Pulse self-switching in optical fiber Bragg gratings

15 January 1995

OPTICS COMMUNICATIONS Optics Communications 114 ( 1995) 170- 180

Full length article

Pulse self-switching in optical fiber Bragg gratings S. Wabnitz Fondazione Vgo Bordoni, via B. Castiglione 59, 00142 Rome, Italy

Received 10 June 1994; revised version received 16 September 1994

Abstract The nonlinear propagation of counterpropagating pulses in a distributed feedback fiber grating is numerically analysed. We discuss the role of the input pulse time width, mean wavelength and peak power on the self-switching, time compression and break-up of the transmitted and reflected pulses from the grating.

1. Introduction Since the tirst demonstration of photo-induced optical fiber Bragg gratings by Hill and coworkers in 1978 [ I], significant progress was made in the fabrication technology of fiber Bragg reflectors [ 2-51. These advances open the way to the extensive use of fiber gratings in fiber optic components and devices such as strain sensors, wavelength division multiplexers, fiber and diode lasers [ 61. In a distributed feedback (DFB) fiber, a periodic corrugation of the linear refractive index matches forward and backward light waves at a certain Bragg wavelength. As a result, for input wavelengths close to the Bragg value, most of the light that is coupled into the grating is back reflected [ 7]. This well known effect is usually described in terms of coupled mode equations [ 8,9]. The presence of an intensity dependent contribution to the linear refractive index [ lo] may shift the Bragg wavelength through self or cross-phase modulation. This permits self-switching and bistability [ 11,121, as well as the control of the the reflection from the grating at the Bragg wavelength by means of an intense pulse at a different wavelength [ 13-161. Moreover, in combination with the large chromatic disper-

sion that is associated with the frequency dependent linear response of the grating [ 171, nonlinearity may lead to the temporal reshaping, compression [ 18,191 and break-up [ 20-221 of intense light pulses. Nonlinearity induced pulse self-switching and break-up has been recently observed in a semiconductor DF’B waveguide [ 231. Another intriguing phenomenon that is peculiar to the nonlinear response of a grating is the possibility, for wavelengths close to the Bragg value, of trapping light as a standing wave (gap soliton) in the grating itself [ 24,251. Although experimental evidence of gap soliton generation has not yet been reported, numerical simulations indicate that zero velocity gap solitons may be generated in a grating of finite length by illuminating both ends of the grating with two counterpropagating equally intense beams 1261. Theoretical analyses of the grating coupled mode equations showed that gap solitons may also propagate inside the grating with any value of the group velocity, provided that its absolute value is less than the group velocity of linear waves [ 27-321. In this work, we characterize by numerical simulations the switching of an intense pulse that self-shifts the Bragg resonance of a fiber grating. We study the

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171

S. Wabnitz /Optics Communications 114 (1995) 170-180

dependence of the switching power and of the fractional changes of transmitted and reflected energy on the input pulse time width and wavelength detuning from the Bragg condition. We also associate the particular switching points in the energy transmission characteristics with the occurrence of temporal reshaping effects in the reflected and transmitted pulses. This may permit for example to infer the occurrence of pulse compression in the transmitted pulses by simply measuring the power dependence of the average transmission from the grating.

F = umexp( t=Ti?,

-ivZ), z = ZRjw’,

ut + uz = iv + ifiu + i( [vi2 + (T[u[~)u, U, - v, = iu + i0v + i( lu12+ alv12)v,

one obtains

Pulse propagation in a nonlinear DFB fiber may be conveniently analysed by means of coupled mode theory. Let us express the electric field E in the periodic waveguide in terms of two counterpropagating modes at frequency w and with complex envelopes El and E2, E(r, Z, T) = [F( Z, T) exp(@Z> + B(Z,T)exp(-i/?Z)]G(r),whereP?27rn/Aisthe modal propagation constant and G(r), r = (X, Y) is the common transverse mode profile. These modes are coupled through a linear refractive index corrugation oftheform&=nlcos(AZ+4).Oneobtains[18]

