16 August 2017 Magneto-optical Kerr effect studies with classical electromagnetic theory
Author Affiliations +
Proceedings Volume 10452, 14th Conference on Education and Training in Optics and Photonics: ETOP 2017; 104525A (2017) https://doi.org/10.1117/12.2269711
Event: 14th Conference on Education and Training in Optics and Photonics, ETOP 2017, 2017, Hangzhou, China
Abstract
Employing the basic law of electromagnetic wave propagation with the constitutive relation of ideal ferromagnetic medium, the magneto-optical Kerr effect of linearly polarized plane wave on the surface of ideal ferromagnetic medium is studied. We also discuss the change law of the Kerr rotation angle with the incident angle and magnetic-field strength in detail, which can provide crucial information for the applications of the magneto-optical Kerr effect in many fields.

1.

INTRODUCTION

The magneto-optical Kerr effect[1] refers to the phenomenon that the vibrating surface of reflected wave rotates while the incidence of the linearly polarized plane wave (LPPW) on the magnetized material, which is discovered in 1876 by J·Kerr. Magneto-optical Kerr effect is widely used in many fields, like magnetic ordering, magnetic anisotropy, coupling between the layers of multilayer film, and the research of the magnetic ultrathin membrane phase transition behavior[2-5]. Magneto-optical Kerr effect is an important experimental methods of magnetism, and is also an effective way to measure the properties of materials, especially the physical properties of film materials[6-7]. Besides, magneto-optical information storage is the technological innovation developed in recent years, and it is a hot research field of developing practical magnetic medium material with much more superior performance for information storage. Magneto-optical Kerr effect has been applied to the high density of computer memory, e.g., there is the recordable compact disc with the storage of hundreds to thousands of megabytes in single piece.

At present, the magneto-optical Kerr effect is heavily studied in experimental measurements[8-11]. The theoretical explanation on this effect is mainly based on complex quantum theories[12-15], while how to employ the classical electromagnetic theory on this effect is rarely discussed. In this paper, with Maxwell’s electromagnetic equations, the boundary conditions[16] and medium constitutive relation of ideal ferromagnetic medium, we analyze the LPPW reflection behavior on the surface of the ideal ferromagnetic medium, which can naturally gives the poloidal, longitudinal and transverse magneto-optic Kerr effect, besides the Kerr rotation angle change law with the incident angle and magnetic-field strength in detail. The studies can provide crucial information for the applications of the magnetooptical Kerr effect in many fields.

2.

PRIMARY ELECTROMAGNETIC THEORY AND IDEAL FERROMAGNETIC MEDIUM MODEL

The effect of the medium on electromagnetic waves is generally described by Maxwell’s differential equations:

00150_PSISDG10452_104525A_page_1_1.jpg

and the corresponding boundary conditions:

00150_PSISDG10452_104525A_page_2_1.jpg

Note here supposed that no free charges and free currents at the interface between two kinds of mediums. Here the ideal ferromagnetic medium referred to in this paper has the following electromagnetic properties:

00150_PSISDG10452_104525A_page_2_2.jpg

where 00150_PSISDG10452_104525A_page_2_2a.jpg is a constant vector. The equation (1) is the approximation model of magnetic hysteresis loop of ferromagnetic medium. From (1) we can see that, an ideal ferromagnetic medium can be taken as a mixture of the linear magnetic medium and permanent magnet. Therefore, when electromagnetic behaviors are relevant to the ideal ferromagnetic medium, the corresponding analysis can be divided into two steps: at first, the studies are made according to the linear magnetic medium of 00150_PSISDG10452_104525A_page_2_3.jpg, and then the influence and modification of the permanent magnet is considered carefully. In the next section, employing the above ideas, we shall deduce the magneto-optical Kerr effects and rotation angles when LPPW is incident to the surface of ideal ferromagnetic medium, in which the 00150_PSISDG10452_104525A_page_2_3a.jpg can be along one of the poloidal, longitudinal and transverse directions.

3.

MAGNETO-OPTICAL KERR EFFECTS ON IDEAL FERROMAGNETIC MEDIUM SURFACE

Figure 1 shows the coordinate system employed in the following derivations. The XOY plane is the interface of two semi-infinite mediums, where 2 is the ideal ferromagnetic medium and 1 is the homogeneous linear medium. The normal direction of the interface is consistent with the Z axis. The incidence of LPPW is from 1 to 2, so the incident plane is XOZ plane. Here the Kerr rotation angle β is defined as the included angle between the vector direction of total magnetic field in reflected wave and the incident plane.

Figure 1.

The coordinate system is adopted.

