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Interaction of optical waveguide modes with magnetostatic surface waves in tangentially magnetized finite samples
Optics Communications 92 (1992) 31-34 North-Holland
OPTICS COMMUNICATIONS
Interaction of optical waveguide modes with magnetostatic surface waves in tangentially magnetized finite samples V.V. M a t y u s h e v , A.A. S t a s h k e v i c h a n d A.I. L u k y a n o v Institute of Electrical Engineering, 197376Saint Petersburg, Russia
Received 12 February 1992; revised manuscript received 30 March 1992
To increase the efficiency of the diffraction of optical wavegnide modes by magnetostatic waves (MSW) and thus to improve the performance of the magnetooptical devices based on this principle one has to confine both waves in a limited sample of a ferrimagnetic film. This paper reports the results of the experimental investigation of the peculiarities of the collinear interaction of guided light with surface MSW in a three-dimensional YIG-GGG waveguide. While measSring the frequency response of the interaction we scanned the light beam across the film. This made it possible to identify the optical response of different transverse modes of surface MSW (n= 1, 2, 3, 4) and of volume MSW (n=6). To excite different numbers of transverse modes including antisymmetric ones we changed the orientation of the transmitting antenna with respect to the direction of the bias magnetic field.
1. Introduction
2. Experiment
It has been shown in a n u m b e r o f experimental papers [ 1-4] that efficient magnetooptical ( M O ) interaction o f optical waveguide modes with magnetostatic waves ( M S W ) in thin ferrimagnetic films holds promise for the development o f integrated-optical signal-processing and light modulating devices operating at frequencies as high as 3 - 4 0 GHz. This interaction can also be utilized as a probing instrument to study magnetostatic waves in thin films. In all earlier works it was taken that a MSW is propagating in a ferrimagnetic film o f infinite dimensions. For tangentially magnetized films it means that the classical relations obtained in ref. [ 5 ] may be directly applied. On the other hand, as shown in ref. [ 6 ] the spectrum o f MSW in a tangentially magnetized finite sample o f a ferrimagnetic film is radically changed owing to the additional quantization along the transverse coordinate (transverse modes). Besides, narrowing o f a ferrimagnetic waveguide is one o f possible ways to increase the efficiency o f the M O interaction. The purpose o f this paper is the experimental investigation o f the peculiarities o f the collinear interaction o f guided optical modes with MSW in a three-dimensional ferfirnagnetic waveguide magnetized in the tangential transverse direction.
A simplified schematic diagram of the experimental set-up is given is figi 1. An yttrium iron garnetgadolinium gallium garnet ( Y I G - G G G ) layered structure was utilized as a waveguide for MSW and optical waves. The film was 3.8 ~tm thick and 3 m m wide. The sample was magnetized to saturation by a transverse bias field H of 800 Oe. We used a HeNe laser ( 2 = 1.15 ~tm) as an optical source. A lock-in amplifier was used to increase the sensibility o f the photodetection. MSW were excited and received by two antennas made o f gold wire 50 ~tm in diameter (antenna 1 and antenna 2, respectively). It has been shown in ref. [ 6 ] that MSW in a fer-
SIGNAL
Fig. I. Simplified scheme of the experimental set-up.
0030-4018/92/$05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
31
Volume 92, number 1,2,3
OPTICS COMMUNICATIONS
15 August 1992
rimagnetic film magnetized in the transverse tangential direction is a multimode wave process owing to the quantization along the transverse coordinate. The wavenumber of transverse modes Ky is given by the following expression K 2 = (nrc/w)2/ltl +K 2 ,
n=l
( 1) f, Gnz
where w= 3 m m is the width of the waveguide,/z, is the diagonal component of the tensor of magnetic permeability, n = 1, 2, 3, .... The distribution of the magnetostatic potential under the condition of pinned spins at both edges of the waveguide can be written as
b n=l
~0= [A exp (Kxx) + B e x p ( - K ~ x ) ] ×exp(iKyy) s i n ( n n z / w ) ,
f, GHz
0
(2)
where A and B are two constants; Ky is the wavenumber of a propagating MSW, Kx is a transverse wavenumber characterizing the MSW field distribution across the film thickness. A focused laser beam 200 rtm in diameter was propagating-along the MSW waveguide. The beam was scanned across the film along the z coordinate together with the photodetector. Figure 2 is illustrating the procedure of the measurement of the distribution of magnetization in the transverse modes ( n = 1, 2, 3...). The results of the measurements of the frequency dependence of the intensity of light wave diffracted by MSW are presented in fig. 3. For the probing of a MSW a collinear regime of TMo~TEo diffraction has been utilized. This means that the incident opa
Z
[
n=l
ANTENNA 1
t
2.or LIGHT
n=3
n=l
//
ANTENNA 2
n=2 n=3
b
T
Fig. 2. Excitation of transverse MSW modes (a) by a parallel antenna, (b) by a tilted antenna.