U,+U,

BT - OJ’B.~= iii exp( 2ivZ - i4) F

i- ix( IFI + (+IB12)B ,

(1)

where w’ = dw( p) /d/? is the fiber mode group velocity, Y = p - A/2 = p - PO = 2m( & - A) /Ai is proportional to the wavelength detuning from the Bragg wavelength Ao. Whereas L?and x = 47rnzo’/( AA,ff) are linear [33] and nonlinear coupling coefficients, and A,e is the effective area of the fiber mode. Note that R is expressed in unitsof s-’ , whereas K = R/w’ is the coupling coefficient in units of m-’ . Moreover, CT is the ratio between the self-phase modulation (SPM) and cross-phase modulation (CPM) coefficients. The subscripts denote derivatives. In silica fiber gratings, (+ = 0.5. We also set, without loss of generality, 4 = 0 in Eqs. ( 1) . In terms of the dimensionless variables

(3)

where L?= z&/i? is the dimensionless detuning. Note that if one rewrites Eqs. (3) in terms of the frequency shifted variables

2. Equations

+ ix( IB12+ (+IFj2) F ,

(2)

one obtains the equations

U=uexp(-iat),

FT + W’FZ = iiiexp( -2ivZ + i4)B

B = ofiexp(ivZ),

V=uexp(-iL!t),

(4)

=iV+i(IV12+alUj2)U,

V, - V, =iU+i(lU12+alVj2)V.

(5)

Eqs. (5), with u = 0, are well known in field theory as the massive ‘Ihirring model, which is completely integrable by means of the inverse scattering transform [ 34,351. In general (i.e., with nonzero a), the Thirring solitons may be continuously deformed into exact solitary wave solutions of Eqs. (5) [ 291. The group velocity of these solitary pulses is a free parameter that may take any value between zero (steady gap soliton) and the velocity of linear waves in the grating. Note that other solutions of Eqs. (5) exists, for example two-soliton breathers [ 361 and nonlinear continuous waves [ 261. In particular, as reviewed in the appendix, the nonlinearity permits the existence of continuous waves whose wavelength lies within the linear stop band of the grating. These solutions express the intensity induced shift of the Bragg resonance which is at the origin of the self-switching phenomena.

3. Self-switching We intend to analyse here the effects of varying the mean wavelength and the time width of the input pulses on the intensity-dependent energy reflection and transmission characteristics of the grating. Therefore we numerically solved Eqs. (3) for various values of the wavelength detuning 0 within the linear stop-band of the grating. Whereas we considered pulse widths that are either longer or shorter than

172

S. Wabnitz /Optics Communications 1I4 (1995) 170-180

the round trip time delay through the grating. We restrict our attention here to the case of a single incident pulse at the z = 0 end face of the grating. We consider an hyperbolic secant pulse with amplitude A and full time width tf,,,hm= 1.763 W. This yields the boundary conditions for Eqs. (3) U(Z =O,t) U(Z =

KL,

u(z,t=O)

=Asech[(t-rc)/W], t)

=

0, =o,

(6)

where to >> W. The round trip time delay through the grating is equal to n = 2L/w’, or 6t = ~KL. Therefore & = tfwhm for W = 1.13KL. In the shulations, we have considered a set of parameters that are appropriate for existing photo-induced fiber gratings [ 371. We set the grating strength equal to KL = 1.9, which yields about 91 percent intensity reflection at the Bragg wavelength. On the other hand, we assumed a grating length Z = L = 3.7 mm. The condition KL = 1.9 yields K = 5.1 x lo* m-‘. Let us define the full bandwidth of the grating as the range of wavelengths within the first two zeroes of the linear reflectivity (which correspond to transmission resonances [ 221) K*

f&3=

sinh* (SL)

2? cash* ( SL) + v2 sinh* (SL) ’

where S* = K* grating bandwidth

(7)

One obtains the dimensionless [8,18,15]

v*.

2JW KL

= J lu(z J lu(z +CC

=u(z,t=O)

r(A) =

Fig. 1 shows the calculated energy switching characteristics of the transmitted and reflected light, for different values of the frequency detuning 0 from the Bragg resonance condition. Fig. 1 also shows the dependence on the input dimensionless power p = A2 of the energy fractional transmission (dashed line) and reflection (solid line) from the grating. The fractional transmission T and reflection R are calculated as

.

