0 ) cos θ ( 0 ) + N z 0 ( 2 ) ) ( N ( 0 ) N z 0 ( 2 ) + ε 0 ( 2 ) cos θ ( 0 ) )
r pp = r pp ( ε 1 ( 1 ) = 0 ) t pp ( 02 ) t pp ( 20 ) ω c d ( 1 ) ε 1 ( 1 ) ε 0 ( 1 ) 1 N y sin θ M ( 1 ) cos ϕ M ( 1 )
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