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Coupled mode aspects of echo amplification in a ferrimagnet
Volume 53A, number 2
PHYSICS LETTERS
2 June 1975
COUPLED MODE ASPECTS OF ECHO AMPLIFICATION IN A FERRIMAGNET* D.E. KAPLAN Lockheed Palo Alto Research Laboratory, Palo Alto, California, USA 94304 Received 24 March 1975 The formation of amplified ferrimagnetic echoes due to localized parametric coupling among spin modes is interpreted using the dispersion relation for an axially magnetized slab and compared with experimental results.
Echo phenomena with temporal properties similar to the paramagnetic spin echo [1] can be observed in narrow linewidth ferrimagnets such as yttrium iron garnet (YIG) under conditions for which the echo is strongly amplified [2, 3] with respect to a weak initial pulse in a two-pulse sequence. Various models for the amplification, which is unique within this class of echo phenomena, have been presented [4—6].However, they are not specific to any particular experimental configuration. In this letter we show that echo amplification due to local parametric coupling [4] within the spin mode manifold of a convenient experimental geometry can be related to properties of the dispersion relation in the neighborhood of the local resonance frequency. The wave numbers of the spin modes producing the echo may then be specified and the dependence of amplified echo parameters upon the sample geometry interpreted. Fig. 1 illustrates a configuration in which amplified echoes can readily be observed. A single crystal YIG wafer of thickness ~l mm is incorporated near the end face of an axially magnetized polycrystalline YIG slab, in a region where the internal magnetic field has a slow axial variation [7] The equivalence of the sat.
uration magnetization, 4nM5, for the single and polycrystalline material prevents discontinuities in the internal field. The relatively broad linewidth of the polycrystalline YIG restricts echo formation to the single crystal wafer, which is excited by a transverse microwave field from a folded tape slowwave structure. The experiment is performed at room tempera. ture in the frequency range of 4 to 11 GHz. For this geometry, Vasile and LaRosa [8] find the
~II
POLYC~E~4~jj~
CRYSTAL VIG WAFER hrf ~
WAVE
CIRCUIT Fig. 1. Schematic diagram of experimental configuration, with
single crystal YIG wafer
incorporated near polycrystalline slab endface. Slow-wave coupling structure is shown separated from YIG slab for clarity.
dispersion relation to be characterized by a coupling region of frequency breadth ~wc centered at the local resonance frequency o~~(z) = ‘yH~(z)where H~(z)= internal magnetic field. ~wc is given by the expression (1) 2(wmwexa2k~)”2. = ‘I4irM 5, ~~ex = exchange frequency = 7Hex, a = lattice constant, and k~= transverse wave number of minimum value k~ n/b, where b = slab width. Within the coupling region defined by ~Wc, the wave numbers of the backward magnetostatic wave and forward spin wave, km2 and k52, respectively, are cornplex with kmz = k2, 1k2 I 1 O~.This satisfies the conditions for active parametric coupling [9] and gives rise to a local standing wave. For values of b in the range 1 to 4mm in a YIG slab with ~ —3 X lOb sec~,~ 8.8 X 1013 sec~,anda = 5 X 10—8 cm, we find ~Wc 106 sec~.A pulse sequence with spectral breadth ~x ~ ~wc then excites simultaneousHere
=
(s.)m
‘~
~ Supported by the Naval Research Laboratory and the Lockheed Independent Research Fund.
‘~
149
Volume 53A, number 2
PHYSICS LI~TTERS
PROPAGATING MAGNETOSTATIC REGION LOCAL K FREQUENCIES
PROPAGATING EXCHANGE REGION
L~WC~~ wj~yH ~cOU~LIN61GION 1~ j~
Km~K~1) 1) —1o~ K~(CM Fig. 2. 1)ispersion relation diagram for axially magnetized slab illustrating coupling region continuum for amplified echo formation.
