Dynamical conversion of optical modes in garnet films induced by ferrimagneticresonanceB. Neite and H. Dötsch Citation: Journal of Applied Physics 62, 648 (1987); doi: 10.1063/1.339794 View online: http://dx.doi.org/10.1063/1.339794 View Table of Contents: http://scitation.aip.org/content/aip/journal/jap/62/2?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Dynamic conversion of optical modes in magnetic garnet films induced by resonances of periodic stripe domains J. Appl. Phys. 76, 3272 (1994); 10.1063/1.357448 Ferrimagnetic resonance in garnet films at large precession angles J. Appl. Phys. 62, 4839 (1987); 10.1063/1.338988 Simple technique for observing ferrimagnetic resonance in magneticbubble garnet films Rev. Sci. Instrum. 46, 1188 (1975); 10.1063/1.1134441 Dynamic Conversion Effects in Epitaxial Garnet Films AIP Conf. Proc. 18, 217 (1974); 10.1063/1.2947321 Ferrimagnetic Resonance in Gadolinium Iron Garnet J. Appl. Phys. 29, 427 (1958); 10.1063/1.1723167

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Dynamical conversion of optical modes in garnet films induced by ferrimagnetic resonance

B. Neite and H. Dotsch University a/Osnabrock, P. O. Box 4469,4500 Osnabrock, West Germany

(Received 30 December 1986; accepted for publication 10 March 1987)

It is demonstrated that ferrimagnetic resonance can be applied to induce a dynamical conversion of optical modes in a magnetic garnet film. The fundamental and first-harmonic Fourier components of the converted mode are measured as a function offrequency and bias field; a dynamical conversion efficiency of 2 % is obtained. The amplitude of the first harmonic depends very critically on the orientation of the magnetization with respect to the film normal. The results are confirmed by calculations on the basis of Faraday rotation and Cotton-Mouton effect.

I. INTRODUCTION

Magnetic garnet films can be applied as optical wave­guides in the near-infrared region where they exhibit a high Faraday rotation combined with low optical losses. These properties are very attractive to realize nonreciprocal com­ponents for integrated optics, such as isolators and circula­tors. The basis ofthose applications is the conversion of opti­cal modes induced by the Faraday rotation and the Cotton­Mouton effect, which depend on the orientation of the mag­netization with respect to the direction oflight propagation. Changing this component by external magnetic fields yields a change of the mode conversion efficiency; thus optical switches and modulators can be realized. The modulation may be performed at high frequencies if the ferrimagnetic resonance in low-loss garnet films is applied. In that case the magnetization precesses about the axis of an effective mag­netic field at a frequency in the GHz range, which can be tuned by external fields.

This technique was used by Dillon et al, 1 and by Takeu­chi et al. 2 in bulk crystals. In garnet films Tien et al.3 ob­tained light modulation at frequencies up to 300 MHz. Fish­er et al.4 and Solomko et af.5-7 excited magnetostatic waves in garnet films in order to achieve modulation and deflection of optical modes by Bragg scattering. In this paper dynami­cal mode conversion induced by ferrimagnetic resonance is reported.

II. GEOMETRY OF THE GARNET FILM

The geometry considered in this paper is sketched in Fig. 1. The film normal is paranel to a [Ill] crystallograph­ic direction; it is chosen as the x axis. Light propagates in the film plane parallel to the z direction, which is paranel to [ - 1,1,0J. The direction of the magnetization M is specified by the angles 9 and W. The equilibrium values 9 0 and ¢o are determined by the minimum of the total energy density Fto!

given by

P tot = - ,uo(MH) + (Ku - ';0 M2)sin2 e

+ K j [! sin4 e + ~ cos4 e + (.J2/3 )sin3 e cos e cos(3!/')], (1)

where H is the applied bias field and Po = 41T1O-- 7 V siCA

m). The first term in Eq. (1) is the Zeeman energy, the second the uniaxial anisotropy energy, including the demag­netization energy, and the third is the cubic anisotropy ener­gy; Ku and Kr are the respective anisotropy constants.

m. COUPLING OF OPTICAL MODES

The propagation and coupling of optical modes is deter­mined by the dielectric tensors of the cover, film, and sub­strate. Cover and substrate are assumed isotropic, having refractive indices n,. = 1 (air) and ns = 1.95 (gadolinium gallium garnet at A = 1.3 ftm), respectively.

The dielectric tensor of the film can be split into three parts:

(2)

where ~ takes into account the isotropic refractive index and the contributions due to stress and growth-induced anisotro­pies. It is given by

(3)

where nyy = nzz for a (111 )-oriented film. E

CM is the contribution due to the Cotton-Mouton ef­fect, and it depends on the products M;Mj' where M; are the components of the magnetization M, as shown by Prok­horov etal. H From the tensor components GIl' Gw and G44,

z _ k-.l-~H---o-[110J

FIG. L Geometry of the garnet film.

