1. Introduction to Magneto-Optics
Katsuaki Sato
Department of Applied Physics
Tokyo University of Agriculture & Technology
ISOM2000 Tutorial

2. CONTENTS Introduction Light and Magnetism
What is the Magneto-Optical Effect?
Electromagnetism and Magneto-Optics
Electronic Theory
Measurement of Magneto-Optical Effect
Magneto-Optical Spectra
Recent Advances in Magneto-Optics
Summary

3. 1. Introduction Magneto-Optical Effect：Discovered by Faraday on 1845
Phenomenon：Change of Linear Polarization to Elliptically Polarized Light Accompanied by Rotation of Principal Axis
Cause：Difference of Optical Response between LCP and RCP
Application：
Magneto-Optical Disk
Optical Isolator
Current Sensors
Observation Technique

4. 2. Light and Magnetism Light→Magnetism：Photomagentic Effect
Thermomagnetic Effect：Curie pt. recording→MO disk
Light-induced Magnetization：ruby, DMS
Light-induced spin reorientation→Optical motor
Magnetism→Light：Magneto-Optical Effect
Shift or splitting of optical absorption line(Zeeman eff.)
Magnetic resonance：ESR, magneto-plasma effect
Magneto-optical effect(Faraday, Kerr, Cotton Mouton)

5. 3.What is the Magneto-Optical Effect?
MO Effect in Wide Meaning
Any change of optical response induced by magnetization
MO Effect in Narrow Meaning
Change of intensity or polarization induced by magentization
Faraday effect
MOKE(Magneto-optical Kerr effect)
Cotton-Mouton effect

6. 3.1 Faraday & Voigt Configurations
(a) Faraday Configuration:
Magnetization // Light Vector
(b)Voigt Configuration:
Magnetization  Light Vector

7. 3.2 Faraday Effect MO effect for optical transmission
Magnetic rotation（Faraday rotation）F
Magnetic Circular Dichroism（Faraday Ellipticity）F
Comparison to Natural Optical Rotation
Faraday Effect is Nonreciprocal (Double rotation for round trip)
Natural rotation is Reciprocal (Zero for round trip)
Verdet Constant
F=VlH (For paramagnetic and diamagnetic materials）

8. Illustration of Faraday Effect
For linearly polarized light incidence,
　Elliptically polarized light goes out (MCD)
With the principal axis rotated (Magnetic rotation)
Rotation of
Principal axis
Elliptically
Polarized light
Linearly polarized
light

9. 3.3 Faraday rotation of magnetic materials
(deg)
　figure of merit(deg/dB)
wavelength
(nm)
temperature
(K)
Mag. field
(T)
literature Fe
3.825･105
578 RT
2.4
1.11) Co
1.88･105
546 〃 2 Ni
1.3･105
826
120 K
0.27
Y3Fe5O12
250
1150
100 K
1.12)
Gd2BiFe5O12
1.01･104 44
800
1.13)
MnSb
2.8･105
500
1.14)
MnBi
5.0･105
1.43
633
1.15)
YFeO3
4.9･103
1.16)
NdFeO3
4.72･104
1.17)
CrBr3
1.5K
1.18)
EuO
5･105
104
660
4.2 K
2.08
1.19)
CdCr2S4
3.8･103
35(80K)
1000 4K
0.6
1.20)

11. 3.4 Magneto-Optical Kerr Effect
Three kinds of MO Kerr effects
Polar Kerr（Magnetization is oriented perpendicular to the suraface）
Longitudinal Kerr（Magnetization is in plane and is parallel to the plane of incidence）
Transverse Kerr （Magnetization is in plane and is perpendicular to the plane of incidence）

13. 3.5 MO Kerr rotation of magnetic materials
Photon energy
temperature
field
literature
(deg)
(eV)
(K)
(T) Fe
0.87
0.75 RT
1.21) Co
0.85
0.62 〃 Ni
0.19
3.1 Gd
0.16
4.3
1.22)
Fe3O4
0.32 1
1.23)
MnBi
0.7
1.9
1.24)
PtMnSb
2.0
1.75
1.7
1.8)
CoS2
1.1
0.8
4.2
0.4
1.25)
CrBr3
3.5
2.9
1.26)
EuO 6
2.1 12
1.27)
USb0.8Te0.2
9.0 10
4.0
1.28)
CoCr2S4
4.5 80
1.29)
a-GdCo *
0.3
1.30)
CeSb 90 2
1.31)
* "a-" means "amorphous".