00150_PSISDG10452_104525A_page_2_4.jpg

3.1

S-wave case

In S-wave case, the electric/magnetic field vibration direction of incident wave is perpendicular/parallel to the incident plane. To obtain the magnetic field amplitude ratios of reflected and refracted waves to incident wave, one needs two boundary conditions: tangential components of electric fields are continuous, and tangential components of magnetic fields are continuous on the interface, too. As mentioned in Section 2, these boundary conditions correspond respectively to the following formulas (2)-(3):

00150_PSISDG10452_104525A_page_2_5.jpg
00150_PSISDG10452_104525A_page_2_6.jpg

where E/H is the electric/magnetic field amplitude of the incident wave, and E/H′ (E″/H)″is the electric/magnetic field amplitude of reflected (refracted) wave when supposed 2 is linear medium, so according to TEM wave properties we have

00150_PSISDG10452_104525A_page_3_1.jpg

Here the tangential direction is X or Y axis as shown in Figure 1. With Eq. (2)-(4), we can obtain the equation on H′ of reflected wave:

00150_PSISDG10452_104525A_page_3_2.jpg

According to refraction law: 00150_PSISDG10452_104525A_page_3_3.jpg we can get

00150_PSISDG10452_104525A_page_3_4.jpg

With Eq. (5) and (6), H′ is solved as

00150_PSISDG10452_104525A_page_3_5.jpg

In the above process of derivation, only the linear part 00150_PSISDG10452_104525A_page_3_6.jpg of electromagnetic characteristic law of ideal ferromagnetic medium in Eq. (1), i.e., the medium-2 is supposed just as linear medium. However, the fundamental reason of ideal ferromagnetic medium different from linear medium is that the term of 00150_PSISDG10452_104525A_page_3_7.jpg exists in its electromagnetic characteristics, which can have an important impact on the boundary conditions and then request the reflected wave to produce extra magnetic field besides linear medium. Furthermore, this extra magnetic field may vary with the direction of 00150_PSISDG10452_104525A_page_3_8.jpg So to get the total magnetic field 00150_PSISDG10452_104525A_page_3_9.jpg of reflected wave, the specific impact of the term of 00150_PSISDG10452_104525A_page_3_10.jpg should be considered and the magnetic field boundary conditions should be modified correspondingly.

When 00150_PSISDG10452_104525A_page_3_11.jpg is in X direction (i.e., lengthwise direction), the boundary condition that the tangential components of magnetic field are continuous can be written as:

00150_PSISDG10452_104525A_page_3_12.jpg

With Eq. (2), (3’), (4) and (6), we get

00150_PSISDG10452_104525A_page_3_13.jpg

In this case, the direction of 00150_PSISDG10452_104525A_page_3_14.jpg is parallel to the incident plane (XOZ plane), so the Kerr rotation angle is zero. When 00150_PSISDG10452_104525A_page_3_15.jpg is in the Y axis (i.e., transverse direction), the reflected wave’s total magnetic field is

00150_PSISDG10452_104525A_page_3_16.jpg

In this case, the direction of 00150_PSISDG10452_104525A_page_4_1.jpg is not parallel to the incident plane, and the Kerr rotation angle is

00150_PSISDG10452_104525A_page_4_2.jpg

When 00150_PSISDG10452_104525A_page_4_3.jpg is along Z direction (i.e., poloidal direction), one needs not only the boundary conditions of (2) and (3), but also the boundary condition that the normal components of magnetic field are continuous, that is

00150_PSISDG10452_104525A_page_4_4.jpg

Then we can obtain

00150_PSISDG10452_104525A_page_4_5.jpg

Here, like 00150_PSISDG10452_104525A_page_4_6.jpg is in the X direction, 00150_PSISDG10452_104525A_page_4_7.jpg is parallel to the incident plane and the Kerr rotation angle is zero.

3.2

P-wave case

In this case, the electric/magnetic field vibration direction of incident wave is parallel/perpendicular to the incident plane. When we just take account of the linear part of electromagnetic characteristic law of ideal ferromagnetic medium, with the boundary conditions that tangential components of electric fields and magnetic fields are continuous separately on the interface, H′ is solved as

00150_PSISDG10452_104525A_page_4_8.jpg

The derivation with no more detailed description is similar as in the S-wave case.

When 00150_PSISDG10452_104525A_page_4_9.jpg is along the X axis, we can obtain the total magnetic field of the reflected wave as

00150_PSISDG10452_104525A_page_4_10.jpg

and the Kerr rotation angle is

00150_PSISDG10452_104525A_page_4_11.jpg

When 00150_PSISDG10452_104525A_page_4_10b.jpg is in the Y axis, the total magnetic field of the reflected wave is

00150_PSISDG10452_104525A_page_4_13.jpg

so the vector of 00150_PSISDG10452_104525A_page_4_14.jpg is still within the incident plane and the Kerr rotation angle is zero.