ly a plurality of these coupled regions which form a continuum in a smoothly varying axial field as illustrated in fig. 2. The localized active parametric coupling distinguishes echo formation in this mode ensemble from that of the echo from noninteracting oscillators as typified by the familiar spin echo. Herrmann Ct al. [4] have shown that a parametrically coupled oscillator system with a frequency interaction range a, excited by a weak pulse followed after an interval r by an intense pulse, will produce an echo with amplitude growing exponentially to a maximum value at rm~ 1/a, At this time the echo may be several orders of magnitude larger than the initial pulse. The echo amplitude then decays rapidly with further increase of pulse interval due to an increasing phase disparity between the coupled modes. For slap widths b of a few mm, Tma~ is conveniently observable experimentally permitting evaluation of the interaction range a, which may be ii~ terpreted as the coupling range ~ of Vasile and d LaRosa [8]. The following experimental points are particularly noteworthy: (a) The largest values of ~ for a given slap width b are obtained if the single crystal YIG wafer is located sufficiently far from the endface to minimize the inplane variation of the axial field, but where there remains a field gradient of order 100 Oe/rnm. Under these conditions, in a 3-mm slab width for example, we find 0.8 X 106 sec, giving a value for a 1.2 X 106 sec1. This is in remarkably close agreement with the value of ~wc given by eq. (1) for b 3 mm (k~ 10) of~w~ 1.6 X 106 sec1. ‘—j
150
,
upon (b)the ~ amplitude is found of tothe varysecond directly pulse. with slap width, (x~COMPLEX,
‘~‘
Observed values of echo gain at Tmax are in the i04, as predicted by ref. [4] depend-
range of 102 to
ing primarily upon the efficiency of coupling to the electromagnetic field, and at Tm~ the echo amplitude also exhibits an expected exponential dependence
—
‘~
2 June 1975
reflecting an increase in the coupling frequency range as the width diminishes. This is to be expected from eq.(1). tal (c) wafer is located progressively closer slabcrysendTm~ decreases monotonically as to thethe single face. The larger in-plane variation of the axial field in this region also increases the coupling breadth. (d) Coupling to the electromagnetic field is found to be greatly enhanced by the use of a slowwave structure (such as the meanderline illustrated in fig. 1) with parameters adjusted to provide a transverse field cornponent hrf that is approximately phase matched to the transverse magnetic wave number k~.The energy absorbed by the sample, which is governed by an integral of the form [10]
I mt(k~)X hrf dv vol
is maximized under these conditions, resulting in echo gain saturation with a IOns duration pulse of power less than 100 mW. We wish to acknowledge helpful discussions with Drs. W.I. Dobrov and G.F. Herrmann. [I] 12]
E.L. Hahn, Phys. Rev. 80 (1950) 580.
D.E. Kaplan, R.M. Hill and G.F. Herrmann, J. App!. Phys 40 (1969) 1164. 13] J.A. Deryugin, Yu.A. Nechiporuk and A.V. Tychinskii, Soy. Phys. Solid State 14 (1972) 264. 14] G.F. Herrmann, D.E. Kaplan and R.M. Hill, Phys. Rev. 181 (1969) 829. [5] G.F. Herrmann, R.M. Hill and Dli. Kaplan, Phys. Rev. B2 (1970) 2587. 16] Physics Ia.Ia. Asadullin and U.Kh. Kopvillem, Ukranian J. of 18 (1973) 565. 17] D.E. Kaplan, W.I. Dobrov and G.F. Herrmann, AlP Conf. Proc. 5 (1971) 1569. [8] C.F. Vasile and R. LaRosa, J. Appl. Phys. 39 (1968) 1863. [9] W.H. Coupled mode Sections and parametric (JohnLouise!l, Wiley and Sons, 1960) 1.8 andelectronics 5.3. 110] P. Fletcher, I.H. Solt and R. Bell, Phys. Rev. 114 (1959) 739,
PHYSICS LETTERS
2 June 1975
COUPLED MODE ASPECTS OF ECHO AMPLIFICATION IN A FERRIMAGNET* D.E. KAPLAN Lockheed Palo Alto Research Laboratory, Palo Alto, California, USA 94304 Received 24 March 1975 The formation of amplified ferrimagnetic echoes due to localized parametric coupling among spin modes is interpreted using the dispersion relation for an axially magnetized slab and compared with experimental results.
Echo phenomena with temporal properties similar to the paramagnetic spin echo [1] can be observed in narrow linewidth ferrimagnets such as yttrium iron garnet (YIG) under conditions for which the echo is strongly amplified [2, 3] with respect to a weak initial pulse in a two-pulse sequence. Various models for the amplification, which is unique within this class of echo phenomena, have been presented [4—6].However, they are not specific to any particular experimental configuration. In this letter we show that echo amplification due to local parametric coupling [4] within the spin mode manifold of a convenient experimental geometry can be related to properties of the dispersion relation in the neighborhood of the local resonance frequency. The wave numbers of the spin modes producing the echo may then be specified and the dependence of amplified echo parameters upon the sample geometry interpreted. Fig. 1 illustrates a configuration in which amplified echoes can readily be observed. A single crystal YIG wafer of thickness ~l mm is incorporated near the end face of an axially magnetized polycrystalline YIG slab, in a region where the internal magnetic field has a slow axial variation [7] The equivalence of the sat.
uration magnetization, 4nM5, for the single and polycrystalline material prevents discontinuities in the internal field. The relatively broad linewidth of the polycrystalline YIG restricts echo formation to the single crystal wafer, which is excited by a transverse microwave field from a folded tape slowwave structure. The experiment is performed at room tempera. ture in the frequency range of 4 to 11 GHz. For this geometry, Vasile and LaRosa [8] find the
~II
POLYC~E~4~jj~
CRYSTAL VIG WAFER hrf ~
WAVE
CIRCUIT Fig. 1. Schematic diagram of experimental configuration, with
single crystal YIG wafer
incorporated near polycrystalline slab endface. Slow-wave coupling structure is shown separated from YIG slab for clarity.
dispersion relation to be characterized by a coupling region of frequency breadth ~wc centered at the local resonance frequency o~~(z) = ‘yH~(z)where H~(z)= internal magnetic field. ~wc is given by the expression (1) 2(wmwexa2k~)”2. = ‘I4irM 5, ~~ex = exchange frequency = 7Hex, a = lattice constant, and k~= transverse wave number of minimum value k~ n/b, where b = slab width. Within the coupling region defined by ~Wc, the wave numbers of the backward magnetostatic wave and forward spin wave, km2 and k52, respectively, are cornplex with kmz = k2, 1k2 I 1 O~.This satisfies the conditions for active parametric coupling [9] and gives rise to a local standing wave. For values of b in the range 1 to 4mm in a YIG slab with ~ —3 X lOb sec~,~ 8.8 X 1013 sec~,anda = 5 X 10—8 cm, we find ~Wc 106 sec~.A pulse sequence with spectral breadth ~x ~ ~wc then excites simultaneousHere
=
(s.)m
‘~
~ Supported by the Naval Research Laboratory and the Lockheed Independent Research Fund.
