Characteristics of microwave magneto-optic modulation based on magnetostatic waves

Optics Communications 282 (2009) 1724–1727

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Characteristics of microwave magneto-optic modulation based on magnetostatic waves Bao-Jian Wu *, Cheng-You Luo, Kun Qiu Key Lab of Broadband Optical Fiber Transmission and Communication Networks of the Ministry of Education, University of Electronic Science and Technology of China, Chengdu 610054, People’s Republic of China

a r t i c l e

i n f o

Article history: Received 21 November 2008 Received in revised form 8 January 2009 Accepted 8 January 2009

a b s t r a c t According to the magneto-optic (MO) perturbation theory, the coupled-mode equations for guided optical waves (GOWs) with microwave magneto-static waves (MSWs) in MO film waveguides are presented, which cannot only be used to analyze the MO modulation induced by MSW pulses, but also explain the optical pulse compression caused by chirped MSWs. The rectangular pulse modulation of continuous guided optical waves by magnetostatic forward volume waves (MSFVWs) excited under normal bias magnetic field is theoretically studied in detail. It is shown that: (1) the diffracted light pulse approximates to an isosceles trapezoid, which is in agreement with the experimental results, and the duration time and the flat-top response time are equal to the sum and difference between the width and transit time of the rectangular MSFVW pulse, respectively; (2) utilizing a small duty factor of MSFVW pulse helps to improve the modulating data rate, which is in inverse proportion to the transit time, and the relative peak intensity of the diffracted light pulse is less than the corresponding Bragg diffraction efficiency when the MSW duty factor is less than 0.5. Ó 2009 Elsevier B.V. All rights reserved.

1. Introduction With the rapid development and wide application of optical communication technologies for several decades, more and more signals are transmitted and processed in the domain of optics or photonics. For example, in the radio-over-fiber (ROF) systems, RF signal processing implemented in optical domain is often involved in the subcarrier multiplex modulation; the data information carried by the RF carriers may also be directly transformed into light waves by using a new class of magnetostatic wave (MSW)-based magneto-optic (MO) Bragg modulators. The microwave MO Bragg cells are capable of providing all standard functions implemented by using the acousto-optic Bragg cells and operating at higher carrier frequencies of microwave signals (typically from 0.5 to 40 GHz by simply varying bias magnetic fields) and also at faster modulation speeds due to higher velocities of MSWs [1]. By means of the MSW pulse modulation on continuous guided optical waves (GOWs) input to the bismuth-doped-YIG-based MO Bragg cell, a triangular pulse of diffracted light has been demonstrated experimentally [2]. However, to the best of our knowledge, the characteristics of the MO pulse modulation have not been quantitatively analyzed until now. In this paper, the microwave MO pulse coupling theory is presented by the coupled-mode and perturbation analysis. The modulation of rectangular magnetostatic forward * Corresponding author. Tel.: +86 28 66141580; fax: +86 28 83201084. E-mail address: [email protected] (B.-J. Wu). 0030-4018/$ - see front matter Ó 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2009.01.012

volume wave (MSFVW) pulses is taken into account in term of the MSFVW pulse width and transit time, and our theoretical results are in agreement with the experimental ones [2,3]. The MO coupled-mode equations obtained here can also be applied to the MO pulse compression [4,5]. 2. The MO coupled-mode equations For simplicity, we consider the noncollinear interaction between incident TM0-mode GOWs and MSFVWs excited in a perpendicularly magnetized MO film as shown in Fig. 1. In the case, there exist the undiffracted TM0-mode and diffracted TE0-mode GOWs at the output. The MO effects as a perturbation may lead to the additional relative permittivity at normal magnetization as follows:

Derij ðr; xÞ ¼ Deð1Þ rij



   1 ð1Þ 1  ~m ðr; x  xm Þ þ Derij ~m ðr; x  xm Þ g g 2 2

ð1Þ ð1Þ rij ½

where De denotes the linear function of effective dynamic mag~m ðr; xÞ in frequency ~m ðr; xÞ and its complex conjugate g netization g domain [6,7] and i,j = y,z; r ¼ x^i þ y^j is position vector; xm is the carrier frequency of MSWs. By substituting Eq. (1) into the perturbation equations with the MO effects [2] and neglecting the MO coupling in the y direction, according to the coupled-mode theory [8] and the deduction similar to that used in [2,7], the microwave MO coupled-mode equations which can describe the modulation