14. 4. Electromagnetism and Magnetooptics
Light is the electromagnetic wave.
Transmission of EM wave：Maxwell equation
Medium is regareded as continuum→dielectric permeability tensor
Effect of Magnetic field→mainly to off-diagonal element
Eigenequation
→Complex refractive index：two eigenvalues
eigenfunctions：right and left circularpolarization
Phase difference between RCP and LCP→rotation
Amplitude difference →circular dichroism

15. 4.1 Dielectric tensor Isotromic media；M//z
Invariant C4 for 90°rotation around z-axis

16. 4.2 MO Equations (1) Maxwell Equation Eigenequation Eigenvalue
Eigenfunction：LCP and RCP
Without off-diagonal terms：No difference between LCP & RCP
No magnetooptical effect

17. MO Equations (2) Both diagonal and off-diagonal terms contribute to
Magneto-optical effect

18. 4.3 Phenomenology of MO effect
Linearly polarized light can be decomposed to LCP and RCP
Difference in phase causes rotation of
the direction of Linear polarization
Difference in amplitudes makes
Elliptically polarized light
In general, elliptically polarized light
With the principal axis rotated

23. 5. Electron theory of Magneto-Optics
Magnetization→Splitting of spin-states
No direct cause of difference of optical response between LCP and RCP
Spin-orbit interaction→Splitting of orbital states
Absorption of circular polarization→Induction of circular motion of electrons
Condition for large magneto-optical response
Presence of strong (allowed) transitions
Involving elements with large spin-orbit interaction
Not directly related with Magnetization

24. 5.1 Microscopic concepts of electronic polarization+ + -
Wavefunction
perturbed by
electric field
Unperturbed
wavefunction + - = + +
+　・・
S-like
P-like
Expansion by unperturbed orbitals

25. 5.2 Orbital angular momentum-selection rules and circular dichroism
py-orbital
px-orbital
p+=px+ipy
Lz=+1
Lz=-1
p-=px-ipy
Lz=0
s-like

26. 5.3 Role of Spin-Orbit Interaction
Jz=-3/2
Jz=-1/2
L=1
Jz=+1/2
LZ=+1,0,-1
Jz=+3/2
Jz=-1/2
L=0
Jz=+1/2
LZ=0
Exchange
+spin-orbit
Without magnetization
Exchange splitting

27. 5.4 MO lineshapes (1) 1.Diamagnetic lineshape Excited state
Ground state 0 1 2 
Without
magnetization
With
Lz=0
Lz=+1
Lz=-1
1+2
Photon energy
’xy
”xy
1.Diamagnetic lineshape

28. 5.4 MO lineshapes (2) excited state ground state f+ f-  f=f+ - f- 0
without magnetic
field
with magnetic
’xy
”xy
photon energy
(a)
(b)
dielectric constant

29. 6. Measurement of MO effect
Cross-polarizer technique
Vibrating polarizer technique
Rotating analyzer technique
Faraday modulation technique
Optical retardation modulation
Measuring system for MO spectrum
Measurement of elleipticity

30. 6.1 Cross-Nicol techniqueP B A D P F A I
P=A+/2
/4 rotation
/2 rotation
 rotation
(a)
(b) S

31. 6.2 Vibrating polarizer techniqueF
+F A D ID S

32. 6.3 Rotating analyzer techniqueP A D S B E F
A=pt ID

33. 6.4 Faraday modulation technique
Faraday modulator P
=0+sin pt B S A D
I=I0+ I sin pt F ID
Zero method

34. 6.5 Retardation modulation techniquej
/4 P
PEM A D
quartz
Isotropic
medium B
fused silica
CaF2
Ge etc.
Piezoelectric
crystal
amplitude
position l
Retardation
=(2/)nl sin pt
=0sin pt

35. 6.6 Spectral measurement M1 L MC PEM S P C (f Hz) M2 A D LA1 (f Hz)
(p Hz) S
Electromagnet D
Preamplifier
LA1 (f Hz)
LA2 (p Hz)
LA3 (2p Hz)

36. 6.7 Measurement of ellipticityx’ y’ E’
E0sinh y E0 h E x h x
Optic
axis
E0cosh
l/4plate

37. 7. MO spectra of materials
Magnetic garnets
Metallic ferromagnet：Fe, Co, Ni
Intermetallic compounds and alloys：PtMnSb etc.
Magnetic semiconductor：CdMnTe etc.
Superlattices：Pt/Co, Fe/Au etc.
Amorphous：TbFeCo, GdFeCo etc.
Granular：Al2O3:Coなど

38. Theory and experiment of MO spectra in Fe
Katayama
theory

39. MO spectra of PtMnSb
カー回転と楕円率
誘電率対角成分
誘電率非対角成分
(a)
(b)
(c)

40. MO spectra in RE-TM (1)
Wavelength (nm)
Polar Kerr rotation (min)

41. MO spectra in RE-TM(2) Photon Energy (eV) -0.2 -0.4 -0.65 4 3 2
Photon Energy (eV)
-0.2
-0.4
-0.6
Polar Kerr rotation (deg)
Wavelength (nm)
300
400
500
600
700

42. Recent Advances in Magneto-Optics
Scanning Near Field Magneto-Optical Microscope (MO-SNOM)
Nonlinear Magneto-Optics
Sagnac Magneto-Optical Microscope
X-ray Magneto-Optical Imaging

43. SUMMARY Basic concept of magneto-optics is described.
Macroscopic and microscopic origins of magneto-optics are described.
Some of the recent development of magneto-optics is also given.