- Email: [email protected]
Analysis by magnetic phase grating model for domain wall observation
Journal of Magnetism and Magnetic Materials 140-144 (1995) 1845-1846
~14 Journalof renalneusm magnetic ~i materials
ELSEVIER
Analysis by magnetic phase grating model for domain wall observation M. Ohkoshi *, K. Utsunomiya, K. Tsushima Faculty of Computer Science, Kyushu Institute of Technology, Fukuoka 820, Japan Abstract
The wall image formation associated with the light diffraction in Bi-substituted iron garnets is analyzed by a magnetic phase grating model. Thin wall width with high Faraday rotation yields the wall image with high contrast both in bright field and dark field configurations.
A polarized anisotropic dark field optical imaging technique (PADO) has been demonstrated to be able to reveal the location of Bloch lines within the bubble or stripe domain walls [1]. In spite of continued efforts on the diffraction effects due to magnetic domains [2], the subject is not completed yet and an optical diffraction contrast model has been proposed recently [3]. In this paper, the wall image formation in magnetic garnets with a perpendicular easy axis is analyzed by a magnetic phase grating model. Let the domain structure be a stripe form along the y-axis and the distribution of magnetization be trapezoidal along the x-axis in the film plane. The stripe domain period is d, and the wall width is w. The widths of up and down domains are s - w and d - s w, respectively, s and d are intended to be changed by a magnetic field applied normal to the film plane. Let the incident plane wave of unit amplitude with wavelength A propagate along the z-axis, the film normal. The Faraday rotation O(x) of the light transmitted through the film changes periodically along the x-axis across the antiparallel domains as + 0 F in the up domains and - 0 F in the down domains. In the wall, the Faraday rotation is assumed to change linearly between the domains alternately [4]. The optical phase difference between the adjacent antiparallel domains is 20 F after transmitted the film. The Faraday ellipticity is small and has little contribution to the following analysis. Then, the amplitude of the undiffracted 0th order light is
respectively, to the incident linear polarization direction, and w' and s' are normalized widths by the period d. In a demagnetized state where s / d = 1/2, the perpendicular component is zero as recognized from Eq. (1), and the undiffracted light is polarized completely parallel to the incident linear polarization. The perpendicular component of the 0th light intensity I 0_ is maximum of sin20F in the magnetic saturation state ( s / d = 0 or 1). Since the wall width is negligible compared to the domain width ( w ' = 0) in a standard uniaxial garnet (the quality factor Q > 1), we can safely suppose the parallel component of the 0th light intensity 1011 to be COS20F,independent of the applied field although it increases for thinner wall widths. Therefore, the intensity change of the undiffracted light by the applied field comes mainly from the perpendicular component. In a garnet of 90 ° Faraday rotation, the intensity of the undiffracted beam changes from 0 to 1 by the applied field without crossed polarizers. The diffracted ruth order lights due to the magnetic phase grating of the stripe form appear along x-axis at angles ~bm = s i n - l ( m A / d ) from the z-axis. The amplitude of the diffracted mth order light (m :~ 0) is obtained straightforwardly as u m = i [ w ' / ( m ~ r w ' - 0F) sin(marw' -- 0F) + w ' / ( m ~ r w ' + 0F) sin(m'trw' + 0F) - - ( 2 / m l r ) sin(m~rw') COS0F] cos(turfs') + j [ w ' / ( m ~ r w ' + 0F) sin(m'trw' + 0F) - - w ' / ( m ~ w ' - - 0F) sin(m'rrw' -- 0 r )
Uo = i[(2W'/OF) sin 0 F + (1 -- 2w') cos 0F] + j ( 1 - - 2 s ' ) sin 0F,
where i and j are unit vectors parallel and perpendicular,
* Corresponding author. Fax: +81-948-29-7651; [email protected].
- - ( 2 / m ~ ) cos(m'rrw') sin0F] sin(m'n's').
