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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 3 — Feb. 11, 2013
  • pp: 3400–3416
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Analytical analysis of a multilayer structure with ultrathin Fe film for magneto-optical sensing

Š. Višňovský, E. Lišková-Jakubisová, I. Harward, and Z. Celinski  »View Author Affiliations


Optics Express, Vol. 21, Issue 3, pp. 3400-3416 (2013)
http://dx.doi.org/10.1364/OE.21.003400


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Abstract

Magneto-optic (MO) response in nanostructures with ultrathin Fe considered for the MO mapping of current pulses with a two-dimensional diffraction limited resolution is investigated in detail. The structures consist of an ultrathin Fe layer sandwiched with dielectric layers, deposited on a reflector and covered by a noble metal protecting layer. The structures are modeled as five-layer systems with abrupt interfaces. Analytical expressions are provided that are useful in the search for the maximum of MO reflected wave amplitude polarized perpendicular to the incident linearly polarized wave, |ryx(05)|. The procedure of finding the maximal |ryx(05)| is illustrated on the structures with ultrathin Fe at the laser wavelength of 632.8 nm. The maximal |ryx(05)| of 0.018347 was achieved in the structure AlN(52 nm)/Fe(15 nm)/AlN(26 nm)/Au. The deposition of a 5 nm protecting Au layer reduced |ryx(05)| by 6 per cent.

© 2013 OSA

1. Introduction

It has been proposed to employ magneto-optical (MO) effects in bismuth and gallium substituted iron garnet films, (YBiLuPr)3(FeGa)5O12, (Bi-YIGaG) epitaxially grown on optically transparent gadolinium gallium garnet Gd3Ga5O12 (GGG) substrates as probes for contactless and non-invasive metrology of microwave (mw) currents in integrated circuits with diffraction limited resolution [1

1. A. Y. Elezzabi and M. R. Freeman, “Ultrafast magneto-optic sampling of picosecond current pulses,” Appl. Phys. Lett. 68(25), 3546–3548 (1996). [CrossRef]

3

3. J. Y. Hwang, M. Ferrera, L. Razzari, A. Pignolet, and R. Morandotti, “The role of Bi3+ ions in magneto-optic Ce and Bi comodified epitaxial iron garnet films,” Appl. Phys. Lett. 97(16), 161901 (2010). [CrossRef]

]. The mw currents generate fringing magnetic fields which cause the magnetization in an adjacent Bi-YIGaG film to precess. The precessional angle is proportional to the magnitude of the mw magnetic field [4

4. H. Suhl, “Origin and use of instabilities in ferromagnetic resonance,” J. Appl. Phys. 29(3), 416–421 (1958). [CrossRef]

]. The magnetization component perpendicular to the film surface can be detected with linear (in magnetization) magneto-optical (MO) effects leading to linearity between the currents and MO response. As the current-carrying circuits are optically non-transparent, it is more convenient to perform the MO detection in reflection. The operation of such a MO probe requires applied in-plane static magnetic flux density fields, Bappl, biasing garnet films close to the ferromagnetic resonance at a desired microwave frequency. With the present semiconductor chips the bandwidth of which exceeds 100 GHz, Bappl may reach several Tesla.

The cost, total probe thickness and requirements for Bappl can be reduced by using ultrathin ferromagnetic iron layer with the saturation magnetization, µ0Ms = 2.158 Tesla [5

5. S. Chikazumi, Physics of Magnetism (Oxford Science Publications, Clarendon Press, 1997).

], instead of Bi-YIGaG films (about 3 µm thick) on 500 µm GGG substrates where µ0Ms is more than two orders in magnitude smaller [1

1. A. Y. Elezzabi and M. R. Freeman, “Ultrafast magneto-optic sampling of picosecond current pulses,” Appl. Phys. Lett. 68(25), 3546–3548 (1996). [CrossRef]

,6

6. R. Lopusnik, J. Correu, I. Harward, S. Widuch, P. Maslankiewicz, S. Demirtas, Z. Celinski, E. Liskova, M. Veis, and S. Visnovsky, “Optimization of magneto-optical response of FeF2/Fe/ FeF2 sandwiches for microwave field detection,” J. Appl. Phys. 101(9), 09C516 (2007). [CrossRef]

]. Indeed, according to Kittel [7

7. C. Kittel, “On the theory of ferromagnetic resonance absorption,” Phys. Rev. 73(2), 155–161 (1948). [CrossRef]

], for a plate with in-plane µ0Ms the required in-plane Bappl for the ferromagnetic resonance (FMR) frequency, f, is given by
f=γ2π[Bappl(Bappl+μ0Ms)]1/2,
(1)
where γ≈2π × 28 GHz/Tesla. Figure 1
Fig. 1 Ferromagnetic resonance frequency, f, vs. magnetic flux density field, Bappl, applied in-plane on the plates with the saturation magnetizations µ0Ms = 0.0160 Tesla (gallium substituted yttrium iron garnet, GaYIG) and µ0Ms = 2.158 Tesla (Fe).
illustrates the achieved reduction of Bappl with respect to that in Bi YIGaG films [1

1. A. Y. Elezzabi and M. R. Freeman, “Ultrafast magneto-optic sampling of picosecond current pulses,” Appl. Phys. Lett. 68(25), 3546–3548 (1996). [CrossRef]

]. At zero input, the strong shape anisotropy in ultrathin Fe (with anisotropy fields induced by growth, magnetostriction, surface effects, etc., negligible) stabilizes in-plane magnetization, M||. The low Fe coercivity allows for an easy saturation of the Fe film and manipulation of the in-plane orientation of M|| with Bappl. The MO activity in iron is reasonably high, but unlike in iron garnet oxides [8

8. C. F. Buhrer, “Faraday rotation and dichroism of bismuth calcium vanadium iron garnet,” J. Appl. Phys. 40(11), 4500–4502 (1969). [CrossRef]

,9

9. S. Wittekoek, T. J. A. Popma, J. M. Robertson, and P. F. Bongers, “Magneto-optic spectra and the dielectric tensor elements of bismuth-substituted iron garnets at photon energies between 2.2-5.2 eV,” Phys. Rev. B 12(7), 2777–2788 (1975). [CrossRef]

], always accompanied with a strong optical absorption [10

10. G. S. Krinchik and V. A. Artem’ev, “Magneto-optical properties of Ni, Co and Fe in ultraviolet visible and infrared parts of spectrum,” Sov. Phys. JETP 26, 1080–1085 (1968).

12

12. P. B. Johnson and R. W. Christy, “Optical constants of transition metals: Ti, V, Cr, Mn, Fe, Co, Ni, and Pd,” Phys. Rev. B 9(12), 5056–5070 (1974). [CrossRef]

].