With KL = 1.9, one obtains & = 3.9. In real units, with n = 1.5 and A = 1.064 pm, the full bandwidth of the grating is equal to AAn = 0.24 nm (or 104 GHz). In the numerical solutions of Eqs. (3) and (6)) we considered first the case of a 100 ps input pulse full width, which yields W = 5.8. Eqs. (3) and (6) were numerically solved by integrating along the characteristics with a .fourth-order Runge-Kutta scheme. Note that the spectral width of the hyperbolic secant input pulses in Eq. (6) is, in the dimensionless units of Eqs. (3), equal to fib = 0.19, that is approximately 20 times smaller than the full spectral width an of the grating reflectivity.

T=E,-’

L,t)]*dt,

-ca

+CC

R=E,-’

=O,t)/*dt,

(9)

--oo

where EO = 2A2W is the input pulse energy. The fractional energy reflection and transmission are directly related to the corresponding quantities that may be measured by a detector whose response time is slow with respect to the temporal width of the pulses. In Fig. 1a the frequency detuning 0 = 1, which corresponds to an input wavelength that is 0.06 nm lower than the Bragg wavelength, i.e., just at the edge of the linear stop band of the grating. As can be seen, input power shifts the stop band to higher wavelengths. Therefore the grating becomes progressively more transparent as the input power grows larger. Note that the reflectivity rapidly switches down to R N 0.2 whenever p reaches the unit value. Whereas the reflectivity maintains a nearly constant value for high powers. As we shall see, this residual reflection is due to the nonlinear pulse reshaping and compression that occurs within the grating. This leads to an intense compressed pulse that sits on low power background wings. Fig. lb corresponds to the Bragg resonance input wavelength R = 0. As can be seen, the low power reflectivity attains its maximum value R N 0.91. The energy reflectivity is initially nearly constant with power, and then it shows a sudden drop to R N 0.6 whenever p crosses the switching value p = 2. On the other hand, for p > 2 the reflectivity R oscillates while it decreases slowly. As we shall see later, for p > 2 the oscillations are due to strong reshaping of the fields inside the grating, namely, a temporal compression of the transmitted pulse and a break-up of the reflected pulse.

S. Wabnitz /Optics Communications 114 (I 995) I TO-180 I

III

l-

-

III1

173 I

II11

/cL=1.9

R=O -r=lOO

III ps

-

.0 -

.6 .4 .2 --_-_--0

IIll

1

IllI

III1

nL=1.9

0=-l

III

T=loo

Illl’lill’lrll’lill 2 3 1 INPUT PEAK POWER p

0

4

2 3 1 INPUT PEAK POWER p

III

I

1

ps

1

L=1.9

\ I-

.4

: I

_.’

III

1

l-l=-1.4

4

III-

7=100

ps-

‘\

.2

0

INP:T

3 2 PEAK POWER p

4

0

11

I

‘.__

-----

I

II

III

1

III

l-

- nL=1.9 a I, .’ \ .6‘\

\\

-0

INPUT2PEAK P:WER

6 p

I

l

I

I

l

IrI

2 4 INPUT PEAK POWER p

0

1

n=-3

III

1

III

r=lOO

ps -

(9 + \\

0 lll’lll’iiI’IIJ 0

6 2 4 INPUT PEAK POWER p

Rg. 1. Fraction of transmitted (dashedlines) and reflected(solid lines) energy from a nonlinear DFB grating with detunings R of the input pulse from the

6

a

KL

=

1.9 and different

Bragg resonance. The grating length is 3.7 mm and the input pulse width is 100 ps.

Fig. lc is obtained with R = - 1, that is the input wavelength is 0.06 nm longer than the Bragg value. In this case, by increasing the input power one shifts the

Bragg resonance towards the input wavelength, which explains the initial slow increase of the reflectivity R when the power p grows larger. Whenever p N 3.2,