(1)
email:
(2)
Since w << d, the diffracted lights are all in perpendicular polarization. When s / d = 1 / 2 at zero field, even order (m ~ 0) diffractions with parallel polarization are absent in the limit of zero wall width. With increasing the applied field, the perpendicular component of the mth order disap-
0304-8853/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0304-8853(94)01387-X
M. Ohkoshiet al. /Journal of Magnetism and MagneticMaterials 140-144 (1995) 1845-1846
1846
0.3 ~
t.rm=+l--+~
eF(deg)24__ 18---
o
Fig. 1. Diffraction patterns representing thick wall and high Faraday rotation effects in crossed and parallel polarizations. pears when s / d is multiple integer of 1/m and takes maximum intensity when s / d = (integer + ½)/m. As an example of wall width contribution to the parallel diffraction, Fig. 1 shows isotropic diffraction patterns observed in a (BiTm)3Fe5012 garnet film with demagnetized maze-domains (Q = 0.3, d = 6.2 ixm, 0 F = 75°). In addition to the perpendicular polarized odd order diffracted lights, parallel polarized even order diffractions due to thick walls are separated by an analyzer for an linearly polarized incident beam. The magnetization distribution in this sample may be sinusoidal [5] rather than trapezoidal because of the low Q-value. Intensity distribution of the outgoing light on the image plane is calculated from
I(x)
=luo12+ E
nm exp(j2m~rx/d) 2.
(3)
m~0
By collecting all the diffracted lights including the undiffracted light (ideal bright field observation), the parallel component of the domain intensity /all is ~ COS20F, and the perpendicular component I a ± is ~ sin20F less depending on the domain width and wall width. Therefore, the wall contrast is bright for parallel (Iwl I = 1) and dark (I,,,± = 0) for crossed nicol as is well known. However, depending on the numerical aperture N A of the objective lens used for imaging, the image is constructed usually by at most a few tens of diffracted lights for domains of txm-size (I m i < N A d/A). Although IwH is nearly 1 for ,
0.85
0.8
,
,
,
,
,
,
I 0.02
,
I
I
I
0.04
I o.oq~
i
L 0.08
i 0.t
w/d
Fig. 2. Dependence of the wall intensity I w and the domain intensity I a on the normalized wall width w~ d for - 20 < ra < 20;
s/d=l/2.
i
i
0.1 w/d
0.15
0,2
Fig. 3. Dependence of the wall intensity 1w and the domain intensity I a on the normalized wall width w / d for one-sided diffracted beams 1 < m < 20; s / d = 1/2.
m > d/w, I w II decreases with the decrease of wall width, and reaches to COS20F at w / d = 0 as shown in Fig. 2. On the other hand, I w ± is always zero. Therefore, even for the unpolarized incident beam, the intensity of wall image Iw = Iwll + lw ± is always smaller than that of domains Ia =/all "l- I a ± ~ 1. Namely, domain walls can be observed as dark lines by conventional bright field microscopy without polarizer and analyzer. The intensity ratio Iw/I a decreases with decrease of w / d and increase of 0 F. When forming the image by using a part of diffracted beams, for instance, one-sided diffracted lights excluding 0th beam (PADO location), the walls can be observed as bright lines because I w > I a in this case. The light at wall is almost perpendicularly polarized. The perpendicular component of the wall image intensity I w.L is larger than that of domains I a ±. The I w ± increases with the decrease of w / d and the increase of 0 F as shown in Fig. 3. The parallel components lwlt and /all are much smaller than Iw± and I a ± , and both reach to zero for w = 0 . The intensity ratio Iw/I a increases with the decrease of w/d. In conclusion, domain walls can be observed as dark lines by conventional bright field microscopy without polarizer and analyzer, while they can be observed as bright lines when forming the image by using one-sided diffracted light excluding the 0th beam. Thin wall width with high Faraday rotation yields a wall image with high contrast both in bright field and dark field configurations. Acknowledgements: Discussions with Drs. J.F. Dillon, Jr and Y. Imai are appreciated very much.
References
18--24 - -
~
i
,
i 0.05
[1] A. Thiaville, L. Amaud, F. Boileau, G. Sauron, and J. Miltat, IEEE Trans. Magn. 24 (1988) 1722; J. Theille and J. Engemann, Appl. Phys. Lett. 53 (1988) 713. [2] G.B. Scott and D.E. Lacklison, IEEE Trans. Magn. 12 (1976) 292; P. Paroli, Thin Solid Films, 114 (1984) 187. [3] A. Thiaville, J. Ben Youssef, Y. Nakatani, and J. Miltat, J. Appl. Phys. 69 (1991) 6090; K. Patek, A. Thiaville, and J. Miltat, Phys. Rev. B 49 (1994) 6678. [4] G. Tielian and H. Yuchuan, J. Magn. Magn. Mater. 35 (1983) 161. [5] H.M. Haskal, IEEE Trans. Magn. 6 (1970) 542.