32
c
~d.
4.01
4.02
f GHz
Fig. 3. Diffraction efficiency and insertion loss as a function of the MSW frequency in a tangentially magnetized finite sample of a YIG fdm in the case of symmetric excitation. The unity at the vertical coordinate in (a) and (b) corresponds to the diffraction efficiency of 0.05% and it corresponds to tenfold attenuation of the microwave signal in (c). The power of the microwave signal at the input antenna 1 was of the order of 2 mW.
tical wave was a TMo mode and the diffracted wave was a TEo mode. The curve in fig. 3a corresponds to the case when the light beam is positioned in the central point z = 1.5 m m (see fig. 2a). Two clearly seen peaks are the result of the diffraction of light by the first and the third transverse modes (symmetric even modes n = l, 3). No optical signals corresponding to antisymmetric odd modes have been observed. Since the distribution of the current across the exciting antenna was close to symmetrical, antisymmetrical modes were not excited. When the laser beam was displayed by 0.5 m m (points z = 1.0 ram, z = 2.0 m m ) the peak of diffraction by the third transverse mode disappeared for these points coinciding with the nulls of the third mode (see fig. 2a). This is demonstrated in fig. 3b ( z = 1.0 m m ) . Figure 3c shows the insertion loss plotted on a linear scale, i.e. the frequency
Volume 92, number 1,2,3
OPTICS COMMUNICATIONS
dependence of the signal at the output of antenna 2. To radiate antisymmetric modes an asymmetric exciting field is required. For this purpose we used a tilted transmitting antenna which constituted an angle 45 ° with respect to the direction of the bias magnetic field H (see fig. 2b). In this case the antenna is not parallel to the z axis which coincides with the plane of the wavefronts of the excited transverse MSW modes. This means that the phase distribution of the exciting microwave field across the antenna is antisymmetric while the amplitude distribution is symmetric. Such a function contains symmetrical as well as antisymmetrical terms in its Fourier decomposition. On the contrary, for a parallel antenna both amplitude and phase functions are even and the corresponding Fourier series contains no antisymmetrical terms. The transverse eigenmodes play the role of the Fourier components and thus the efficiency of excitation of a transverse mode is determined by the value of the coefficient in the Fourier decomposition of the field distribution across the antenna. Now it becomes clear why a tilted antenna excites the full spectrum of modes while the parallel one excites only even modes. The results of the measurements of the efficiency of diffraction in this geometry of excitation are presented in fig. 4a. Peaks of diffraction by antisymmetric as well as symmetric tranverse modes (n = 1, 2, 3, 4) were detected. In fig. 4b is shown the frequency dependence of the insertion loss plotted on a linear scale in the case considered. The position of peaks of diffraction by surface transverse modes in fig. 3 and fig. 4 is in good agreement with the theory. Theoretical dispersion curves calculated according to the expressions given in ref. [ 6 ] for first four modes (small Ky) are plotted in fig. 5. It should be mentioned here that the Bragg condition for collinear diffraction is satisfied only when the difference of optical wavenumbers in the couple of modes Afta'~ is equal to the MSW wavenumber Ky. The difference of wavenumbers between a TEo optical mode and a TMo optical mode Ap a'E is equal to 36 c m - ~ hence the peaks of diffraction would occur for the values of MSW wavenumbers Ky= 36 cm-1. This value was obtained experimentally by means of direct measurements of the spectrum of optical waveguide modes with optically isotropic SrTiO3 prisms. This figure is in good agreement with the
15 August 1992 n=3
n=2 =
f.t
4.0l
4.02 f, GHz
ft.