‘~
149
Volume 53A, number 2
PHYSICS LI~TTERS
PROPAGATING MAGNETOSTATIC REGION LOCAL K FREQUENCIES
PROPAGATING EXCHANGE REGION
L~WC~~ wj~yH ~cOU~LIN61GION 1~ j~
Km~K~1) 1) —1o~ K~(CM Fig. 2. 1)ispersion relation diagram for axially magnetized slab illustrating coupling region continuum for amplified echo formation.
ly a plurality of these coupled regions which form a continuum in a smoothly varying axial field as illustrated in fig. 2. The localized active parametric coupling distinguishes echo formation in this mode ensemble from that of the echo from noninteracting oscillators as typified by the familiar spin echo. Herrmann Ct al. [4] have shown that a parametrically coupled oscillator system with a frequency interaction range a, excited by a weak pulse followed after an interval r by an intense pulse, will produce an echo with amplitude growing exponentially to a maximum value at rm~ 1/a, At this time the echo may be several orders of magnitude larger than the initial pulse. The echo amplitude then decays rapidly with further increase of pulse interval due to an increasing phase disparity between the coupled modes. For slap widths b of a few mm, Tma~ is conveniently observable experimentally permitting evaluation of the interaction range a, which may be ii~ terpreted as the coupling range ~ of Vasile and d LaRosa [8]. The following experimental points are particularly noteworthy: (a) The largest values of ~ for a given slap width b are obtained if the single crystal YIG wafer is located sufficiently far from the endface to minimize the inplane variation of the axial field, but where there remains a field gradient of order 100 Oe/rnm. Under these conditions, in a 3-mm slab width for example, we find 0.8 X 106 sec, giving a value for a 1.2 X 106 sec1. This is in remarkably close agreement with the value of ~wc given by eq. (1) for b 3 mm (k~ 10) of~w~ 1.6 X 106 sec1. ‘—j
150
,
upon (b)the ~ amplitude is found of tothe varysecond directly pulse. with slap width, (x~COMPLEX,
‘~‘
Observed values of echo gain at Tmax are in the i04, as predicted by ref. [4] depend-
range of 102 to
ing primarily upon the efficiency of coupling to the electromagnetic field, and at Tm~ the echo amplitude also exhibits an expected exponential dependence
—
‘~
2 June 1975
reflecting an increase in the coupling frequency range as the width diminishes. This is to be expected from eq.(1). tal (c) wafer is located progressively closer slabcrysendTm~ decreases monotonically as to thethe single face. The larger in-plane variation of the axial field in this region also increases the coupling breadth. (d) Coupling to the electromagnetic field is found to be greatly enhanced by the use of a slowwave structure (such as the meanderline illustrated in fig. 1) with parameters adjusted to provide a transverse field cornponent hrf that is approximately phase matched to the transverse magnetic wave number k~.The energy absorbed by the sample, which is governed by an integral of the form [10]
I mt(k~)X hrf dv vol
is maximized under these conditions, resulting in echo gain saturation with a IOns duration pulse of power less than 100 mW. We wish to acknowledge helpful discussions with Drs. W.I. Dobrov and G.F. Herrmann. [I] 12]
E.L. Hahn, Phys. Rev. 80 (1950) 580.
D.E. Kaplan, R.M. Hill and G.F. Herrmann, J. App!. Phys 40 (1969) 1164. 13] J.A. Deryugin, Yu.A. Nechiporuk and A.V. Tychinskii, Soy. Phys. Solid State 14 (1972) 264. 14] G.F. Herrmann, D.E. Kaplan and R.M. Hill, Phys. Rev. 181 (1969) 829. [5] G.F. Herrmann, R.M. Hill and Dli. Kaplan, Phys. Rev. B2 (1970) 2587. 16] Physics Ia.Ia. Asadullin and U.Kh. Kopvillem, Ukranian J. of 18 (1973) 565. 17] D.E. Kaplan, W.I. Dobrov and G.F. Herrmann, AlP Conf. Proc. 5 (1971) 1569. [8] C.F. Vasile and R. LaRosa, J. Appl. Phys. 39 (1968) 1863. [9] W.H. Coupled mode Sections and parametric (JohnLouise!l, Wiley and Sons, 1960) 1.8 andelectronics 5.3. 110] P. Fletcher, I.H. Solt and R. Bell, Phys. Rev. 114 (1959) 739,