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B.-J. Wu et al. / Optics Communications 282 (2009) 1724–1727

z

[111]

where g^lx ¼ cl  g lx , and cl is dependent on the transit time Tt, i.e.,

Diffracted light pulses Undiffracted light x H0

Output transducer

x [11 2 ]

1 cl ¼ D

y

o

[ 1 10]

TM 0 light pulses

Input transducer

y o

MO film

GGG substrate

of MSFVW on GOWs for the anti-Stokes interaction can be given in time domain as follows [5]:

 ðdÞ ð1Þ  ¼ jzy 12 gm ðy; tÞ C TE ðr; tÞejðbuTM bdTE þkMSW Þr ðdÞ  ðuÞ : @C ðdÞ ðr;tÞ @C TE ðr;tÞ ð1Þ  TE þ V 1 ¼ jyz 12 gm ðy; tÞ C TM ðr; tÞejðbdTE buTM kMSW Þr dTEx @x @t @C TM ðr;tÞ @t

ð2Þ ðuÞ C TM

ðdÞ C TE

where and designate, respectively, the complex amplitudes of the undiffracted TM0-mode and diffracted TE0-mode GOWs in time domain; kMSW, buTM and bdTE are the wave vectors of the MSFVW, TM0- and TE0-modes at the corresponding carrier frequencies xm, xuTM and xdTE, respectively; VuTMx and VdTEx are the group velocities of the GOWs in the x direction,

   1 jk0 1 k0 fM ¼ pffiffiffiffi g mx gm ¼  pffiffiffiffi Deð1Þ g 2 2 er ryz 2 m 4 er     1  jk0 k0 fM ð1Þ 1  ffiffiffiffi p ¼  ¼  pffiffiffiffi g mx jð1Þ D e g g zy 2 m 2 er rzy 2 m 4 er

jð1Þ yz



ð3aÞ ð3bÞ

where gmx and gmy are the x and y components of MSW dynamic magnetization gm; fM ¼ f1 þ rm M0z ð2f 44 þ 23 Df Þ in which f1, f44 and Df are the MO factors related to the MO effects [2,6] and rm = jgmy/gmx. Let

8 > < x  xuTM ¼ mDx x  xdTE ¼ nDx > : x  xm ¼ lDx

ð4Þ

where Dx is the angular frequency resolution, and m, n and l take integer values. Expand the complex amplitudes into following Fourier series:

8 þ1 P ðuÞ > > Am ðxÞejðmDxtbm;TM xÞ > C TM ðx; tÞ ¼ > > m¼1 > > > þ 1 > P > ðdÞ > Bn ðxÞejðnDxtbn;TE xÞ > < C TE ðx; tÞ ¼ n¼1

ð5Þ

þ1 P > > > gm ðy; tÞ ¼ gl ejðlDxtkl yÞ > > > l¼1 > > > þ 1 > P >  > : gm ðy; tÞ ¼ gl ejðlDxtkl yÞ

where bm,TM = mDx/VuTMx, bn,TE = nDx/VdTEx, kl = lDx/Vgm, and Vgm is the group velocity of MSWs. By substituting Eq. (5) into Eq. (2), using the anti-Stokes frequency relationship xdTE = xuTM + xm and the perfectly phase-matching condition at the carrier frequencies bdTE  buTM  kMSW = 0, and considering the MSW transit delay in the light aperture D, the microwave MO coupled-mode equations can be rewritten as follows:

m¼1

þ 1 P > > > :