While the chemically inert refractory garnet oxides can be exposed to the ambient, the ultrathin iron layer requires sandwiching between protecting layers. In addition to their protecting function, the sandwiching layers must enable efficient transfer of the MO signal to the detector. Consequently, they should be transparent at the operating laser wavelength and the sandwiched Fe layer must be deposited on an appropriate reflector. The material choice for the sandwiching layers is restricted to dielectrics, preferably oxygen free. The dielectrics must be compatible with Fe and the reflector as for chemical inertness and deposition temperature. Optically thick Au, Ag, Cu, Pt, or Al layer can be used as the reflectors. Substrates for the probe must be reasonable flat and stable. Commercially available Si or GaAs chips represent a suitable choice. Adding a noble metal (Au) capping layer on top of the probe improves the protection against ambient effects. The probe then may consist of a sequence of five layers (Fig. 2
Fig. 2 Diagram of the multilayer system operating as a magneto-optic probe.
).

Two multilayers were grown and their MO response was tested experimentally. One system designed for the operating wavelength of 810 nm (Ti:sapphire laser) employed the layer sequence Au/FeF2/Fe/FeF2/Ag/GaAs [6

6. R. Lopusnik, J. Correu, I. Harward, S. Widuch, P. Maslankiewicz, S. Demirtas, Z. Celinski, E. Liskova, M. Veis, and S. Visnovsky, “Optimization of magneto-optical response of FeF2/Fe/ FeF2 sandwiches for microwave field detection,” J. Appl. Phys. 101(9), 09C516 (2007). [CrossRef]

,13

13. J. Pištora, M. Lesňák, E. Lišková, Š. Višňovský, I. Harward, P. Maslankiewicz, K. Balin, Z. Celinski, J. Mistrík, T. Yamaguchi, R. Lopusnik, and J. Vlček, “The effect of FeF2 on the magneto-optic response in FeF2/Fe/FeF2 sandwiches,” J. Phys. D Appl. Phys. 43(15), 155301 (2010). [CrossRef]

]. Note that FeF2 required the protection with an Au capping. The second one designed for the operating wavelength of 410 nm employed the layer sequence AlN/Fe/AlN/Cu/Si [14

14. E. Liskova, S. Visnovsky, R. Lopusnik, I. Harward, M. Wenger, T. Christensen, and Z. Celinski, “Magneto-optical AlN/Fe/AlN structures optimized for operation in the violet spectral region,” J. Phys. D Appl. Phys. 41(15), 155007 (2008). [CrossRef]

]. Prior the deposition, the MO response was evaluated using a transfer matrix model [15

15. P. Yeh, “Electromagnetic propagation in birefringent layered media,” J. Opt. Soc. Am. 69(5), 742–756 (1979). [CrossRef]

,16

16. Š. Višňovský, “Magneto-optical ellipsometry,” Czech. J. Phys. B 36(5), 625–650 (1986). [CrossRef]

]. The degree to which the desired MO performance was achieved in the grown sandwiches was evaluated with MO spectroscopy. The sandwiching of the ultrathin iron with dielectrics transformed the smooth MO polar Kerr effect spectrum with broad lines of bulk Fe [6

6. R. Lopusnik, J. Correu, I. Harward, S. Widuch, P. Maslankiewicz, S. Demirtas, Z. Celinski, E. Liskova, M. Veis, and S. Visnovsky, “Optimization of magneto-optical response of FeF2/Fe/ FeF2 sandwiches for microwave field detection,” J. Appl. Phys. 101(9), 09C516 (2007). [CrossRef]

,13

13. J. Pištora, M. Lesňák, E. Lišková, Š. Višňovský, I. Harward, P. Maslankiewicz, K. Balin, Z. Celinski, J. Mistrík, T. Yamaguchi, R. Lopusnik, and J. Vlček, “The effect of FeF2 on the magneto-optic response in FeF2/Fe/FeF2 sandwiches,” J. Phys. D Appl. Phys. 43(15), 155301 (2010). [CrossRef]

,14

14. E. Liskova, S. Visnovsky, R. Lopusnik, I. Harward, M. Wenger, T. Christensen, and Z. Celinski, “Magneto-optical AlN/Fe/AlN structures optimized for operation in the violet spectral region,” J. Phys. D Appl. Phys. 41(15), 155007 (2008). [CrossRef]

] to a MO spectrum with sharp peaks in both MO polar Kerr rotation and ellipticity located close to a desired laser wavelength.

In the past, quadri-layers formed by a magnetic layer sandwiched between dielectrics on an Al reflector received attention as media for high capacity MO recording disks [17

17. Y. Tomita and T. Yoshino, “Optimum design of multilayer-medium structures in a magneto-optical readout system,” J. Opt. Soc. Am. A 1(8), 809–817 (1984). [CrossRef]

20

20. M. Mansuripur, Physical Principles of Magneto-optical Recording (Cambridge University Press, 1996).

]. The magnetic layer was most often an amorphous 3d-4f alloy with magnetic perpendicular anisotropy and a strong temperature dependence of coercivity. Two opposite orientations of the local magnetization on the disk represented logic zeros and ones. The magnetic and MO properties had to be uniform in the whole disk area. There were studies on the enhancement of MO effects in magnetic sandwiches motivated by other applications [21

21. N. Qureshi, H. Schmidt, and A. R. Hawkins, “Cavity enhancement of the magneto-optic Kerr effect for optical studies of magnetic nanostructures,” Appl. Phys. Lett. 85(3), 431–433 (2004). [CrossRef]

,22

22. M. Moradi and M. Ghanaatshoar, “Cavity enhancement of the magneto-optic Kerr effect in glass/Al/SnO2/PtMnSb/SnO2 structure,” Opt. Commun. 283(24), 5053–5057 (2010). [CrossRef]

]. Here we discuss similar layer sequences, but with different requirements and objectives.

2. Figure of merit

We look for a set of individual layer thicknesses in a sandwich consisting of five layers Au(1)/dielectrics(2)/Fe(3)/dielectrics(4)/reflector(5) at which the Fe saturation magnetization z-component normal to the sandwich surface provides the maximal MO response (Fig. 2). Upon reflection at normal light incidence, an incident wave propagating parallel to the z-axis and linearly polarized parallel to the x-axis is partially reflected with the same x-polarization, partially absorbed in Fe, and partially transmitted into the optically thick reflector. A fraction of electron transitions responsible for the absorption in Fe are MO active. They produce the desirable mode conversion and generate transmitted and reflected waves with linear y-polarization.