~14 Journalof renalneusm magnetic ~i materials
ELSEVIER
Analysis by magnetic phase grating model for domain wall observation M. Ohkoshi *, K. Utsunomiya, K. Tsushima Faculty of Computer Science, Kyushu Institute of Technology, Fukuoka 820, Japan Abstract
The wall image formation associated with the light diffraction in Bi-substituted iron garnets is analyzed by a magnetic phase grating model. Thin wall width with high Faraday rotation yields the wall image with high contrast both in bright field and dark field configurations.
A polarized anisotropic dark field optical imaging technique (PADO) has been demonstrated to be able to reveal the location of Bloch lines within the bubble or stripe domain walls [1]. In spite of continued efforts on the diffraction effects due to magnetic domains [2], the subject is not completed yet and an optical diffraction contrast model has been proposed recently [3]. In this paper, the wall image formation in magnetic garnets with a perpendicular easy axis is analyzed by a magnetic phase grating model. Let the domain structure be a stripe form along the y-axis and the distribution of magnetization be trapezoidal along the x-axis in the film plane. The stripe domain period is d, and the wall width is w. The widths of up and down domains are s - w and d - s w, respectively, s and d are intended to be changed by a magnetic field applied normal to the film plane. Let the incident plane wave of unit amplitude with wavelength A propagate along the z-axis, the film normal. The Faraday rotation O(x) of the light transmitted through the film changes periodically along the x-axis across the antiparallel domains as + 0 F in the up domains and - 0 F in the down domains. In the wall, the Faraday rotation is assumed to change linearly between the domains alternately [4]. The optical phase difference between the adjacent antiparallel domains is 20 F after transmitted the film. The Faraday ellipticity is small and has little contribution to the following analysis. Then, the amplitude of the undiffracted 0th order light is
respectively, to the incident linear polarization direction, and w' and s' are normalized widths by the period d. In a demagnetized state where s / d = 1/2, the perpendicular component is zero as recognized from Eq. (1), and the undiffracted light is polarized completely parallel to the incident linear polarization. The perpendicular component of the 0th light intensity I 0_ is maximum of sin20F in the magnetic saturation state ( s / d = 0 or 1). Since the wall width is negligible compared to the domain width ( w ' = 0) in a standard uniaxial garnet (the quality factor Q > 1), we can safely suppose the parallel component of the 0th light intensity 1011 to be COS20F,independent of the applied field although it increases for thinner wall widths. Therefore, the intensity change of the undiffracted light by the applied field comes mainly from the perpendicular component. In a garnet of 90 ° Faraday rotation, the intensity of the undiffracted beam changes from 0 to 1 by the applied field without crossed polarizers. The diffracted ruth order lights due to the magnetic phase grating of the stripe form appear along x-axis at angles ~bm = s i n - l ( m A / d ) from the z-axis. The amplitude of the diffracted mth order light (m :~ 0) is obtained straightforwardly as u m = i [ w ' / ( m ~ r w ' - 0F) sin(marw' -- 0F) + w ' / ( m ~ r w ' + 0F) sin(m'trw' + 0F) - - ( 2 / m l r ) sin(m~rw') COS0F] cos(turfs') + j [ w ' / ( m ~ r w ' + 0F) sin(m'trw' + 0F) - - w ' / ( m ~ w ' - - 0F) sin(m'rrw' -- 0 r )
Uo = i[(2W'/OF) sin 0 F + (1 -- 2w') cos 0F] + j ( 1 - - 2 s ' ) sin 0F,
where i and j are unit vectors parallel and perpendicular,
* Corresponding author. Fax: +81-948-29-7651; [email protected].
- - ( 2 / m ~ ) cos(m'rrw') sin0F] sin(m'n's').