174

S. Wabnitz /Optics Communications 114 (1995) 170-180

the stop band shifts past the input wavelength and the reflectivity shows an abrupt down switching of about 20 percent. Figs. Id-f have been obtained for fi = - 1.4, -2, -3 respectively, which correspond to an initial wavelength that is longer than the Bragg value by 0.084,O. 12, and 0.18 nm, respectively. As can be seen, in all cases the reflectivity initially increases with power until it reaches a maximum value and then R tends to drop slowly and oscillate with p. From the overall examination of the input wavelength behavior of the selfswitching curves in Fig. 1, one can see that optimal switching (in terms of both relatively low switching power and high switching fraction) is obtained at the wavelength that corresponds to the first zero on the high wavelength side of the reflection spectrum (i.e., n-a/2= -2). Note that the unit power value p = 1 corresponds, in real units, to the peak power PC = Ae~~A/(hn2). By taking the silica fiber value n2 = 2.4 x 10e2’ m2/W, A = 1 pm, and A,# = 1.2 x 10-t’ m2, one obtains the relatively high power PC = 13 kW. On Bragg resonance, Fig. lb shows that the switching power is P, N 2P,. This is precisely the power that shifts the grating reflection spectrum by fin( KL = 1.9) /2 2~ 2 [ 151. Note that at the switching power, self-phase modulation leads to the nonlinear phase shift Ac$ = PSxL/(2w’) = KL through a grating of length L. Moreover, it may be interesting to compare the on resonance self-switching power in the fiber grating with the switching power P,’ = 47rw’/( Lx) = in a nonlinear directional cougrPs/(KL&(KL)) pler [ 381 whose linear coupling length is L (we used the relation P, = P,&/2). Fig. la and Fig. le show that the off-resonance switching power in the grating may be further reduced to P,/2. Therefore a 27r reduction of switching power may be typically obtained in a Bragg grating when compared with a directional coupler of the same length. As long as the input pulse width is longer than the round trip time delay 6T, the behavior of the wavelength dependent switching of the nonlinear reflectivity is very similar to the case that we considered above. On the other hand, whenever tf,vhm < 8t (or the spectral width of the input pulses is of the same order of the grating bandwidth), a substantial increase of the switching power results, along with a corresponding decrease of the switching fraction. In fact, Fig. 2

shows the input wavelength dependence of the selfswitching energy reflection (solid lines) and transmission (dashed lines) characteristics, whenever the input pulse width is equal to 10 ps (i.e., W = 0.58). Note that 6T = 38 ps. The comparison between Fig. la and Fig. 2a shows that the spectral broadening that is associated with a reduction of the input pulse width from 100 to 10 ps decreases the low-power reflectivity by about 20 percent. In fact, the spectral width of 10 ps pulses is about half the full grating bandwidth firi. As a result of averaging the grating response over the spectral components of the pulse, a reduction of the total energy reflectivity is observed. Note that in the case of initial Bragg resonance, the comparison between Fig. lb and Fig. 2b shows that whenever the pulse fills half of the grating length the switching power doubles to p = 4. From Figs. 1, 2, one can see that R(p) + T(p) = 1, which means that all of the incident pulse energy is either reflected in the backward propagating mode or it is transmitted in the forward propagating mode. In other words, no generation of stationary gap solitons is observed in the present simulations. Nevertheless, as we shall see, a slowing down is observed in the transfer of energy between input and output pulses (see also Ref. [ 191) . For all-optical switching purposes, it is important to complement the information that is provided by the power dependence of the energy transmission and reflection coefficients of Figs. 1, 2 with the details on the transmitted and reflected pulse profiles. Figs. 3a-c show the temporal dependence of the transmitted and reflected pulse powers for different values of the input peak power p. Here the input pulse witdth is 60 ps, KL = 1.9, the grating length is L = 3.7 mm, the mean wavelength of the input pulse is centered at the Bragg value (i.e., 0 = 0) as in Fig. lb, whereas to = 30. Note that the averaged switching characteristics of Figs. 1, 2 remain nearly unchanged whenever the input pulse width is reduced from 100 down to 60 ps. In Fig. 3a the input power is p = 1.8, that is slightly lower than the energy switching value (see Fig. lb). As can be seen, both the transmitted and reflected pulses show some temporal asymmetry but no break-up. In particular, the trailing (leading) edge of the transmitted (reflected) pulse is sharper than the leading (trailing) edge. This temporal asymmetry may also be interpreted as the result of optical bistability in the switching response of the nonlinear DFB grating [ 111. Note that the time

175

S. Wabnitz /Optics Communications 114 (1995) 170-180 ‘III

l-

I’ll

-

III’

rcL=1.9 n=o

IIll

T=lo

ps

-

.6 .4 __--

__-C

/’ /’

/‘_

/

.2 ------0-

II&T 1

1’ ’ ’ ’ I ’ ’ ’ ’ I ’ ’ ’ ’ I ’ ’ ’ ‘-I

F

nL=1.9

n=-1

T=lo

ps

I-

.4 I-

1

-l

-__ ---_

-________----

$

c

II

t

-I

.2

@I

k 0

1

t’ ’ ’ ’ ’ ’ ’ ’ ’ ’ ‘3 nL=1.9

.4 .2

rI=-2

7=10

ps

INPUT2PEAK P:WER I,,

III

1

d

I,,

FcL=1.9 R=-3

I/ 1 ..