4.0l
4.02 f. GHz
Fig. 4. Diffraction efficiency and insertion loss as a function of the MSW frequency in a tangentially magnetized finite sample of a YIG film in the case of asymmetric excitation. The unity at the vertical coordinate in (a) cOrresponds to the diffraction efficiency of 0.05% and it corresponds to tenfold attenuation of the microwave signal in (b). The power of the microwave signal at the input antenna 1 was of the order of 2 roW.
f,, GHz 4.02
4.01
o
35
~I
36
w
n=l
"-
n=2
g
n=3
•
n=4 u
1
Ky, cm -1
Fig. 5. Frequencyseparationof transverseMSW modesfor small wavenumbersKy. theoretical value. As well known (see, for example ref. [ 7 ] ), it is comprised of the photoelastic contribution because of the mismatch of the lattice parameters of the film and the substrate (of the order of 25 c m - 1 for a Y I G - G G G structure of the ( 111 ) crystallographic orientation) and of the usual splitting of the indices of refraction for the modes of orthogonal polarization in a dielectric isotropic waveguide. The frequencies of the circles in fig. 5 correspond to experimental peaks in fig. 3 and fig. 4. They are located on a vertical line corresponding to the Bragg condition Ky= 36 cm- i. 33
Volume 92, number t ,2,3
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We have also optically detected the volume transverse MSW modes. These are the modes existing below the frequency f±=[fu(fu+fM)] 1/2, where fN=yH, fM=y4xM; 4~M is the saturation magnetization of the YIG film; 7 is the gyromagnetic ratio. In our case f± =4.00 GHz. Since we used the zero couple of optical modes we were able to detect only volume MSW modes really existing in the domain Ky> 36 c m - i. For the parameters of our experiment these are modes with n > 4. A very strong peak corresponding to the volume mode n = 6 has been observed in the case of asymmetric excitation (see fig. 2b). It is very clearly seen in fig. 4a (marked with an arrow ). We did not succeed in detecting transverse MSW modes when using higher couples of optical waveguide modes. The value of AftTE for optical modes in a YIG film increases with the mode number what corresponds to efficient diffraction by MSW with greater values of wavenumbers. It varies from 80 cm-~ for the first couple of optical modes to 580 c m - i for the sixth couple of optical modes (the top couple). For greater values of Ky dispersion curves for transverse MSW modes go closer to one another. This is illustrated in fig. 6: for Ky= 36 cm -m (the zero couple of modes) the dispersion curves are separated by approximately 10 MHz while for Ky=80 cm-~ (the first couple of optical modes) this separation is already of the order of 5 MHz. As a consequence in our case the frequency resolution of the optical technique was not sufficient to separate optical responses of different MSW transverse modes. Certainly for other experimental geometries one
15 August 1992
can succeed in optical probing of separate transverse MSW modes when using higher couples of optical modes. For example, greater values of the length of the magnetooptical interaction lead to higher resolution of the optical technique. On the other hand, the frequency separation due to the transverse quantization is the more pronounced the less is the width of the film sample. Parameters that can considerably change the spectrum of optical modes, such as the film thickness and the mismatch of the lattice parameters of the film and the substrate, should also be taken into account when estimating the applicability of higher couples of optical modes for MSW optical probing.
3. Conclusions The experiments on the collinear diffraction of optical waveguide modes in a YIG film have proven that this technique is an extremely efficient instrument for optical probing of MSW. Its resolution is sufficient to separately detect transverse surface and volume MSW modes and measure the distribution of the magnetic field in them even in comparatively wide samples of magnetic films with the width of the order of several millimeters. The optical measurements of the spectrum of transverse MSW modes have shown that the use of an exciting antenna inclined with respect to the external bias magnetic field gives rise to efficient radiation of antisymmetric modes.
References f, G//z n=l
4.04 n=3 n=4 4.02
'l 20
40
60
80 Ky, cm-!
Fig. 6. Dispersion curves for transverse MSW modes.
34
[ ! ] A.D. Fisher, J.N. Lee, E.S. Gaynor and A.B. Tveten, Appl. Phys. Lett. 41 (1982) 779. [2] C.S. Tsai, D. Young, W. Chen, L. Adkins, C.C. Lee and H. Glass, Appl. Phys. Len. 47 (1985) 651. [3] A.A. Stashkevich, B.A. Kalinikos, N.G. Kovshikov, O.G. Rutkin, A.N. Sigaev and A.N. Ageev, Soy. Tech. Phys. Lett. 13 (1987) 20. [4] S.H. Talisa, IEEE Trans. Magn. MAG-24 (I988) 2811. [5] R.W. Damon and J.R. Eshbach, J. Phys. Chem. Solids 19 ( 1961 ) 308. [6 ] T.W. O'Keeffe and R.W. Patterson, J. Appl. Phys. 49 (1978) 4886. [ 7 ] A.M. Prokhorov, G.A. Smolenskii and A.N: Ageev, Usp. Fiz. Nauk 143 (1984) 33.