The efficiency of the MO mode conversion in a magnetic layer is evaluated with |ryx|, defined by Mansuripur as MO figure of merit (FOM) [23

23. M. Mansuripur, “Figure of merit for magneto-optical media based on the dielectric tensor,” Appl. Phys. Lett. 49(1), 19–21 (1986). [CrossRef]

]. Here ryx, an off-diagonal element of the Cartesian Jones reflection matrix,
[Ex(r)Ey(r)]=[rxxryxryxrxx][Ex(i)Ey(i)],
(2)
measures the y-polarized reflected wave amplitude induced with an incident linearly x-polarized wave of unit amplitude. At a specific thickness of the magnetic layer, d(m), the element ryx becomes maximal and can be further enhanced by sandwiching magnetic layer with dielectric layers and by depositing the sandwich on an appropriate reflector. Similarly, the Jones transmission matrix characterizes the optical response in transmission

[Ex(t)Ey(t)]=[txxtyxtyxtxx][Ex(i)Ey(i)].
(3)

The matrices relate the electric field x- and y-components, Ex and Ey of the incident (i) and reflected (r) or transmitted (t) waves propagating parallel to the z-axis.

An ideal |ryx| corresponds to the situation where the incident x-polarized wave is absorbed in Fe (rxx→0) and converted to y-polarized wave with no power transmitted into the reflector. Mansuripur estimated this ideal upper limit for the optical frequency (ω) dependent |ryx(ω)| assuming linear dependence on the thickness d(m) of the ultrathin ferromagnetic layer (m) of both |ryx| and A, the absorbed power [23

23. M. Mansuripur, “Figure of merit for magneto-optical media based on the dielectric tensor,” Appl. Phys. Lett. 49(1), 19–21 (1986). [CrossRef]

]

|ryx(ω)|max=|εxy(m)(ω,M)2Im[εxx(m)(ω)]|.
(4)

The criterion depends exclusively on the material parameters of the magnetic layer but due to the approximations involved overestimates the upper limit for FOM. A more realistic upper limit for |ryx(ω)| would account for the nonlinear dependence on d (m) of both |ryx| and A. We show below that with our restriction to the structures with five layers at most, we cannot completely eliminate the power leak into the reflector.

3. Reflected and transmitted waves

The Helmholtz equation for the optical wave electric field vector E
2E(.E)+ω2c2ε(m)E=0
(5)
in a uniformly magnetized medium (m) with the magnetization vector, M, parallel to the z-axis and characterized by the optical permittivity tensor,
ε(m)=[εxx(m)εxy(m)0εxy(m)εxx(m)000εzz(m)],
(6)
becomes
(Nz2Exεxx(m)Exεxy(m)Ey)x^+(Nz2Ey+εxy(m)Exεxx(m)Ey)y^εzz(m)Ezz^=0
(7)
for plane waves propagating parallel to M||z^, with the propagation vector, z^ωcNz(m).

The four values for the propagation vector, z^ωcNz(m), i.e., ±z^ωcN+(m)and ±z^ωcN(m)of the four circularly polarized (CP) waves follow from the eigen value equationN±(m)2=εxx(m)±jεxy(m).Their vector electric fields are given by [24

24. Š. Višňovský, Optics in Magnetic Multilayers and Nanostructures (CRC Taylor & Francis, 2006), Chapter 4, pp. 198–211.

]
E1,2(m)=21/2(x^+jy^)E1,2(m)exp[jω(τN+(m)z/c)],E3,4(m)=21/2(x^jy^)E3,4(m)exp[jω(τN(m)z/c)].
(8)
Here 21/2(x^+jy^) and 21/2(x^jy^) are the unit vectors of the + and CP wave electric fields, respectively. The scalar eigen mode amplitudes of the + and CP waves are represented as E1,2(m) andE3,4(m), respectively; τ and c denote the time and the light velocity in a vacuum. In the non-magnetic media (n), the solutions may also be taken as + and CP waves where the complex index of refraction N+(n)=N(n)=N(n).

With the Fresnel reflection and transmission coefficients
r±(i,i+1)=N±(i)N±(i+1)N±(i)+N±(i+1)=r±(i+1,i),t±(i,i+1)=2N±(i)N±(i)+N±(i+1),t±(i+1,i)=2N±(i+1)N±(i)+N±(i+1),
(9)
and β±(k)=N±(k)d(k)ω/c, where d(k) denotes the thickness of the k-th layer, the reflection, r±(05)=E±(r)/E±(i), and transmission, t±(05)=E±(t)/E±(i), coefficients for the electric field of + and CP waves in the five-layer sandwich can be written as [25

25. F. Abelès, “Recherches sur la propagation des ondes électromagnétiques sinusoïdales dans les milieux stratifiés. Application aux couches minces,” Ann. Phys. (Paris) 5, 596–640, 706–782 (1950).

27

27. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (Elsevier, 1987).

]
r±(05)=r(01)+e2jβ(1)r(12)+r±(25)e2jβ(2)1+r(12)r±(25)e2jβ(2)1+r(01)e2jβ(1)r(12)+r±(25)e2jβ(2)1+r(12)r±(25)e2jβ(2),
(10)
and
t±(05)=t(01)ejβ(1)t(12)t±(25)ejβ(2)1+r(12)r±(25)e2jβ(2)1+r(01)e2jβ(1)r(12)+r±(25)e2jβ(2)1+r(12)r±(25)e2jβ(2).
(11)
Here
r±(25)=r±(23)+e2jβ±(3)r±(34)+r(45)e2jβ(4)1+r±(34)r(45)e2jβ(4)1+r±(23)e2jβ±(3)r±(34)+r(45)e2jβ(4)1+r±(34)r(45)e2jβ(4),
(12)
and
t±(25)=t±(23)ejβ±(3)t±(34)t(45)ejβ(4)1+r±(34)r(45)e2jβ(4)1+r±(23)e2jβ±(3)r±(34)+r(45)e2jβ(4)1+r±(34)r(45)e2jβ(4).
(13)
It is useful to express the transmission coefficients for the magnetic field, t±(50)=H±(t)/H±(i), of + and CP waves
t±(50)=t(10)ejβ(1)t(21)t±(52)ejβ(2)1+r(12)r±(25)e2jβ(2)1+r(01)e2jβ(1)r(12)+r±(25)e2jβ(2)1+r(12)r±(25)e2jβ(2),
(14)
where
t±(52)=t±(32)ejβ±(3)t±(43)t(54)ejβ(4)1+r±(34)r(45)e2jβ(4)1+r±(23)e2jβ±(3)r±(34)+r(45)e2jβ(4)1+r±(34)r(45)e2jβ(4).
(15)
The elements of the Cartesian Jones reflection (r) and transmission (t) matrices are given by
rxx(05)=12(r+(05)+r(05)),ryx(05)=12j(r+(05)r(05)),txx(05)=12(t+(05)+t(05)),tyx(05)=12j(t+(05)t(05)),txx(50)=12(t+(50)+t(50)),tyx(50)=12j(t+(50)t(50)).
(16)
To first order in (N+(3)N(3)), the off-diagonal element of the Jones reflection matrix, ryx(05), becomes
ryx(05)εyx(3)4εxx(3)t(03)t(30)[4jβ(3)e2jβ(3)r(35)+(1e2jβ(3))(1+r(35)2e2jβ(3))](1+r(30)r(35)e2jβ(3))2,
(17)
where
t(03)t(30)=(1r(01)2)(1r(12)2)(1r(23)2)e2jβ(1)e2jβ(2)[(1+r(01)r(12)e2jβ(1))+r(23)e2jβ(2)(r(01)e2jβ(1)+r(12))]2,
(18)
r(30)=r(23)+r(01)e2jβ(1)+r(12)1+r(01)r(12)e2jβ(1)e2jβ(2)1+r(01)e2jβ(1)+r(12)1+r(01)r(12)e2jβ(1)r(23)e2jβ(2),
(19)
and
r(35)=r(34)+r(45)e2jβ(4)1+r(34)r(45)e2jβ(4)
(20)
with t±(i,i+1)t±(i+1,i)=1r±(i,i+1)2.