.6

4

2 3 PEAK POWER p

0

0 0

‘. . .

.-..

-_ ----___--

2 INPUT PEAK P:WER

7=10

p ,I

ps

I-

p

6

0

INP:T

6

-l

PEA:

POW:R

p

8

fig. 2. Same as in Fig. 1, for an inputpulsewidthof 10 ps.

delay rd between the peaks of the transmitted and refleeted pulses is slightly larger than the transit time through the grating. On the other hand, in Fig. 3b p = 2.2, which is right above the switching power: a sharp

increase from p = 0.3 up to p = 2 of the transmitted pulse peak power is observed. Note the enhanced asymmetry and temporal compression (with respect to the input pulse width) of the up-switched trans-

S. Wabnitz /Optics Communications I14 (1995) 170-180

176

I I I

2-ql

.3 -t

:

(4

N

.2 -

2 3 .l

-

1.5

I

i

II

I

I

(,

I

I

I

r

(W

1:

.5 E

O_!

i

i

I

I

I

I

i

i

i

0

I I I II 20

40

60 III

0-i

i

i

I I I I1:

I I

0

20

0

20

40

60

40

60

-

l-

f

0

I

40

20

0

Time

1.5

0 -l

I

,

,

I

I

I I

60

t

Time

40

20 I

I

,,I

t

60 III

-

l-

I

.5 0.;

I I I II 40 60

I I , 0

20

Time

t

mitted pulse. Also, the time delay between reflected and transmitted pulses increases up to 2.6 times the transit time through the grating. Correspondingly, the reflected pulse breaks up and a delayed subpulse appears. If the input peak power is further increased up to p = 3, the transmitted energy remains nearly unchanged. Finally, Fig. 3c shows that the transmitted pulse is strongly compressed by the combined action of nonlinearity and grating dispersion [ 181. Whereas multiple peaks appear in the reflected pulse envelope.

Fig. 3. Intensitiesoftransmitted (lu(L)12) andreflected pulses from a nonlinear DFB fiber grating of length L Here (T = 0.5, KL = 1.9, the input pulse FWHM is detuning L? = 0, and the input pulse power is (a) A2 (b) p = 2.2, and (c) p = 3.

(10(0)1~) = 3.7 mm. 60 ps, the = p = 1.8,

Figs. 4a-c show the pulse reshaping that occurs in the course of the self-switching of a 60 ps pulse that is initially detuned to a wavelength which is 0.12 nm higher than the Bragg value (i.e., fi = -2). The comparison between Fig. 4a and Fig. 4b, where p = 1.8 and p = 4.8, respectively, shows that in this case it is the reflected pulse that is switched up by increasing the input power p. Note that the temporal profile of the reflected pulse remains in this case almost unchanged until p = 5, whereas a tail appears in the trailing edge

S. Wabnitz /Optics Communications I1 4 (1995) I70- 180

I

6-

I

I

I

(4

.8

I

I

I

I

I

(

I

I

I

:-

(b)

?

.6 c

4-

.4 =.2 -

.2 t

I I I

o-1

20

0

20

Time

81 6_ I

I

I

, I I

0:;

0

I

I

I

40

60

0

20

40

60

O

20

t I

Time 1

I

I

40

60

40

60

t

I

Cc)

4r 2k

0

20

0

20

Time

40

60

40

60

t

of the transmitted pulse. Fig. 4c shows that significant temporal reshaping occurs for p > 5. As can be seen by comparing Figs. 4b, 4c , although the transmitted and reflected energies remain almost constant for p > 5, the transmitted pulse width is compressed by about ten times, whereas its peak power exhibits a tenfold increase. The simulations show that the pulse widths of these strongly compressed transmitted pulses remain relatively stable whenever the input wavelength is varied

Fig. 4. Same as in Fig. 3, with f2 = -2, and (a) p = 1.8, (b) p = 4.8, and (c) p = 6.