648 J. Appl. Pi1ys. 82 (2),15 July 1987 0021-8979/87/140648-05$02.40 ® 1987 American Institute of Physics 648 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP:

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which describe the Cotton-Mouton eirect, only G44 and the combination ilG = Gil - Gl2 - 2G44 appear in (OCM. The additional isotropic term Gl2 ha<; to be taken into account by E' because it does not depend on the magnetization and thus cannot be distinguished from Ea. The €eM tensor is given in Ref. 8 and will not be reproduced because of its length.

10m represents the contribution due to the Faraday rota­tion and is given by

(

0 sin9sinI/J Em=iKM -sin8sinif; 0 cosS .

- sin e cos 1/1)

sin 8 cos I/J - cos e o

(4)

The constant K is related to the specific Faraday rotation 8 F

by approximately K = 2n08 F I(k(y.'I1), where ko is the vacu­um wave number of the light and no is the average refractive index. Losses are neglected in the following so that E" and ",eM are real and symmetrical, while Em is purely imaginary and antisymmetrica1.

In order to calculate the optical mode conversion, the method described by Yamamoto et al. 9 is applied. The TEo and TMo modes are represented by their electrical fields

ETE = A TE(O,E;E,O), E'I'M = A TM(E;M,O,E;M) (5)

and by their propagation constants {:J TE and f3 'IM, respec­tively. The corresponding magnetic fields HTE and HTM can be derived from Maxwell's equations. A TE and A 'I'M are the normalized amplitUdes of the modes with

(6)

wherejz is a unit vector in the~direction and I denotes TEor TM. The fields and propagation constants are determined by the refractive indices no and 11., of the cover and substrate, respectively, by the diagonal components of the € tensor of the film, and by the film thickness d. A vacuum wavelength ..1,0 of 1.3 pm is used in this analysis.

The coupling constant N of these modes is given by ~ + 00

N = € iJ I (t" ETPE™ + € ETE*E™)dx (7) ali xy y x yz y z ,"-~ 00

where to is the dielectric constant of vacuum. The most effi­cient coupling occurs through the component Exy because it couples the strong transverse field components, whereas tyz

couples longitudinal- and transverse-field components; Exz

does not induce TE-TM-mode coupling at aU.'J The contri­bution ofthe tensor Em to the tensor component t xy is given by Eq. (4), while that of the tensor E" vanishes [Eq. (3)]. The tensor ,-,CM contributes to € xy a term that is proportional to sin(290 ) by using the approximation !J.G = 08

; thus, l;yM reaches its maximum at So = 45". For tlG #Othismaximum occurs at a slightly different angle.

For TE input with amplitude A TE(O), the coupling effi­ciency rt is given by

IA™(Z)12

7f = ATE(o)

~INI2 oSin2(\~.Jf::.f32+4INr), (8) AB~+4INI- 2

649 J. Appl. Phys., Vol. 62, No.2, ; 5 July 1987

where AB = f3 'IE - f3 TM. This equation determines the cou­pling of a TE to a TM mode via Faraday rotation and Cot­ton-Mouton effect by assuming that the magnetization is fixed in space. In a simple approximation Eq. (8) will also be applied to describe the dynamical coupling during ferrimag­netic resonance, where M precesses about its equilibrium direction at a frequency of approximately 1 GHz. For a cou­pling length z of 3.5 mm, the light takes about 0.02 ns to pass this region, which is 2 % ofthe time the magnetization needs for one revolution. Thus, even for the coupling process at ferrimagnetic resonance, the magnetization can be regarded as static.

In order to obtain the dynamic coupling efficiency, first the equilibrium position (8o,l/Jo) is calculated according to Eg. (1). Then for 9 0 # 0 the ferrimagnetic resonance fre­quency f is given bylO

(9)

where r is the gyromagnetic ratio, and F eo' F "'if" and F 8", are the partial derivatives of F with respect to the angles e and if;. For 9 0 = 0 the resonance frequency is

f 4KJ 2Ku )

f = 1\ H - it{ - 3fhoM + /-lo'\! . (0)

For a given precession amplitude $ the illstantaneous position of the magnetization can be calculated, which yields, according to Eq. (8), the coupling efficiency 1] (q;). The angle q; (O<q;<3600) denotes the position of M along the precession cone, as shown in Fig. 1. From rt (q;) the mod­ulation spectra of the light can be derived by Fourier analy­sis.