The absolute value |ryx(05)| represents the MO reflected wave amplitude y-polarized perpendicular to the incident linearly x-polarized wave. We give also the off-diagonal element of the Cartesian Jones transmission matrix, tyx(05), in the same approximation, i.e., to first order in (N+(3)N(3)),
tyx(05)t(01)t(12)ejβ(1)(1+r(01)r(12)e2jβ(1))t(34)t(45)ejβ(4)(1+r(34)r(45)e2jβ(4))×t(23)ej(β(2)+β(3))(1+r(30)r(35)e2jβ(3))2(1+r(20)r(23)e2jβ(2))×εyx(3)4εxx(3)[2jβ(3)(1r(30)r(35)e2jβ(3))(1e2jβ(3))(r(30)r(35))],
(21)
where
r(20)=r(01)e2jβ(1)+r(12)1+r(01)r(12)e2jβ(1).
(22)
For the considered sandwich profiles, its MO FOM, |ryx(05)|,depends on d(1) of the capping layer, d(2) of upper sandwiching dielectrics and on thicknesses of the ultrathin ferromagnetic Fe, d(3), and lower dielectrics, d(4). The problem of finding a set of thicknesses {d(1), d(2), d(3), d(4)} which provides the maximal |ryx(05)| can be solved with a computer. In practice, the analytical approach proposed here can provide a better insight into the realistic probe performance where individual layer thicknesses and their optical constants may deviate from the nominal ones due to the uncertainties in the deposition procedure and unavoidable roughness at interfaces.

We observe that the linearized ryx(05)in Eq. (17) consists of four factors. The optical and MO characteristics of the ultrathin Fe layer at the operating wavelength are given asF0=εyx(3)4εxx(3). The factor F1=t(03)t(30)depends on d(1) and d(2). Note that the product t(03)t(30)*of t(03) with the complex conjugated t(30)provides the power flux delivered across the front interface of an optically thick magnetic layer (3) per unit incident power flux. The factor F2=4jβ(3)e2jβ(3)r(35)+(1e2jβ(3))(1+r(35)2e2jβ(3)) depends on d(3) and d(4). At the other parameters fixed, there exist d(3) of the order of the penetration depth for which the absolute value of this factor reaches its maximum. This is due to the differences in the d(3) dependence of the terms 4jβ(3)e2jβ(3)r(35) and (1e2jβ(3))(1+r(35)2e2jβ(3)). Note that r(35) in Eq. (20) characterizes the reflection at the back interface of the magnetic layer. The denominator factor F3=(1+r(30)r(35)e2jβ(3))2 depends on the thicknesses of all layers. The structure of the linearized tyx(05)can be understood in a similar way.

The azimuth rotation, θr(05),and ellipticity, εr(05), in reflection are defined with help of the MO ellipsometric ratio in reflection χr(05)=ryx(05)/rxx(05)as [27

27. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (Elsevier, 1987).

]
θr(05)=12arctan2{χr(05)}1|χr(05)|2,
(23)
εr(05)=12arcsin2Im{χr(05)}1+|χr(05)|2,
(24)
at the linear x-polarized incident wave specified with χi(05)0. The parameters θr(05) and εr(05)are linear inχr(05) in a restricted range of small values of χr(05). We shall therefore evaluate them at 10% of the saturation value of Ms, i.e., at 10% of the saturation value of ryx(05)Ms.

4. Response at 632.8 nm

The procedure leading to the maximal value of the MO response, ryx(05), will be now illustrated on a sandwich Au(d(1))/AlN(d(2))/Fe(d(3))/AlN(d(4))/Au(d(5)) at the wavelength, λ = 632.8 nm. The thickness of the capping Au will be set to d(1) = 0 for dielectrics inert in the ambient. In the other cases, it is assumed that a reasonable protection against ambient effects can be achieved with d(1) = 5 nm. We set the Au substrate optically thick with d(5) = 1000 nm. At λ = 632.8 nm, we characterize AlN as a nonabsorbing dielectrics with a real index of refraction, N(AlN) = ε(AlN)1/2 = 2.00. The actual value of N(AlN) in sandwiches depends on the deposition conditions [28

28. D. Brunner, H. Angerer, E. Bustarret, F. Freudenberg, R. Höpler, R. Dimitrov, O. Ambacher, and M. Stutzmann, “Optical constants of epitaxial AlGaN films and their temperature dependence,” J. Appl. Phys. 82(10), 5090–5096 (1997). [CrossRef]

]. In Table 1

Table 1. Optical and magneto-optical parameters of Fe

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, we summarize the optical and magneto-optical parameters of Fe [10

10. G. S. Krinchik and V. A. Artem’ev, “Magneto-optical properties of Ni, Co and Fe in ultraviolet visible and infrared parts of spectrum,” Sov. Phys. JETP 26, 1080–1085 (1968).

12

12. P. B. Johnson and R. W. Christy, “Optical constants of transition metals: Ti, V, Cr, Mn, Fe, Co, Ni, and Pd,” Phys. Rev. B 9(12), 5056–5070 (1974). [CrossRef]

], i.e., the complex relative permittivity, εxx(Fe)/ε(vac), the complex index of refraction, N(Fe) = εxx(Fe)1/2/ε(vac)1/2, the penetration depth, d(Fe)penetr, reflectivity R(Fe) at d(Fe) = 1000 nm, and |ryx|max0.0188computed from Eq. (4). The corresponding optical parameters of Au [29

29. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972). [CrossRef]

] are summarized in Table 2

Table 2. Optical parameters of Au

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.

This is a considerable improvement with respect to |ryx(05)|=0.006413 at a planar interface between air and optically thick Fe where the incident power is nearly equally distributed between the R and A parts (Table 3). The figure of merit of the thick Fe can be improved in two ways, either with an ultrathin Fe layer (d(3) = 23 nm) on an Au substrate with |ryx(05)|=0.00958, or better by depositing an AlN layer (d(2) = 61 nm) on the optically thick Fe |ryx(05)|=0.0126. The combination of these remedies in the structure AlN(57 nm)/Fe(23 nm)/Au(1000 nm) with the Fe layer situated below the AlN dielectrics provides |ryx(05)|=0.01797, a value rather close to the maximal |ryx(05)| achieved in the sandwiches. With the Fe layer situated above the AlN dielectrics, i.e., with the Fe layer exposed to the ambient, in the structure Fe(6 nm)/AlN(56 nm)/Au(1000 nm), the figure of merit reaches |ryx(05)|=0.01804. The details are listed in Table 3 along with other remarkable examples of the sandwich structures with |ryx(05)|>0.0183.