n¼1

¼

0

1; je

ðl ¼ 0Þ

jlDxT t 1

lDxT t

; ðl–0Þ

ð7Þ

2

3  AðxÞ 0 where y ¼ 4 BðxÞ 5 and M ¼ jBA

ð8Þ  jAB ; A(x) = [Am(x)] and 0

dAm ðxÞ jðmDxtbm;TM xÞ e dx

¼

dBn ðxÞ jðnDxtbn;TE xÞ e dx

k 0 fM pffiffiffi 4 er

¼

B(x) = [Bn(x)] are column vectors; the elements of coupling coefficient matrices jAB and jBA are, respectively, ½jAB m;n ¼ ðjnm Þ M ^ 0 fffiffiffi ejDmn x and ½jBA n;m ¼ jnm ejDnm x with jl ¼ 4kp er g lx and Dmn ¼ bm;TM  bn;TE ¼ ðm=V uTMx  n=V dTEx ÞDx. 3. Calculation steps The key step in analyzing the MSFVW-based MO pulse modulation from Eq. (8) as well as the boundary conditions is to calculate the coupling coefficient matrices jAB and jBA for further evaluation of the complex envelopes of diffracted light pulses. The detailed calculation steps are described as follows: (i) Calculate the phase mismatching factors Dnm = Dmn. (ii) Calculate the MO coupling coefficients jl from the Fourier P expansion coefficients g lx ¼ N1 N=2 k¼N=2  1g mx ðy ¼ 0; t ¼ kDtÞ k ej2plN , where gmx(y = 0,t) is the complex envelope of MSW pulses and Dt = T/N in which T is the calculated time window and N is the number of sampling points. (iii) Calculate the coupling coefficient matrix M(x) in Eq. (8). (iv) Determine the boundary conditions. For TM-mode incident light, the boundary conditions are expressed as Ak(x = 0)=A0k R T=2 and Bk(x = 0)=0, where A0k ¼ T1 T=2 wðtÞejlDxt dt and wðtÞ ¼ ðuÞ C TM ðx ¼ 0; tÞ is the incident optical pulse waveform in the time window T. (v) Resolve the differential equations of Eq. (8) according to the above boundary conditions and get the frequency-domain coefficients Am and Bn at the output end (x = L). Thus, the time domain envelopes of output optical pulses can be obtained by using the fast inverse Fourier transform.

4. Theoretical calculation and analysis For continuous TM-mode light, the incident boundary conditions are expressed as follows:

l¼1

8 þ1 P > > > <

e

ðuÞ

uTMx

@x

(

jlDxy=V gm dy

dyðxÞ ¼ MðxÞyðxÞ dx

Fig. 1. Modulation of rectangular MSFVW pulses on continuous GOWs.

8 ðuÞ < @C TM ðr;tÞ þ V 1

D

in which Tt = D/Vgm and glx is the x component of the Fourier expansion coefficient gl in Eq. (5). From the law of energy conservation, the time-dependent phase factors on the left- and right-sides of Eq. (6) should be equal, i.e., n = m + l. Eq. (6) can also be expressed in the matrix form as follows:

MSFVW pulses

z

Z

M 0 fffiffiffi  4kp er

þ 1 P

þ1 P

þ1 P

n¼1 l¼1 þ 1 P

Bn ðxÞg^lx ej½ðnlÞDxtbn;TE x

Am ðxÞg^lx ej½ðmþlÞDxtbm;TM x

m¼1 l¼1

ð6Þ



A0k ¼ 1; ðk ¼ 0Þ A0k ¼ 0;

ðk–0Þ

ð9Þ

For a rectangular MSW pulse with the complex envelope of t g mx ðy ¼ 0; tÞ ¼ g 0 P sMSW , the corresponding Fourier expansion Þ, where g0 is constant and l takes coefficients are g lx ¼ glp0 sinðlp sMSW T integer. So the MO coupling coefficients can be expressed as:

jl ¼

 s jc cl MSW sin lp lp T

ð10Þ

0 fffiffiffi M where jc ¼ 4kp er g 0 is the MO coupling coefficient in the interaction between continuous MSWs and continuous optical waves. Consider the coupling of fundamental mode GOWs in the Bi-doped YIG (Bi:YIG) MO Bragg cell with the light beam aperture

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D = 0.5 mm. The other parameters used here take from [9] as follows: the wavelength of incident guided light is 1.303 lm; for the Bi:YIG film, the thickness, refractive index and saturation magnetization are 3.9 lm, 2.47 and 4pMs = 1800G, respectively; the refractive index of GGG substrate is 2.01; the length and width of the microstrip line are 5 mm and 50 lm respectively; the perpendicularly applied magnetic field and the effective anisotropy field are 3600 Oe and 1660 Oe, respectively; and the MSFVW carrier frequency in the first pass band is 10 GHz. In this case, according to the dispersion characteristics of the MSFVW and the GOWs in the waveguide [8,10], the calculated group velocities of the GOWs are VuTMx = 1.2113  108 m/s and VdTEx = 1.2116  108 m/s, and the transit time of the MSFVW pulse is Tt = 17.84 ns, which is larger than the GOW delay time Td = 0.4 ns (negligible). According to the transit time and width of MSFVW pulse signals, the input and output waveforms can theoretically be illustrated as shown in Fig. 2(a)–(c). (a) The diffracted light pulses are approximately trapezoidal when the width of rectangular MSFVW pulses is larger than the transit time (sMSW > Tt). The rise-and-fall time Trf is equal to the transit time Tt, i.e., Trf = Tt, and the flat-top response time Tft is identical with the completely overlapping time between the modulating MSW pulse and the light beam aperture, i.e., Tft = sMSW – Tt. The ratio of the peak intensity