(1)
email:
(2)
Since w << d, the diffracted lights are all in perpendicular polarization. When s / d = 1 / 2 at zero field, even order (m ~ 0) diffractions with parallel polarization are absent in the limit of zero wall width. With increasing the applied field, the perpendicular component of the mth order disap-
0304-8853/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0304-8853(94)01387-X
M. Ohkoshiet al. /Journal of Magnetism and MagneticMaterials 140-144 (1995) 1845-1846
1846
0.3 ~
t.rm=+l--+~
eF(deg)24__ 18---
o
Fig. 1. Diffraction patterns representing thick wall and high Faraday rotation effects in crossed and parallel polarizations. pears when s / d is multiple integer of 1/m and takes maximum intensity when s / d = (integer + ½)/m. As an example of wall width contribution to the parallel diffraction, Fig. 1 shows isotropic diffraction patterns observed in a (BiTm)3Fe5012 garnet film with demagnetized maze-domains (Q = 0.3, d = 6.2 ixm, 0 F = 75°). In addition to the perpendicular polarized odd order diffracted lights, parallel polarized even order diffractions due to thick walls are separated by an analyzer for an linearly polarized incident beam. The magnetization distribution in this sample may be sinusoidal [5] rather than trapezoidal because of the low Q-value. Intensity distribution of the outgoing light on the image plane is calculated from
I(x)
=luo12+ E
nm exp(j2m~rx/d) 2.
(3)
m~0
By collecting all the diffracted lights including the undiffracted light (ideal bright field observation), the parallel component of the domain intensity /all is ~ COS20F, and the perpendicular component I a ± is ~ sin20F less depending on the domain width and wall width. Therefore, the wall contrast is bright for parallel (Iwl I = 1) and dark (I,,,± = 0) for crossed nicol as is well known. However, depending on the numerical aperture N A of the objective lens used for imaging, the image is constructed usually by at most a few tens of diffracted lights for domains of txm-size (I m i < N A d/A). Although IwH is nearly 1 for ,
0.85
0.8
,
,
,
,
,
,
I 0.02
,
I
I
I
0.04
I o.oq~
i
L 0.08
i 0.t
w/d
Fig. 2. Dependence of the wall intensity I w and the domain intensity I a on the normalized wall width w~ d for - 20 < ra < 20;
s/d=l/2.
i
i
0.1 w/d
0.15
0,2
Fig. 3. Dependence of the wall intensity 1w and the domain intensity I a on the normalized wall width w / d for one-sided diffracted beams 1 < m < 20; s / d = 1/2.
m > d/w, I w II decreases with the decrease of wall width, and reaches to COS20F at w / d = 0 as shown in Fig. 2. On the other hand, I w ± is always zero. Therefore, even for the unpolarized incident beam, the intensity of wall image Iw = Iwll + lw ± is always smaller than that of domains Ia =/all "l- I a ± ~ 1. Namely, domain walls can be observed as dark lines by conventional bright field microscopy without polarizer and analyzer. The intensity ratio Iw/I a decreases with decrease of w / d and increase of 0 F. When forming the image by using a part of diffracted beams, for instance, one-sided diffracted lights excluding 0th beam (PADO location), the walls can be observed as bright lines because I w > I a in this case. The light at wall is almost perpendicularly polarized. The perpendicular component of the wall image intensity I w.L is larger than that of domains I a ±. The I w ± increases with the decrease of w / d and the increase of 0 F as shown in Fig. 3. The parallel components lwlt and /all are much smaller than Iw± and I a ± , and both reach to zero for w = 0 . The intensity ratio Iw/I a increases with the decrease of w/d. In conclusion, domain walls can be observed as dark lines by conventional bright field microscopy without polarizer and analyzer, while they can be observed as bright lines when forming the image by using one-sided diffracted light excluding the 0th beam. Thin wall width with high Faraday rotation yields a wall image with high contrast both in bright field and dark field configurations. Acknowledgements: Discussions with Drs. J.F. Dillon, Jr and Y. Imai are appreciated very much.
References
18--24 - -
~
i
,
i 0.05
[1] A. Thiaville, L. Amaud, F. Boileau, G. Sauron, and J. Miltat, IEEE Trans. Magn. 24 (1988) 1722; J. Theille and J. Engemann, Appl. Phys. Lett. 53 (1988) 713. [2] G.B. Scott and D.E. Lacklison, IEEE Trans. Magn. 12 (1976) 292; P. Paroli, Thin Solid Films, 114 (1984) 187. [3] A. Thiaville, J. Ben Youssef, Y. Nakatani, and J. Miltat, J. Appl. Phys. 69 (1991) 6090; K. Patek, A. Thiaville, and J. Miltat, Phys. Rev. B 49 (1994) 6678. [4] G. Tielian and H. Yuchuan, J. Magn. Magn. Mater. 35 (1983) 161. [5] H.M. Haskal, IEEE Trans. Magn. 6 (1970) 542.