We now return to the sandwich structure AlN(50 nm)/Fe(14 nm)/AlN(26 nm)/Au(1000 nm) with |ryx(05)|=0.018344 and consider the effect of d(2), d(3), and d(4) on the probe performance. Figure 3
Fig. 3 Effect of the thickness of the Fe layer, d(3), on the parameters of the structure AlN(50 nm)/Fe(d(3))/AlN(26 nm)/Au(1000 nm): (a) power reflection (R), transmission (T), and absorption (A) coefficients, (b) off-diagonal amplitude reflection coefficientryx(05)=|ryx(05)|exp[jarg(ryx(05))], (c) diagonal amplitude reflection coefficientrxx(05)=|rxx(05)|exp[jarg(rxx(05))], and (d) magneto-optic azimuth rotation, θr(05), and ellipticity, εr(05)at 10% magnetic saturation.
shows the effect of d(3) (thickness of Fe layer) on the parameters of the structure AlN(50 nm)/Fe(d(3))/AlN(26 nm)/Au(1000 nm).

In Figs. 4
Fig. 4 Effect of the thickness of the upper AlN layer, d(2), on the parameters of the structure AlN(d(2))/Fe(14 nm)/AlN(26)/Au(1000 nm): (a) power reflection (R), transmission (T), and absorption (A) coefficients, (b) off-diagonal amplitude reflection coefficientryx(05)=|ryx(05)|exp[jarg(ryx(05))], (c) diagonal amplitude reflection coefficientrxx(05)=|rxx(05)|exp[jarg(rxx(05))], and (d) magneto-optic azimuth rotation, θr(05), and ellipticity, εr(05)at 10% magnetic saturation.
and 5
Fig. 5 Effect of the thickness of the lower AlN layer, d(4), on the parameters of the structure AlN(50 nm)/Fe(14 nm)/AlN(d(4))/Au(1000 nm): (a) power reflection (R), transmission (T), and absorption (A) coefficients, (b) off-diagonal amplitude reflection coefficient ryx(05)=|ryx(05)|exp[jarg(ryx(05))], (c) diagonal amplitude reflection coefficientrxx(05)=|rxx(05)|exp[jarg(rxx(05))], and (d) magneto-optic azimuth rotation, θr(05), and ellipticity, εr(05)at 10% magnetic saturation.
, we illustrate the effect of d(2) (thickness of upper AlN layer) and d(4) (thickness of lower AlN layer) on the parameters of the structures AlN(d(2))/Fe(14 nm)/AlN(26 nm)/Au(1000 nm) and AlN(54 nm)/Fe(14 nm)/AlN(d(4))/Au(1000 nm), respectively. From the inspection of Table 3 and Figs. 3-5 we observe that |ryx(05)| in the structure AlN(d(2))/Fe(d(3))/AlN(d(4))/Au(1000 nm) is enhanced by a factor of 2.86 with respect to air/Fe(1000 nm) interface. In the region of the maximal|ryx(05)|, R can be made vanishingly small, but T cannot be reduced below 1.5%. As a result, the computed minimum of R, Rmin≈2 × 10−5, at {d(2), d(3), d(4)} = {54nm, 16 nm, 25 nm} is displaced from the position {50 nm, 14 nm, 25 nm} of the maximal |ryx(05)|. At Rmin (“antireflection condition”), i.e., at the minimum of |rxx(05)|, where arg(rxx(05)) changes rapidly, θr(05)and εr(05)display strongly localized maxima (minima). On the other hand, in the region of the maximal|ryx(05)| both|ryx(05)| and arg(ryx(05)) are slowly varying functions of {d(2), d(3), d(4)} which makes the tolerances for the sandwich deposition easily achievable. If required, a fine tuning of d(2), d(3), d(4) or λ can shift the structure to zero εr(05) and/or to maximal θr(05) at negligible reduction of |ryx(05)| with respect to its maximal value.

From the inspection of Tables 3 and 4, we observe an enhancement of |ryx(05)| by a factor of 2.64 with respect to air/Fe(1000 nm) interface, a value slightly reduced with respect to that achieved in the absence of the protecting Au layer (d(1) = 0). The addition of the protecting Au layer may considerably reduce Rmin, corresponding to “antireflection,” and enhance in this way magneto-optic azimuth rotation, θr(05), and ellipticity, εr(05). The situation is illustrated in Table 4 for the structure Au(5 nm)/AlN(75 nm)/Fe(18 nm)/AlN(12 nm)/Au(1000 nm). The modifications of d(3) and d(4) led to |ryx(05)| reduced from the maximal value of 0.0172 by less than 1% to |ryx(05)|0.0171, only. A comparison of the structures AlN(61 nm)/Fe(1000 nm)/Au(1000 nm) and AlN(57 nm)/Fe(23 nm)/Au(1000 nm) in Table 3 with Au(5 nm)/AlN(82 nm)/Fe(1000 nm)/Au(1000 nm) and Au(5 nm)/AlN(78 nm)/Fe(26 nm)/Au(1000 nm) in Table 4, respectively, represent another examples. A similar enhancement of the MO azimuth rotation was reported by Querishi et al. on Au/SiO2/Ni structures [21

21. N. Qureshi, H. Schmidt, and A. R. Hawkins, “Cavity enhancement of the magneto-optic Kerr effect for optical studies of magnetic nanostructures,” Appl. Phys. Lett. 85(3), 431–433 (2004). [CrossRef]

].

5. Discussion

The figure of merit, |ryx(05)|, was evaluated for the case of a complete saturation of ultrathin Fe perpendicular to the interfaces, which would require the perpendicular field, Bμ0Ms. On the other hand, the MO probes are considered to detect much lower B at which both precessional angles of FMR and MO angles θr(05) and εr(05)remain within the range of linearity in B. In the ideal case, the maximal figure of merit |ryx(05)|maxof Eq. (4) is achieved where an incident x-polarized wave (χi(05)0) is completely converted to a MO reflected y-polarized wave (χr(05)) corresponding to |rxx(05)|0, i.e., the maximum of |ryx(05)| and the minimum of |rxx(05)| coincide and the corresponding MO azimuth rotation reaches 90 degrees. There would be no reflected wave with the incident polarization (R = 0) and there would be no power leak, into the optically thick reflector (T = 0). Because of our restriction to five-layer structures T≠0 and the positions of the maximal |ryx(05)|and those of the minimal |rxx(05)|as well as those of the maximal A are slightly displaced. The maximum of |ryx(05)|is broad, while the minimum of |rxx(05)| is sharp. Around the minimum of |rxx(05)|, i.e., around the “antireflection,” Rmin, the ratioχr(05)varies rapidly with d(2), d(3), or d(4) producing sharp maxima and minima of θr(05) and εr(05) often exceeding the region of linearity in Eq. (23) and Eq. (24). This illustrates the important effect of rxx on the observed quantities θr(05)and εr(05)which should be accounted for even in the ultrathin film limit where |ryx| can be assumed as linear in the thickness of the magnetic layer [24

24. Š. Višňovský, Optics in Magnetic Multilayers and Nanostructures (CRC Taylor & Francis, 2006), Chapter 4, pp. 198–211.