a

V =1.2289X107m/s uTMx 7 VdTEx=1.2292X10 m/s =20%, L=5mm η

=30ns > T =17.84ns

MSW

t

1.2

The above-mentioned conclusions may also be explained by means of the spatial distribution of MSFVWs. Four critical distribution states (S1–S4) in the case are sketched in Fig. 3(a) and their complex envelopes are expressed as follows:

   sMSW y þ LMSW =2 ¼ g0 P S1 : g mx y; t ¼  LMSW 2    sMSW y  ðD  LMSW 2Þ ¼ g0 P S2 : g mx y; t ¼ T t  LMSW 2    sMSW y  LMSW =2 ¼ g0 P S3 : g mx y; t ¼ LMSW 2    sMSW y  ðD þ LMSW =2Þ S4 : g mx y; t ¼ T t þ ¼ g0 P LMSW 2  yV gm t where g mx ðy; tÞ ¼ g 0 P LMSW and LMSW = VgmsMSW > D. From Fig. 3, the MSFVW pulses travel through the light beam aperture via the four critical states from no overlap (S1) to the start and end of complete overlap (S2 and S3), then to the disappearance

b

1.4 τ

of diffracted pulses to the input light intensity (also called relative peak intensity) is up to the matching diffraction efficiency (DE) gmatch = sin2(|jl|L), which is defined as the Bragg DE of continuous GOWs induced by the corresponding continuous MSFVWs (with the same peak envelope power) under perfectly phase-matching condition.

1.4 τ

7 VuTMx=1.2289X10 m/s 7 VdTEx=1.2292X10 m/s ηmatch=20%, L=5mm

=T =17.84ns

MSW

t

1.2

match

0.8

0.6

Relative intensity

1

Undiffracted GOW Normalized incident MSW pulse

0.4 t

0.2

0 −5

0.8

0.6

Undiffracted GOW

Normalized incident MSW pulse

0.4

Diffracted pulse

T

Diffracted pulse

0.2

0

0 −5

5

c

0

−8

t (s)

t (s)

x 10 1.4

τMSW=10ns < Tt=17.84ns

7 VuTMx=1.2289X10 m/s 7 VdTEx=1.2292X10 m/s ηmatch=20%, L=5mm

1.2

1

Relative intensity

Relative intensity

1

0.8

0.6

Undiffracted GOW Normalized incident MSW pulse

0.4 Tt

0.2

0 −5

Diffracted pulse

0

t (s)

5 −8

x 10

Fig. 2. Modulation of three typical MSFVW rectangular pulses on the continuous GOW.

5 −8

x 10

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B.-J. Wu et al. / Optics Communications 282 (2009) 1724–1727 2

D

( a ) L MSW >D S1

pwf

1.8

R /R

S2

b

1.6

S3 S4

( b ) L MSW =D

1.2

S 2, S 3

1

S4 ( c ) L MSW
τMSW/Tt

1.4

S1

t

Rm/Rt

S1

0.8

MSW pulse

S3

0.6

S2 S4

y

0.4

O 0.2

Fig. 3. The spatial distributions of rectangular MSFVW pulses.