through the stop-band. Moreover, Fig. 5 shows that by injecting into the grating relatively long pulses (here the input pulse width is 0.7 ns), one observes a modulational instability which breaks up the field into a train of highly compressed pulses whose time width is of about 10 ps (see also Refs. [ 22,21,26 ] ). In Fig. 5, the input pulse peak power is constant and equal to p = 4, whereas the input frequency detuning varies from 0 = 0 (Fig. 5a) to 0 = -1 (Fig. 5b) and R = - 1.5 (Fig. 5~). As can be seen, the peak power

S. Wabnitz /Optics Communications 114(1995) 170-180

178

N

x

+I

Y 100

120

140

160

180

200

JO0

120

140

160

180

200

120

140

160

180

200

100

120

140 160 Time t

180

200

2 1.5 1 .5 0 100

Time t III~III

III

III

III_

61 4-

@) 1

20~,,1~,1,~,,1~~1,~111~ 100

120

140

160

180

200

100

120

140

160

180

200

Time t and the time width of the transmitted pulses remains nearly constant with 0. Whereas the pulse repetition rate decreases from 40 GHz in Fig. 5a to 30 GHz in Fig. 5b and 10 GHz in Fig. 5c.

4. Conclusions We have analysed by numerical simulations the nonlinear detuning of the Bragg resonance in optical fiber gratings. The detuning is achieved by means of the

Fig. 5. Same as in Fig. 3,with an input pulse FWHM of 0.7 ns, the input powerp = 4, and the detunings (a) fi = 0, (b) L! = - 1, and (c) R= -1.5. self and cross-phase modulation inside the grating. We have characterized the variation with input power of the energy transmission and reflection from the grating, when a single unchirped hyperbolic secant pulse is incident at one end of the grating. In particular, we have studied the dependence of the switching power and of the fractional energy switching as functions of the input pulse detuning from the Bragg wavelength. We found that optimal switching conditions may be achieved for pulses that are longer than the round trip

S. Wabnitz /Optics Communications 114 (1995) 170-180

time through the grating, whenever the input wavelength is up-shifted with respect to the Bragg value, and corresponds to the first zero of the linear reflectivity spectrum. Whenever the input pulse width is reduced down to less than the travel time through the grating, a significant decrease of the switching fraction and increase of the switching power are predicted. For input peak powers so large that the stop band is shifted above the input wavelength, the transmitted field reorganizes in the form of highly compressed solitary-like pulses. Although the switching powers are typically large, as we have seen they compare favourably with the switching power of a nonlinear directional coupler of the same length. Moreover, the switching powers that we have derived here might be reduced by orders of magnitude by employing the fast refractive index changes in a semiconductor DFB laser biased at transparency [ 391. Several potential applications of nonlinear Bragg switches may be conceived. For example, both wavelength stabilization and fast saturable absorption may be provided by a nonlinear Bragg element inserted in the cavity of a mode locked fiber or diode laser. Acknowledgements

The Author would like to thank R. Kashyap and M. Tamburrini for several helpful discussions. This work was carried out in the framework of a project of european cooperation in the field of science and technology (COST project 241), and under the agreement between the Fondazione Ugo Bordoni and the Istituto Superiore Poste e Telecomunicazioni. Appendix

Continuous wave solutions of Eqs. (5) may be easily obtained by direct substitution. In the linear regime, that is neglecting the intensity dependent phase shifts, one obtains the solutions U=Aexp[i(kz v=-

A

k-R

-tit>],

exp[i(kz - at)],

where k2 = 0’ - 1. This dispersion relation yields that, whenever the frequency lies in the gap - 1 < J1 < 1,

179

the amplitudes of the fields experience an exponential decay (proportional to exp ( -d-z ) ) as they propagate in the grating. In the nonlinear regime, we search for solutions of Eqs. (5) in the form U=Aexp[i(kz

- 0t) +i&],

V = Bexp[i(kz

- fit) + i&_r].

One easily obtains that, for k = 0, A = fB R=~fl

- (1 +(+)A*.

1261 (10)

The above relationship shows that the frequency stop

band of the grating shifts down as the input intensity grows larger. With u = 0.5, shifting half of the stop band is obtained for A2 = 2/3 (note that the stop band -1 < L2< 1 does not coincide, in general, with the full band -a/2 < 0 < G/2), In terms of the physical parameters that have been introduced in the text, this corresponds to a peak power of 8.7 kW.

References

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