]. In multilayer structures with ultrathin Fe, the roughness and mixing at interfaces, layer uniformity, etc., do not allow to achieve Rmin as deep as predicted by the calculations [6

6. R. Lopusnik, J. Correu, I. Harward, S. Widuch, P. Maslankiewicz, S. Demirtas, Z. Celinski, E. Liskova, M. Veis, and S. Visnovsky, “Optimization of magneto-optical response of FeF2/Fe/ FeF2 sandwiches for microwave field detection,” J. Appl. Phys. 101(9), 09C516 (2007). [CrossRef]

,13

13. J. Pištora, M. Lesňák, E. Lišková, Š. Višňovský, I. Harward, P. Maslankiewicz, K. Balin, Z. Celinski, J. Mistrík, T. Yamaguchi, R. Lopusnik, and J. Vlček, “The effect of FeF2 on the magneto-optic response in FeF2/Fe/FeF2 sandwiches,” J. Phys. D Appl. Phys. 43(15), 155301 (2010). [CrossRef]

,14

14. E. Liskova, S. Visnovsky, R. Lopusnik, I. Harward, M. Wenger, T. Christensen, and Z. Celinski, “Magneto-optical AlN/Fe/AlN structures optimized for operation in the violet spectral region,” J. Phys. D Appl. Phys. 41(15), 155007 (2008). [CrossRef]

]. The MO azimuth rotation of 90 degrees were observed on single crystals only where a nearly perfect atomic layering may produce antireflection and χr=ryx/rxx>1 [30

30. R. Pittini, J. Schoenes, O. Vogt, and P. Wachter, “Discovery of 90 degree magneto-optical polar Kerr rotation in CeSb,” Phys. Rev. Lett. 77(5), 944–947 (1996). [CrossRef] [PubMed]

]. For the present purpose, it was then sufficient to evaluate the d(2), d(3), or d(4) dependence of the parameters, including rxx, in 1 nm steps and evaluate θr(05), and εr(05)at 10% saturation magnetization using the proportionality χr(05)ryx(05)εyx(3)μ0Ms(Fe).Our focus was not on θr(05) and εr(05) but on the figure of merit, |ryx(05)|, as the most important parameter for the probe operation. This can alternatively be evaluated using the first order expression of Eq. (17) with a practically sufficient precision. The corresponding expression for transmission,|tyx(05)|, given in Eq. (22), can be used in the same way.

Due to the chemical instability of pure Fe, special measures must be used to imbed an ultrathin Fe into a dielectric resonator. The present results (Table 3 and 4) indicate that the resort to structures with a single dielectric layer and ultrathin Fe may already provide a reasonable performance with |ryx(05)| reduced by a few percent only with respect to that achieved in five-layer sandwiches. For example, in the structure AlN(57 nm)/Fe(23 nm)/AlN(0 nm)/Au(1000 nm), |ryx(05)|=0.0172 and in the structure with the Fe layer exposed to the ambient, AlN(0 nm)/Fe(6 nm)/AlN(56 nm)/Au(1000 nm), even |ryx(05)|=0.0180 (Table 3). The latter case is of less practical use but it suggests that a replacement of pure Fe with a chemically stable alloy, e.g., FePt, represents a promising solution. In addition, the reduced theoretical |ryx(05)|can be compensated by a better growth perfection achievable in simpler structures.

The illustrations were performed at the wavelength λ = 632.8 nm. The probe performance can be equally well evaluated in any other wavelength, e.g., in blue region with a better diffraction limited resolution. Here the ultrathin Fe with low |ryx|max0.008at λ = 310 nm would be better replaced, e.g., with an ultrathin FePt alloy.

Acknowledgment

This project was supported by the Ministry of Education, Youth and Sport of the Czech Republic (#ME09045).

References and links

1.

A. Y. Elezzabi and M. R. Freeman, “Ultrafast magneto-optic sampling of picosecond current pulses,” Appl. Phys. Lett. 68(25), 3546–3548 (1996). [CrossRef]

2.

M. C. Sekhar, J. Y. Hwang, M. Ferrera, Y. Linzon, L. Razzari, C. Harnagea, M. Zaezjev, A. Pignolet, and R. Morandotti, “Strong enhancement of the Faraday rotation in Ce and Bi comodified epitaxial iron garnet thin films,” Appl. Phys. Lett. 94(18), 181916 (2009). [CrossRef]

3.

J. Y. Hwang, M. Ferrera, L. Razzari, A. Pignolet, and R. Morandotti, “The role of Bi3+ ions in magneto-optic Ce and Bi comodified epitaxial iron garnet films,” Appl. Phys. Lett. 97(16), 161901 (2010). [CrossRef]

4.

H. Suhl, “Origin and use of instabilities in ferromagnetic resonance,” J. Appl. Phys. 29(3), 416–421 (1958). [CrossRef]

5.

S. Chikazumi, Physics of Magnetism (Oxford Science Publications, Clarendon Press, 1997).

6.

R. Lopusnik, J. Correu, I. Harward, S. Widuch, P. Maslankiewicz, S. Demirtas, Z. Celinski, E. Liskova, M. Veis, and S. Visnovsky, “Optimization of magneto-optical response of FeF2/Fe/ FeF2 sandwiches for microwave field detection,” J. Appl. Phys. 101(9), 09C516 (2007). [CrossRef]

7.

C. Kittel, “On the theory of ferromagnetic resonance absorption,” Phys. Rev. 73(2), 155–161 (1948). [CrossRef]

8.

C. F. Buhrer, “Faraday rotation and dichroism of bismuth calcium vanadium iron garnet,” J. Appl. Phys. 40(11), 4500–4502 (1969). [CrossRef]

9.

S. Wittekoek, T. J. A. Popma, J. M. Robertson, and P. F. Bongers, “Magneto-optic spectra and the dielectric tensor elements of bismuth-substituted iron garnets at photon energies between 2.2-5.2 eV,” Phys. Rev. B 12(7), 2777–2788 (1975). [CrossRef]

10.

G. S. Krinchik and V. A. Artem’ev, “Magneto-optical properties of Ni, Co and Fe in ultraviolet visible and infrared parts of spectrum,” Sov. Phys. JETP 26, 1080–1085 (1968).

11.