0

0

5. Optimization of MO pulse modulation According to the foregoing analysis, for the case of rectangular pulse modulation, the resulting diffracted light pulse may almost be regarded as an isosceles trapezoid with the rise-and-fall time Trf = min(Tt,sMSW) and the flat-top response time Tft = |sMSW – Tt|. The diffracted light pulse duration is equal to Tp = sMSW + Tt and then the maximal data rate without intersymbol interference is Rb = 1/Tp. The duty factor of MSW modulating pulses is expressed as dc = sMSWRb. In addition, the waveform of diffracted pulses of interest can be described by the waveform parameter pwf = Tft/Tp. For a rectangular pulse, pwf = 1; for a triangular pulse, pwf = 0. We might as well introduce the parameter Rm = pwfRb to optimize the performance of MO pulse modulators. The dependences of the above-mentioned parameters on the duty factor dc are plotted in Fig. 4, where Rt = 1/Tt.From Fig. 4, dc > 0.5, dc = 0.5 and dc < 0.5 are equivalent to the three cases of (a) sMSW > Tt, (b) sMSW = Tt and (c) sMSW < Tt, respectively. In the range of sMSW > Tt, the MO pulse modulators operating at sMSW = 3Tt (dc = 0.75) have an optimal performance with Rb = 0.25Rt and pwf = 0.5. For the case of sMSW < Tt, the performance parameter Rm can be improved by reducing the

0.4

0.6

0.8

1

MSW pulse duty factor dc

of overlap (S4) again, which are in turn corresponding to the rise edge, flat top response and fall edge of the diffracted light pulses. (a) For the case with sMSW = Tt as shown in Fig. 3(b), which may is regarded as a special case of (a), the diffracted light pulse is close to a triangular waveform and the rise-and-fall time is equal to the transit time. In fact, assuming the transit time of the MSW pulse is set as 15 ns by changing the bias magnetic field, a triangular pulse is theoretically expected as well for the 15 ns square-wave MSFVW pulse, and our calculation result is basically identical to the experimental curve [2]. (b) When sMSW < Tt as shown in Fig. 2(c), the diffracted light pulse is also close to an isosceles trapezoid. The rise-and-fall time is equal to the duration of the rectangular MSFVW pulse and the flat-top response time is determined by Tft = Tt – sMSW. From Fig. 3(c) the state S2 follows the state S3 and the relative peak intensity of the diffracted light pulse is less than the matching diffraction efficiency because LMSW < D.

0.2

Fig. 4. The Characteristics of MO pulse modulation.

MSW pulse width. The data rate corresponding to sMSW = Tt/3 (dc = 0.25) is up to Rb = 0.75Rt with pwf = 0.5. Of course, it is also important for the MO Bragg modulators to decrease the MSW transit time in the light aperture D. 6. Conclusion The modulation of rectangular MSFVW pulses on continuous GOWs propagating in normally magnetized film waveguides is in detail analyzed by means of the noncollinear MO coupled-mode equations. The waveforms of diffracted light pulses obtained theoretically are basically in agreement with the experimental results. The modulating performance can be improved by appropriately reducing the MSW duty factor in the range of less than 0.5, but the relative peak intensity of the diffracted light pulse is less than the corresponding Bragg diffraction efficiency. Similarly, one can also investigate other MSFVW modulating waveforms, such as Gaussian or triangular pulses. Thus, our analysis method is useful for the design and optimization of the MSW-based MO modulators. Acknowledgement This work was supported by the National Natural Science Foundation of China (under Grant No: 60671027) and partly by the Applied and Basic Research Program of Sichuan Province (07JY029-089). References [1] C.S. Tsai, Proc. IEEE 84 (1996) 853. [2] C.S. Tsai, D. Young, IEEE Trans. Microwave Theory Tech. MTT-38 (1990) 560. [3] C.S. Tsai, D. Yong, L. Adkins, C.C. Lee, H. Glass, Topical Meeting on Integrated and Guided-Wave Optics, Orlando, FL, USA, 1984. [4] S.A. Nikitov, J. Magn. Magn. Mater. 196–97 (1999) 400. [5] Bao-Jian Wu, Xiang Gao, Chin. Phys. Lett. 25 (2008) 4006. [6] M.L. Torfeh, Courtois, L. Smoczynski, H. Le Gall, J.M. Desvignes, Physica 89B (1977) 255. [7] Bao-Jian Wu, D. Shang, K. Qiu, Jpn. J. Appl. Phys. 46 (2007) 6710. [8] A. Yariv, IEEE J. Quantum Electron 9 (1973) 919–933. [9] D. Young, C.S. Tsai, Appl. Phys. Lett. 55 (1989) 2242. [10] D.D. Stancil, Theory of Magnetostatic Waves, Springer, New York, 1993.