When 00150_PSISDG10452_104525A_page_5_1.jpg is in the Z axis, after taking account of the boundary condition that the normal components of magnetic field are continuous besides, one gets reflected wave’s total magnetic field as

00150_PSISDG10452_104525A_page_5_2.jpg

then the Kerr rotation angle is

00150_PSISDG10452_104525A_page_5_3.jpg

4.

RESULTS AND DISCUSSION

Based on the analysis in Section 3, we can draw a conclusion that magneto-optical Kerr effect is generated when LPPW propagates on the surface of ideal ferromagnetic medium. The Kerr rotation angle β value is related with the incident angle 00150_PSISDG10452_104525A_page_5_3a.jpg and magnetic field H, and with the different directions of 00150_PSISDG10452_104525A_page_5_3a.jpg the corresponding change laws are different, as shown in Table 1.

Table 1.

The Kerr rotation angle β with along different directions.

directionS-waveP-wave
X (longitudinal)β= 0, no rotation
Y (transverse)β= 0, no rotation
Z (poloidal)β= 0, no rotation

According to Table 1, we can see that for S-wave the Kerr rotation angle β≠0, and on the contrary β=0 for P-wave only when 00150_PSISDG10452_104525A_page_5_6a.jpg along the Y axis (transverse direction). Furthermore, when 00150_PSISDG10452_104525A_page_5_6b.jpg is along the X or Z axis, the expressions of β are same for P-wave.

For simplicity, the following analysis on the variation trends of the Kerr rotation angle with the incident angle θ and magnetic field H, is in the conditions of the LPPW incident on the ideal ferromagnetic medium from vacuum, namely medium-1 is vacuum. Therefore, the magnetic conductivity and dielectric constant of medium-1 and medium-2 are respectively adopted as: μ1 = μ0=4π×10-7H/m, ε1=ε0=8.85×10-12 F/m, and μ2=500×4π×10-7 H/m, ε2=5.51×8.85×10-12 F/m. Here μ2 and ε2 of ideal ferromagnetic medium are taken the values as manganese zinc ferrite, and according to its hysteresis loop of manganese, M0 is estimated as M0=4×104 A/m. After substituting above constant values into Table 1, one can get the functions of β only related with two variables: the incident angle θ and magnetic field H, as shown in Figure 2.

Figure 2.

The 3D Figure of the function of the Kerr rotation angle β with two variables of incident angle θ (0~π/2) and magnetic field H: (a) 00150_PSISDG10452_104525A_page_6_2.jpg direction is along the X(Z) axis, H belongs to 10~104 A/m; (b) 00150_PSISDG10452_104525A_page_6_2.jpg direction is along the X(Z) axis, H belongs to 104~107 A/m; (c) 00150_PSISDG10452_104525A_page_6_3.jpg direction is along the Y axis, H belongs to 10~104 A/m; (d) 00150_PSISDG10452_104525A_page_6_4.jpg direction is along the Y axis, H belongs to 104~107 A/m. Here we ignore other interaction effects of wave with substances.

00150_PSISDG10452_104525A_page_6_1.jpg

In the following we shall give the detailed analysis of β dependence of H(θ) when regarding θ (H) as a constant, see Figure 3 and 4 respectively.

Figure 3.

When 00150_PSISDG10452_104525A_page_6_2.jpg direction is along the X(Z) axis, the variation trend of the Kerr rotation angle β with (a) the incident angle θ (b) the magnetic field H of incident P-wave.

00150_PSISDG10452_104525A_page_7_1.jpg

Figure 4.

When 00150_PSISDG10452_104525A_page_6_2.jpg direction is along the Y axis, the variation trend of the Kerr rotation angle β with (a) the incident angle θ (b) the magnetic field H of incident S-wave.