S. Visnovsky, R. Krishnan, M. Nyvlt, and V. Prosser, “Optical behaviour of Fe in magnetic multilayers,” J. Magn. Soc. Jpn. 20, Supplement No. S1, 41–46 (1996).

12.

P. B. Johnson and R. W. Christy, “Optical constants of transition metals: Ti, V, Cr, Mn, Fe, Co, Ni, and Pd,” Phys. Rev. B 9(12), 5056–5070 (1974). [CrossRef]

13.

J. Pištora, M. Lesňák, E. Lišková, Š. Višňovský, I. Harward, P. Maslankiewicz, K. Balin, Z. Celinski, J. Mistrík, T. Yamaguchi, R. Lopusnik, and J. Vlček, “The effect of FeF2 on the magneto-optic response in FeF2/Fe/FeF2 sandwiches,” J. Phys. D Appl. Phys. 43(15), 155301 (2010). [CrossRef]

14.

E. Liskova, S. Visnovsky, R. Lopusnik, I. Harward, M. Wenger, T. Christensen, and Z. Celinski, “Magneto-optical AlN/Fe/AlN structures optimized for operation in the violet spectral region,” J. Phys. D Appl. Phys. 41(15), 155007 (2008). [CrossRef]

15.

P. Yeh, “Electromagnetic propagation in birefringent layered media,” J. Opt. Soc. Am. 69(5), 742–756 (1979). [CrossRef]

16.

Š. Višňovský, “Magneto-optical ellipsometry,” Czech. J. Phys. B 36(5), 625–650 (1986). [CrossRef]

17.

Y. Tomita and T. Yoshino, “Optimum design of multilayer-medium structures in a magneto-optical readout system,” J. Opt. Soc. Am. A 1(8), 809–817 (1984). [CrossRef]

18.

R. Atkinson, I. W. Salter, and J. Xu, “Design, fabrication, and performance of enhanced magneto-optic quadrilayers with controllable ellipticity,” Appl. Opt. 31(23), 4847–4852 (1992). [CrossRef] [PubMed]

19.

W. R. Hendren, R. Atkinson, R. J. Pollard, I. W. Salter, C. D. Wright, W. W. Clegg, and D. F. L. Jenkins, “The design and optimization of disk structures for MAMMOS/MSR magneto-optic recording,” J. Phys. D Appl. Phys. 38(14), 2310–2320 (2005). [CrossRef]

20.

M. Mansuripur, Physical Principles of Magneto-optical Recording (Cambridge University Press, 1996).

21.

N. Qureshi, H. Schmidt, and A. R. Hawkins, “Cavity enhancement of the magneto-optic Kerr effect for optical studies of magnetic nanostructures,” Appl. Phys. Lett. 85(3), 431–433 (2004). [CrossRef]

22.

M. Moradi and M. Ghanaatshoar, “Cavity enhancement of the magneto-optic Kerr effect in glass/Al/SnO2/PtMnSb/SnO2 structure,” Opt. Commun. 283(24), 5053–5057 (2010). [CrossRef]

23.

M. Mansuripur, “Figure of merit for magneto-optical media based on the dielectric tensor,” Appl. Phys. Lett. 49(1), 19–21 (1986). [CrossRef]

24.

Š. Višňovský, Optics in Magnetic Multilayers and Nanostructures (CRC Taylor & Francis, 2006), Chapter 4, pp. 198–211.

25.

F. Abelès, “Recherches sur la propagation des ondes électromagnétiques sinusoïdales dans les milieux stratifiés. Application aux couches minces,” Ann. Phys. (Paris) 5, 596–640, 706–782 (1950).

26.

M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1997), pp.51–70.

27.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (Elsevier, 1987).

28.

D. Brunner, H. Angerer, E. Bustarret, F. Freudenberg, R. Höpler, R. Dimitrov, O. Ambacher, and M. Stutzmann, “Optical constants of epitaxial AlGaN films and their temperature dependence,” J. Appl. Phys. 82(10), 5090–5096 (1997). [CrossRef]

29.

P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B 6(12), 4370–4379 (1972). [CrossRef]

30.

R. Pittini, J. Schoenes, O. Vogt, and P. Wachter, “Discovery of 90 degree magneto-optical polar Kerr rotation in CeSb,” Phys. Rev. Lett. 77(5), 944–947 (1996). [CrossRef] [PubMed]

31.

A. Berger and M. R. Pufall, “Quantitative vector magnetometry using generalized magneto-optical ellipsometry,” J. Appl. Phys. 85(8), 4583–4585 (1999). [CrossRef]

32.

M. R. Pufall and A. Berger, “Studying the reversal mode of the magnetization vector versus applied field angle using generalized magneto-optical ellipsometry,” J. Appl. Phys. 87(9), 5834–5836 (2000). [CrossRef]

33.

J. A. Arregi, J. B. Gonzales-Diaz, E. Bergaretxe, O. Idigoras, T. Unsal, and A. Berger, “Study of generalized magneto-optical ellipsometry measurement reliability,” J. Appl. Phys. 111(10), 103912 (2012). [CrossRef]

34.

Š. Visnovský, K. Postava, T. Yamaguchi, and R. Lopusník, “Magneto-optic ellipsometry in exchange-coupled films,” Appl. Opt. 41(19), 3950–3960 (2002). [CrossRef] [PubMed]

OCIS Codes
(160.3820) Materials : Magneto-optical materials
(230.3810) Optical devices : Magneto-optic systems
(310.6845) Thin films : Thin film devices and applications

ToC Category:
Thin Films

History
Original Manuscript: November 20, 2012
Revised Manuscript: January 21, 2013
Manuscript Accepted: January 23, 2013
Published: February 4, 2013

Citation
Š. Višňovský, E. Lišková-Jakubisová, I. Harward, and Z. Celinski, "Analytical analysis of a multilayer structure with ultrathin Fe film for magneto-optical sensing," Opt. Express 21, 3400-3416 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-3-3400