IV. EXPERIMENTS

The crystal used for the experiments has the composi­tion y 2.9 Lao.l Fe3.9 Gal.! 0 12 • Its material parameters are listed in Table 1. The experimental setup is shown in Fig. 2. Light of 1.3-/-lm wavelength from a pulse laser (l-ps pulse width, 1-kHz repetition frequency) is coupled into and out of the garnet film by rutile prisms. The converted light is then focused to a fast photodetector, the output of which is amplified and fed into a spectrum analyzer that can operate up to a frequency of 3.5 GHz. A magnetic field can be ap­plied perpendicular to the film plane. The ferrimagnetic res­onance is excited by a microstrip antenna that provides an approximately homogeneous rf magnetic field in the film plane perpendicular to the applied static field. The frequency of the rf generator is tunable between 10 MHz and 2 GHz. By an additional pickup antenna the ferromagnetic reso­nance can be monitored.

TABLE 1. Material parameters of the crystal.

/I xx ~'o 2.1304 Ilyy = 2.1301

d=6.8j<m G •• M 2.= 2.40 X 10 • AGM' = .. ~ 0.73 X 10 4

M= 17030A/m KM ~-, 3.07X 10-4

K, = -76J/m3 Ku = - 107 J/m3

Y= 35.2 kHz/AIm

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PULSE ~ASER

L. z FILM

~~,~~~~~R~:~"~~'~~~ ~~~ ~~ ~~" ~ ~ "'-"'- ~ "'-, " ~ "'- ,~'" "'- ~ <,,~ ~- ~ ~~ ~ ~ "'-FIG. 2. Experimental setup.

V.RESULTS Figure 3 shows the ferrimagnetic-resonance frequency

versus the bias field Hl. applied perpendicular to the film plane. These measurements are obtained by the pickup an­tenna. As Ku is negative, the magnetization is in-plane at zero bias field. For Hl. > Hp = 2.23 X 104 A/m, the magneti­zation is parallel to Hl , and the resonance frequency is given by Eq. (10). From the data of Fig. 3 the material parameters y, K j , and Ku are derived and given in Table 1. The satura­tion magnetization has been measured by a vibrating-sample magnetometer.

At ferrimagnetic resonance the TEo mode is coupled into the garnet film, and the modulation frequencies of the converted TMo mode are determined. The results are repre-

i >. u c Ql ::l CT Q) L. --w u c o c

3.0

fJ3Hz

2.0

5l 1.0 ~

a

l

I

o

I I

I

bias field Hl.-

FIG. 3. Measured frequency of the ferrimagnetic resonance vs bias field H J •

650 J. Appl. Phys., Vol. 62, No.2, i 5 July 1987

i >. u c

3.0

ffl3Hz

3 2.0 0-W '---c ~ o '3 '0 o E

1.0

o o

I fundamental /'

frequency

first harmonic

bias field H.l.-

FIG. 4. Measured modulation frequencies of the converted TMo mode vs bias field H 1 •

serrted in Fig. 4. For Hi >Hp, where M is parallel to the applied field modulation, components at both the funda­mental frequency of the ferrimagnetic resonance and its first harmonic are observed. However, for Hl. < Hp the magneti­zation is no longer aligned parallel to the bias field, and only the fundamental frequency can be seen. A typical spectrum is shown in Fig. 5, where the measured light intensity is plot­ted versus frequency at a fixed bias field. The sidebands at an integer mUltiple of 1 MHz are caused by the i-ItS pulse width of the light.

The dynamical coupling efficiency achieved is about 2%. The ratio between the light intensities at the fundamen­tal frequency and at the first harmonic depends very critical­lyon the equilibrium angles 8 0 and .,po.

For the following discussion the E tensor of Eg. (2) has to be determined. To obtain €", the magnetization is first adjusted by an external magnetic field in the film plane per­pendicular to the propagation direction of light. Thus the Faraday rotation has no influence on the light, while the

t P/dB 0..

I 20

~ 0

'" ~ -20 S

~r\ H,.5.3·10'A/m 1\"""," I ~~~~~"-I

2.3 MHz 2.3 MHz .:

. ___ ------L .. ___ --/ !-----_-----'--_____ .J 0.949 . 1.898 f/GHz

frequency -

FIG. 5. Measured optical spectrum of the converted TMo mode at a fixed bias field HI .