OPTICS COMMUNICATIONS
Interaction of optical waveguide modes with magnetostatic surface waves in tangentially magnetized finite samples V.V. M a t y u s h e v , A.A. S t a s h k e v i c h a n d A.I. L u k y a n o v Institute of Electrical Engineering, 197376Saint Petersburg, Russia
Received 12 February 1992; revised manuscript received 30 March 1992
To increase the efficiency of the diffraction of optical wavegnide modes by magnetostatic waves (MSW) and thus to improve the performance of the magnetooptical devices based on this principle one has to confine both waves in a limited sample of a ferrimagnetic film. This paper reports the results of the experimental investigation of the peculiarities of the collinear interaction of guided light with surface MSW in a three-dimensional YIG-GGG waveguide. While measSring the frequency response of the interaction we scanned the light beam across the film. This made it possible to identify the optical response of different transverse modes of surface MSW (n= 1, 2, 3, 4) and of volume MSW (n=6). To excite different numbers of transverse modes including antisymmetric ones we changed the orientation of the transmitting antenna with respect to the direction of the bias magnetic field.
1. Introduction
2. Experiment
It has been shown in a n u m b e r o f experimental papers [ 1-4] that efficient magnetooptical ( M O ) interaction o f optical waveguide modes with magnetostatic waves ( M S W ) in thin ferrimagnetic films holds promise for the development o f integrated-optical signal-processing and light modulating devices operating at frequencies as high as 3 - 4 0 GHz. This interaction can also be utilized as a probing instrument to study magnetostatic waves in thin films. In all earlier works it was taken that a MSW is propagating in a ferrimagnetic film o f infinite dimensions. For tangentially magnetized films it means that the classical relations obtained in ref. [ 5 ] may be directly applied. On the other hand, as shown in ref. [ 6 ] the spectrum o f MSW in a tangentially magnetized finite sample o f a ferrimagnetic film is radically changed owing to the additional quantization along the transverse coordinate (transverse modes). Besides, narrowing o f a ferrimagnetic waveguide is one o f possible ways to increase the efficiency o f the M O interaction. The purpose o f this paper is the experimental investigation o f the peculiarities o f the collinear interaction o f guided optical modes with MSW in a three-dimensional ferfirnagnetic waveguide magnetized in the tangential transverse direction.
A simplified schematic diagram of the experimental set-up is given is figi 1. An yttrium iron garnetgadolinium gallium garnet ( Y I G - G G G ) layered structure was utilized as a waveguide for MSW and optical waves. The film was 3.8 ~tm thick and 3 m m wide. The sample was magnetized to saturation by a transverse bias field H of 800 Oe. We used a HeNe laser ( 2 = 1.15 ~tm) as an optical source. A lock-in amplifier was used to increase the sensibility o f the photodetection. MSW were excited and received by two antennas made o f gold wire 50 ~tm in diameter (antenna 1 and antenna 2, respectively). It has been shown in ref. [ 6 ] that MSW in a fer-
SIGNAL
Fig. I. Simplified scheme of the experimental set-up.
0030-4018/92/$05.00 © 1992 Elsevier Science Publishers B.V. All rights reserved.
31
Volume 92, number 1,2,3
OPTICS COMMUNICATIONS
15 August 1992
rimagnetic film magnetized in the transverse tangential direction is a multimode wave process owing to the quantization along the transverse coordinate. The wavenumber of transverse modes Ky is given by the following expression K 2 = (nrc/w)2/ltl +K 2 ,
n=l
( 1) f, Gnz
where w= 3 m m is the width of the waveguide,/z, is the diagonal component of the tensor of magnetic permeability, n = 1, 2, 3, .... The distribution of the magnetostatic potential under the condition of pinned spins at both edges of the waveguide can be written as
b n=l
~0= [A exp (Kxx) + B e x p ( - K ~ x ) ] ×exp(iKyy) s i n ( n n z / w ) ,
f, GHz
0
(2)
where A and B are two constants; Ky is the wavenumber of a propagating MSW, Kx is a transverse wavenumber characterizing the MSW field distribution across the film thickness. A focused laser beam 200 rtm in diameter was propagating-along the MSW waveguide. The beam was scanned across the film along the z coordinate together with the photodetector. Figure 2 is illustrating the procedure of the measurement of the distribution of magnetization in the transverse modes ( n = 1, 2, 3...). The results of the measurements of the frequency dependence of the intensity of light wave diffracted by MSW are presented in fig. 3. For the probing of a MSW a collinear regime of TMo~TEo diffraction has been utilized. This means that the incident opa
Z
[
n=l
ANTENNA 1
t
2.or LIGHT
n=3
n=l
//
ANTENNA 2
n=2 n=3
b
T
Fig. 2. Excitation of transverse MSW modes (a) by a parallel antenna, (b) by a tilted antenna.