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References

  1. A. Y. Elezzabi and M. R. Freeman, “Ultrafast magneto-optic sampling of picosecond current pulses,” Appl. Phys. Lett.68(25), 3546–3548 (1996). [CrossRef]
  2. M. C. Sekhar, J. Y. Hwang, M. Ferrera, Y. Linzon, L. Razzari, C. Harnagea, M. Zaezjev, A. Pignolet, and R. Morandotti, “Strong enhancement of the Faraday rotation in Ce and Bi comodified epitaxial iron garnet thin films,” Appl. Phys. Lett.94(18), 181916 (2009). [CrossRef]
  3. J. Y. Hwang, M. Ferrera, L. Razzari, A. Pignolet, and R. Morandotti, “The role of Bi3+ ions in magneto-optic Ce and Bi comodified epitaxial iron garnet films,” Appl. Phys. Lett.97(16), 161901 (2010). [CrossRef]
  4. H. Suhl, “Origin and use of instabilities in ferromagnetic resonance,” J. Appl. Phys.29(3), 416–421 (1958). [CrossRef]
  5. S. Chikazumi, Physics of Magnetism (Oxford Science Publications, Clarendon Press, 1997).
  6. R. Lopusnik, J. Correu, I. Harward, S. Widuch, P. Maslankiewicz, S. Demirtas, Z. Celinski, E. Liskova, M. Veis, and S. Visnovsky, “Optimization of magneto-optical response of FeF2/Fe/ FeF2 sandwiches for microwave field detection,” J. Appl. Phys.101(9), 09C516 (2007). [CrossRef]
  7. C. Kittel, “On the theory of ferromagnetic resonance absorption,” Phys. Rev.73(2), 155–161 (1948). [CrossRef]
  8. C. F. Buhrer, “Faraday rotation and dichroism of bismuth calcium vanadium iron garnet,” J. Appl. Phys.40(11), 4500–4502 (1969). [CrossRef]
  9. S. Wittekoek, T. J. A. Popma, J. M. Robertson, and P. F. Bongers, “Magneto-optic spectra and the dielectric tensor elements of bismuth-substituted iron garnets at photon energies between 2.2-5.2 eV,” Phys. Rev. B12(7), 2777–2788 (1975). [CrossRef]
  10. G. S. Krinchik and V. A. Artem’ev, “Magneto-optical properties of Ni, Co and Fe in ultraviolet visible and infrared parts of spectrum,” Sov. Phys. JETP26, 1080–1085 (1968).
  11. S. Visnovsky, R. Krishnan, M. Nyvlt, and V. Prosser, “Optical behaviour of Fe in magnetic multilayers,” J. Magn. Soc. Jpn.20, Supplement No. S1, 41–46 (1996).
  12. P. B. Johnson and R. W. Christy, “Optical constants of transition metals: Ti, V, Cr, Mn, Fe, Co, Ni, and Pd,” Phys. Rev. B9(12), 5056–5070 (1974). [CrossRef]
  13. J. Pištora, M. Lesňák, E. Lišková, Š. Višňovský, I. Harward, P. Maslankiewicz, K. Balin, Z. Celinski, J. Mistrík, T. Yamaguchi, R. Lopusnik, and J. Vlček, “The effect of FeF2 on the magneto-optic response in FeF2/Fe/FeF2 sandwiches,” J. Phys. D Appl. Phys.43(15), 155301 (2010). [CrossRef]
  14. E. Liskova, S. Visnovsky, R. Lopusnik, I. Harward, M. Wenger, T. Christensen, and Z. Celinski, “Magneto-optical AlN/Fe/AlN structures optimized for operation in the violet spectral region,” J. Phys. D Appl. Phys.41(15), 155007 (2008). [CrossRef]
  15. P. Yeh, “Electromagnetic propagation in birefringent layered media,” J. Opt. Soc. Am.69(5), 742–756 (1979). [CrossRef]
  16. Š. Višňovský, “Magneto-optical ellipsometry,” Czech. J. Phys. B36(5), 625–650 (1986). [CrossRef]
  17. Y. Tomita and T. Yoshino, “Optimum design of multilayer-medium structures in a magneto-optical readout system,” J. Opt. Soc. Am. A1(8), 809–817 (1984). [CrossRef]
  18. R. Atkinson, I. W. Salter, and J. Xu, “Design, fabrication, and performance of enhanced magneto-optic quadrilayers with controllable ellipticity,” Appl. Opt.31(23), 4847–4852 (1992). [CrossRef] [PubMed]
  19. W. R. Hendren, R. Atkinson, R. J. Pollard, I. W. Salter, C. D. Wright, W. W. Clegg, and D. F. L. Jenkins, “The design and optimization of disk structures for MAMMOS/MSR magneto-optic recording,” J. Phys. D Appl. Phys.38(14), 2310–2320 (2005). [CrossRef]
  20. M. Mansuripur, Physical Principles of Magneto-optical Recording (Cambridge University Press, 1996).
  21. N. Qureshi, H. Schmidt, and A. R. Hawkins, “Cavity enhancement of the magneto-optic Kerr effect for optical studies of magnetic nanostructures,” Appl. Phys. Lett.85(3), 431–433 (2004). [CrossRef]
  22. M. Moradi and M. Ghanaatshoar, “Cavity enhancement of the magneto-optic Kerr effect in glass/Al/SnO2/PtMnSb/SnO2 structure,” Opt. Commun.283(24), 5053–5057 (2010). [CrossRef]
  23. M. Mansuripur, “Figure of merit for magneto-optical media based on the dielectric tensor,” Appl. Phys. Lett.49(1), 19–21 (1986). [CrossRef]
  24. Š. Višňovský, Optics in Magnetic Multilayers and Nanostructures (CRC Taylor & Francis, 2006), Chapter 4, pp. 198–211.
  25. F. Abelès, “Recherches sur la propagation des ondes électromagnétiques sinusoïdales dans les milieux stratifiés. Application aux couches minces,” Ann. Phys. (Paris)5, 596–640, 706–782 (1950).
  26. M. Born and E. Wolf, Principles of Optics (Cambridge University Press, 1997), pp.51–70.
  27. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (Elsevier, 1987).
  28. D. Brunner, H. Angerer, E. Bustarret, F. Freudenberg, R. Höpler, R. Dimitrov, O. Ambacher, and M. Stutzmann, “Optical constants of epitaxial AlGaN films and their temperature dependence,” J. Appl. Phys.82(10), 5090–5096 (1997). [CrossRef]
  29. P. B. Johnson and R. W. Christy, “Optical constants of the noble metals,” Phys. Rev. B6(12), 4370–4379 (1972). [CrossRef]
  30. R. Pittini, J. Schoenes, O. Vogt, and P. Wachter, “Discovery of 90 degree magneto-optical polar Kerr rotation in CeSb,” Phys. Rev. Lett.77(5), 944–947 (1996). [CrossRef] [PubMed]
  31. A. Berger and M. R. Pufall, “Quantitative vector magnetometry using generalized magneto-optical ellipsometry,” J. Appl. Phys.85(8), 4583–4585 (1999). [CrossRef]
  32. M. R. Pufall and A. Berger, “Studying the reversal mode of the magnetization vector versus applied field angle using generalized magneto-optical ellipsometry,” J. Appl. Phys.87(9), 5834–5836 (2000). [CrossRef]
  33. J. A. Arregi, J. B. Gonzales-Diaz, E. Bergaretxe, O. Idigoras, T. Unsal, and A. Berger, “Study of generalized magneto-optical ellipsometry measurement reliability,” J. Appl. Phys.111(10), 103912 (2012). [CrossRef]
  34. Š. Visnovský, K. Postava, T. Yamaguchi, and R. Lopusník, “Magneto-optic ellipsometry in exchange-coupled films,” Appl. Opt.41(19), 3950–3960 (2002). [CrossRef] [PubMed]

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