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influence of the Cotton-Mouton effect is sman and can be neglected. The film thickness d and the tensor c are then determined by fitting these quantities to the measured TE­and TM-mode spectra of this configuration. The tensors 6 CM

and em are obtained in the following manner. The magneti­zation of the film is adjusted in two different directions: (1) parallel to the film plane and parallel to light propagation so that mode coupling is induced by Faraday rotation and (2) 45° to the film plane and perpendicular to light propagation so that the Cotton-Mouton effect is responsible for the mode coupling. In both cases the coupling efficiency 17 is measured as a function of z. Fitting Eq. (8) to these results yields £m

and ECM• The values for nxx • nyy = nzz , K, G44, and !J.G ob­

tained in this manner are listed in Table I. The maximum coupling efficiency achieved in these static measurements is 30%.

VI. DISCUSSION

In order to calculate the dynamical coupling efficiency, the precession amplitude <1>, the angle of the precession cone in Fig. 1, has to be known. Fitting the dynamical coupling efficiency to Eq. (8) yields <P = 27°, This amplitude is in reasonable agreement with other measurements on the same crystal, where the precession angle is directly derived from the resonance induced change of the Faraday rotation of light passing through the film perpendicular to the film plane. 11 In these experiments, where higher microwave pow~ er and a stronger coupling between the antenna and the mag­netic film are applied, precession angles up to 48° are ob­served. The dependence of the precession angle q:. on the microwave power, however, is not linear.ll

0.02 0) 6,,~ 0°

0.015

0.01 \ t F O.OOS >-g

0 <1l Tl 0.070 ~ bl e"" 10" c 0.056 \II" = 0° 0 . iii

\ b

~ 0.042 0 ! U r-

0.028~

o~1 ~ a 90 180 270 360

angle 'P-\j¥deg

FIG. 6. Calculated conversion efficiency vs the angle 'P, the p05ition ofM on the precession cone, for a precession angle <I'> = 27"; the angle cp = 0 corre­sponds to if! = O. (a) 8 0 = 0, (b) eo = 1O",,po = O.

651 J. Appl. Phys., Vol. 62, No.2, 15 July 1987

a,=!lK. I llf

\ ~ =0" 4Jo '" 90° I 3.0 " 3.0

I \

2.0 ,

\ 2.0 I

I

-----) 1.0

~ 1.0

15 10 5 0 5 10 15 60fdeg

angle 8"

FIG. 7. Calculated ratio between the conversion efficiencies at the first har­monic and the fundamental frequency Hear 8" = 0 in the two planes ¢" = ()

and lfi" = 90".

By using this amplitUde of 27° for <I> the conversion effi­ciency, 17 has been calculated for 8 0 = 0 and Ht, 1/10 = 0, as a function of cp; the material parameters given in Table I are applied. For such small values of 8 0 the ellipticity of the precession is small, too, and can be neglected, Figure 6 repre­sents the results of the calculation. Fourier analysis of these curves yields the modulation components 17f and 7J2f at the fundamental frequency f and at the first-harmonic frequen­cy 2/ For eo = 0 the modulation at the first-harmonic fre­quency dominates; the ratio a = Tlzf /7Jf is about 3.340 The small contribution to 'flf in Fig. 6(a) is induced by the quan­tity fiG. For So = 10° the ratio a = rt2firlf is 0.48 and thus small compared to the case 8() = o. The strong dependence of a on Sn around 8 0 = 0 is shown in Fig. 7, where a is plotted versus 8 0 for .¢;() = 0 and 90°.

These results explain why for Hl. <Hp' where 8 0 #0, only modulation at the fundamental frequency is observed. A component 7Jzf could not be measured at the sensitivity of the present experiment of - 70 dB m. On the other hand, for HI > Hp a strong component 172/ should occur as is observed experimentally. The magnitude ofthis component, however, depends very critically on the alignment of the bias field Hl. parallel to the film normal.

VII. CONCLUSION

The presented experiments show that light modulation in a magneto-optical waveguide can be performed by ferri­magnetic resonance. The modulation frequencies lie in the GHz range and can be tuned by an external magnetic field. The modulation efficiency obtained is 2%. However, it may be increased by properly adjusting the material parameters. Especially, phase matching of the optical modes is essential, which can be achieved by optical anisotropy. 12.13 For preces­sion angles less than ;:P = 45° at 8 0 = 0, the conversion is

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mainly due to the Cotton-Mouton effect and increases with G44 and i:J.G. To induce dynamical conversion by Faraday rotation, which is stronger than the Cotton-Mouton effect, precession angles .:p> 60· at 8 0 = 0 must be achieved. The necessary conditions to realize such large precession angles will be investigated.

ACKNOWLEDGMENTS

We gratefully acknowledge financial support by the Deutsche Forschungsgemeinschaft, Sonderforschungsber­eich 225. Furthermore, we thank Professor W. Tolksdorf and I. Bartels for the preparation of the crystal and K. Witter for the magnetization measurement.

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