32
c
~d.
4.01
4.02
f GHz
Fig. 3. Diffraction efficiency and insertion loss as a function of the MSW frequency in a tangentially magnetized finite sample of a YIG fdm in the case of symmetric excitation. The unity at the vertical coordinate in (a) and (b) corresponds to the diffraction efficiency of 0.05% and it corresponds to tenfold attenuation of the microwave signal in (c). The power of the microwave signal at the input antenna 1 was of the order of 2 mW.
tical wave was a TMo mode and the diffracted wave was a TEo mode. The curve in fig. 3a corresponds to the case when the light beam is positioned in the central point z = 1.5 m m (see fig. 2a). Two clearly seen peaks are the result of the diffraction of light by the first and the third transverse modes (symmetric even modes n = l, 3). No optical signals corresponding to antisymmetric odd modes have been observed. Since the distribution of the current across the exciting antenna was close to symmetrical, antisymmetrical modes were not excited. When the laser beam was displayed by 0.5 m m (points z = 1.0 ram, z = 2.0 m m ) the peak of diffraction by the third transverse mode disappeared for these points coinciding with the nulls of the third mode (see fig. 2a). This is demonstrated in fig. 3b ( z = 1.0 m m ) . Figure 3c shows the insertion loss plotted on a linear scale, i.e. the frequency
Volume 92, number 1,2,3
OPTICS COMMUNICATIONS
dependence of the signal at the output of antenna 2. To radiate antisymmetric modes an asymmetric exciting field is required. For this purpose we used a tilted transmitting antenna which constituted an angle 45 ° with respect to the direction of the bias magnetic field H (see fig. 2b). In this case the antenna is not parallel to the z axis which coincides with the plane of the wavefronts of the excited transverse MSW modes. This means that the phase distribution of the exciting microwave field across the antenna is antisymmetric while the amplitude distribution is symmetric. Such a function contains symmetrical as well as antisymmetrical terms in its Fourier decomposition. On the contrary, for a parallel antenna both amplitude and phase functions are even and the corresponding Fourier series contains no antisymmetrical terms. The transverse eigenmodes play the role of the Fourier components and thus the efficiency of excitation of a transverse mode is determined by the value of the coefficient in the Fourier decomposition of the field distribution across the antenna. Now it becomes clear why a tilted antenna excites the full spectrum of modes while the parallel one excites only even modes. The results of the measurements of the efficiency of diffraction in this geometry of excitation are presented in fig. 4a. Peaks of diffraction by antisymmetric as well as symmetric tranverse modes (n = 1, 2, 3, 4) were detected. In fig. 4b is shown the frequency dependence of the insertion loss plotted on a linear scale in the case considered. The position of peaks of diffraction by surface transverse modes in fig. 3 and fig. 4 is in good agreement with the theory. Theoretical dispersion curves calculated according to the expressions given in ref. [ 6 ] for first four modes (small Ky) are plotted in fig. 5. It should be mentioned here that the Bragg condition for collinear diffraction is satisfied only when the difference of optical wavenumbers in the couple of modes Afta'~ is equal to the MSW wavenumber Ky. The difference of wavenumbers between a TEo optical mode and a TMo optical mode Ap a'E is equal to 36 c m - ~ hence the peaks of diffraction would occur for the values of MSW wavenumbers Ky= 36 cm-1. This value was obtained experimentally by means of direct measurements of the spectrum of optical waveguide modes with optically isotropic SrTiO3 prisms. This figure is in good agreement with the
15 August 1992 n=3
n=2 =
f.t
4.0l
4.02 f, GHz
ft.