00150_PSISDG10452_104525A_page_8_1.jpg

If the magnetic field H of incident wave is adopted as H=4.59×103A/m (here we ignore other interaction effects of wave with substances) [17], the refraction law 00150_PSISDG10452_104525A_page_7_3.jpg and sin2 θ + cos2 θ=1 are also considered, for 00150_PSISDG10452_104525A_page_7_4.jpg direction along the X(Z) axis β can be simplified as

00150_PSISDG10452_104525A_page_7_5.jpg

according to Table 1. β∈ [0.09, 0.11] because of θ∈ [0, π/2]. The β is defined as the included angle between the vector direction of total magnetic field in reflected wave and the incident plane, it should not be greater than π/2 rad, therefore the absolute value is adopted in (9). Figure 3(a) shows that β is monotone increasing very slowly with θ increasing, which can be easily understood when considering Eq. (9) and tangent function variation curve with cosθ ∈ [0, 1]. On the other hand, when the incident angle θ is considered as a constant π/4 rad, Figure 3(b) indicates β changes with H as Eq. (10)

00150_PSISDG10452_104525A_page_7_6.jpg

It increases very slowly when H belongs to 0~103 A/m, but rapidly increases when H belongs to 103~106 A/m, and it trends to a stable value of π/2 rad when H≥106 A/m.

For 00150_PSISDG10452_104525A_page_8_3.jpg direction along the Y axis and H=4.59×103 A/m, according to Table 1, the Kerr rotation angle β can be simplified as

00150_PSISDG10452_104525A_page_8_4.jpg

When the incident θ∈ [0, π/2], β∈ [1.46, π/2] within a very small range of about 0.1 rad. In Eq. (11), because there is a term of (10cosθ-1) on the denominator, we can see that with the increase of θ, the value of this term changes from positive to 0 and then to negative, i.e., the direction of the total magnetic field of reflected wave will have a reversal. However, according to the definition of β, here we also take the absolute value in Eq. (11). Figure 4(a) shows that when θ increases from 0, according to cosine function properties, β is about 1.48 rad and the increase is not obviously at the beginning. When θ near 1.47 rad, β rapidly increases to π/2 rad, and then sharp decreases to about 1.46 rad after θ continues to increase. The location of peak value of β= π/2 rad is θ=1.47 rad, which is a fixed value due to the peak position is decided by the term of 00150_PSISDG10452_104525A_page_8_5.jpg equal to zero on the denominator of the expression of β list in Table 1. When the incident angle θ is considered as a constant π/4 rad, from Figure 4(b) based on Eq. (12)

00150_PSISDG10452_104525A_page_8_6.jpg

we can see a curve with reverse trend with H comparing to Figure 3(b).

5.

CONCLUSION AND OUTLOOK

Based on the classical electromagnetic theory, we study th e propagation behavior of LPPW incident on the surface of the ideal ferromagnetic medium of constitutive relation 00150_PSISDG10452_104525A_page_8_7.jpg which is different from the general linear homogeneous medium and leads to the magneto-optical Kerr effect. We discuss in detail the change law of the Kerr rotation angle with the incident angle and the magnetic field for 00150_PSISDG10452_104525A_page_9_1.jpg along different directions in S-wave and P-wave. The theoretical analysis indicates that the Kerr rotation angle β has the follow characteristics: (1) for S-wave the Kerr rotation angle β≠0, but β=0 for P-wave only when 00150_PSISDG10452_104525A_page_9_2.jpg along the Y axis (transverse direction); when 00150_PSISDG10452_104525A_page_9_3.jpg is along the X or Z axis (longitudinal or poloidal direction), the expressions of β are r same for P-wave. (2) When the inr cident angle θ increases and magnetic field H is invariant, β has a slow growth with 00150_PSISDG10452_104525A_page_9_4.jpg in the X or Z direction; when 00150_PSISDG10452_104525A_page_9_5.jpg direction is the Y axis, there is a peak value of β=π/2 rad and the position of this peak does not change. (3) With the incident magnetic field H increases and incident angle θ is invariant, β increases from about 0 rad to the maximum value of π/2 rad when 00150_PSISDG10452_104525A_page_9_6.jpg direction is in the X or Z axis; however, when 00150_PSISDG10452_104525A_page_9_7.jpg is along the Y axis β decreases from π/2 rad to 0 rad.

The analysis employing the classical electromagnetic theory on the magneto-optical Kerr effect in this paper can share some similarity with other investigations, e.g., the electro-optic Kerr effect, Faraday effect, and the concerned phenomena of total reflection and Brewster angle. These theoretical studies can shed light on the application of new technology related with the subjects mentioned above.

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© (2017) COPYRIGHT Society of Photo-Optical Instrumentation Engineers (SPIE). Downloading of the abstract is permitted for personal use only.
Rui-Can Ma, Rui-Can Ma, Ling-Ling Gao, Ling-Ling Gao, Ting Zhang, Ting Zhang, Yi Jin, Yi Jin, } "Magneto-optical Kerr effect studies with classical electromagnetic theory", Proc. SPIE 10452, 14th Conference on Education and Training in Optics and Photonics: ETOP 2017, 104525A (16 August 2017); doi: 10.1117/12.2269711; https://doi.org/10.1117/12.2269711
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