4.0l
4.02 f. GHz
Fig. 4. Diffraction efficiency and insertion loss as a function of the MSW frequency in a tangentially magnetized finite sample of a YIG film in the case of asymmetric excitation. The unity at the vertical coordinate in (a) cOrresponds to the diffraction efficiency of 0.05% and it corresponds to tenfold attenuation of the microwave signal in (b). The power of the microwave signal at the input antenna 1 was of the order of 2 roW.
f,, GHz 4.02
4.01
o
35
~I
36
w
n=l
"-
n=2
g
n=3
•
n=4 u
1
Ky, cm -1
Fig. 5. Frequencyseparationof transverseMSW modesfor small wavenumbersKy. theoretical value. As well known (see, for example ref. [ 7 ] ), it is comprised of the photoelastic contribution because of the mismatch of the lattice parameters of the film and the substrate (of the order of 25 c m - 1 for a Y I G - G G G structure of the ( 111 ) crystallographic orientation) and of the usual splitting of the indices of refraction for the modes of orthogonal polarization in a dielectric isotropic waveguide. The frequencies of the circles in fig. 5 correspond to experimental peaks in fig. 3 and fig. 4. They are located on a vertical line corresponding to the Bragg condition Ky= 36 cm- i. 33
Volume 92, number t ,2,3
OPTICS COMMUNICATIONS
We have also optically detected the volume transverse MSW modes. These are the modes existing below the frequency f±=[fu(fu+fM)] 1/2, where fN=yH, fM=y4xM; 4~M is the saturation magnetization of the YIG film; 7 is the gyromagnetic ratio. In our case f± =4.00 GHz. Since we used the zero couple of optical modes we were able to detect only volume MSW modes really existing in the domain Ky> 36 c m - i. For the parameters of our experiment these are modes with n > 4. A very strong peak corresponding to the volume mode n = 6 has been observed in the case of asymmetric excitation (see fig. 2b). It is very clearly seen in fig. 4a (marked with an arrow ). We did not succeed in detecting transverse MSW modes when using higher couples of optical waveguide modes. The value of AftTE for optical modes in a YIG film increases with the mode number what corresponds to efficient diffraction by MSW with greater values of wavenumbers. It varies from 80 cm-~ for the first couple of optical modes to 580 c m - i for the sixth couple of optical modes (the top couple). For greater values of Ky dispersion curves for transverse MSW modes go closer to one another. This is illustrated in fig. 6: for Ky= 36 cm -m (the zero couple of modes) the dispersion curves are separated by approximately 10 MHz while for Ky=80 cm-~ (the first couple of optical modes) this separation is already of the order of 5 MHz. As a consequence in our case the frequency resolution of the optical technique was not sufficient to separate optical responses of different MSW transverse modes. Certainly for other experimental geometries one
15 August 1992
can succeed in optical probing of separate transverse MSW modes when using higher couples of optical modes. For example, greater values of the length of the magnetooptical interaction lead to higher resolution of the optical technique. On the other hand, the frequency separation due to the transverse quantization is the more pronounced the less is the width of the film sample. Parameters that can considerably change the spectrum of optical modes, such as the film thickness and the mismatch of the lattice parameters of the film and the substrate, should also be taken into account when estimating the applicability of higher couples of optical modes for MSW optical probing.
3. Conclusions The experiments on the collinear diffraction of optical waveguide modes in a YIG film have proven that this technique is an extremely efficient instrument for optical probing of MSW. Its resolution is sufficient to separately detect transverse surface and volume MSW modes and measure the distribution of the magnetic field in them even in comparatively wide samples of magnetic films with the width of the order of several millimeters. The optical measurements of the spectrum of transverse MSW modes have shown that the use of an exciting antenna inclined with respect to the external bias magnetic field gives rise to efficient radiation of antisymmetric modes.
References f, G//z n=l
4.04 n=3 n=4 4.02
'l 20
40
60
80 Ky, cm-!
Fig. 6. Dispersion curves for transverse MSW modes.
34
[ ! ] A.D. Fisher, J.N. Lee, E.S. Gaynor and A.B. Tveten, Appl. Phys. Lett. 41 (1982) 779. [2] C.S. Tsai, D. Young, W. Chen, L. Adkins, C.C. Lee and H. Glass, Appl. Phys. Len. 47 (1985) 651. [3] A.A. Stashkevich, B.A. Kalinikos, N.G. Kovshikov, O.G. Rutkin, A.N. Sigaev and A.N. Ageev, Soy. Tech. Phys. Lett. 13 (1987) 20. [4] S.H. Talisa, IEEE Trans. Magn. MAG-24 (I988) 2811. [5] R.W. Damon and J.R. Eshbach, J. Phys. Chem. Solids 19 ( 1961 ) 308. [6 ] T.W. O'Keeffe and R.W. Patterson, J. Appl. Phys. 49 (1978) 4886. [ 7 ] A.M. Prokhorov, G.A. Smolenskii and A.N: Ageev, Usp. Fiz. Nauk 143 (1984) 33.