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2746 J. Opt. Soc. Am. B/Vol. 24, No. 10 /October 2007 Xin-Hong Jia

Performance analysis and design of tapered andchirped nonlinear Bragg gratings for

application to optical isolators

Xin-Hong Jia

School of Physics and Electronic Engineering, Sichuan Normal University, Chengdu 610068, Chinajxh�0@yahoo.com.cn

Received June 26, 2007; revised August 28, 2007; accepted August 29, 2007;posted September 10, 2007 (Doc. ID 84546); published September 27, 2007

Performance analysis of optical isolators based on the nonreciprocal property of optical bistability in taperedand chirped nonlinear Bragg grating (NLBG) has been carried out. The design method for detuning adjust-ment according to input power level and expected on–off ratio is also provided. The results display that, fortapered NLBG, the broad operation range and greater on–off ratio can be achieved by using larger taper slope;there exists an optimization range of chirp coefficient for chirped NLBG; the operation range boundary isshifted toward lower input power with the increased detuning; and the tapered NLBG is more preferable toobtain the larger on–off ratio and wider operation range simultaneously compared with chirped NLBG.© 2007 Optical Society of America

OCIS codes: 050.2770, 190.1450.

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. INTRODUCTIONt is well known that the nonlinear Bragg grating (NLBG)hows the optical bistability behavior when the incidentight is inside its photonic bandgap (PBG) [1]. The opticalistability has wide applications in optical signal process-ng, optical memory, optical limiting, optical switchingnd optical gate operations [2–4]. Many efforts have beenade to offer additional feasibility of NLBG, such as the

witching-on threshold, the switching time, the on–offwitching ratio, and dynamic stability. The technologiesainly include spatial taper, phase shift, and chirp

5–14]. For instance, the phase-shift technology can besed to obtain the lower switching-on threshold owing tohe reduced group velocity and increased interaction time7]; the introduced chirp can extend the available fre-uency range [8].We have analyzed the bistable steady characteristic

nd dynamic stability of linearly tapered nonlinear Braggrating (LT-NLBG) [13]. The different switching-onhreshold and transmittance are shown if the incident di-ection is altered, which is referred to a direction-ependent nonreciprocal characteristic. Maitra et al. fur-her outlined this property and its potential applicationor optical isolator [14]. The feasibility has been verified ifhe material with very large third-order nonlinear suscep-ibility and lower loss is used. In fact, the nonreciprocalharacteristic also occurs in linearly chirped nonlinearragg grating (LC-NLBG) [10].In 1998, Slusher et al. experimentally observed that,

or LC-NLBG, the reflected pulse splits into a pair ofulses where the transmitted pulse is a single narrowedulse evolving into a fundamental soliton [15]. Lenz et al.heoretically analyzed the adiabatic soliton compressionerformance by using the nonuniform fiber Bragg grating,hich can manufacture any dispersion profile [16]. In

0740-3224/07/102746-6/$15.00 © 2

005, Mok et al. demonstrated pulse compression andulse-train generation using kilowatt 580 ps pulses gen-rated by a compact �15 cm�3 cm�3 cm� microchip-switched laser followed by a fiber Bragg grating [17].ecently, Mok et al. observed that in a fiber Bragg grat-

ng, the subnanosecond solitons travel at 16% of the speedf light and remain undistorted [18].

The on–off ratio is a key parameter to evaluate its per-ormance for optical isolators. Since the on–off ratio isower dependent and related to detuning between inputight frequency and Bragg frequency in LT-NLBG and LC-LBG, it is essential to properly design their structuresnd adjust operation condition according to practical in-ut power level to obtain the expected on–off ratio.In this paper, the structure design and operation con-

ition selection of optical isolators based on the bistabilityf tapered and chirped NLBG have been performed by us-ng the nonlinearly coupled-mode equation (NLCME).he performance comparisons of tapered and chirpedLBG are also demonstrated. For the first time to ournowledge, the on–off ratio with the variation of detuningnd input power level for tapered and chirped NLBG haseen outlined, which may provide an instruction for prac-ical device design.

This paper is organized as follows. Section 2 gives theheoretical model and general format. The structure de-ign, operation condition selection, and performance com-arisons of LT-NLBG and LC-NLBG are provided in Sec-ion 3. Finally, Section 4 concludes our results.

. THEORETICAL MODEL. Nonlinearly Coupled-Mode Equationshe axial distribution of refractive index n can be de-cribed by

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Xin-Hong Jia Vol. 24, No. 10 /October 2007 /J. Opt. Soc. Am. B 2747

n�z� = n0 + n1�z�cos�2�

�z + ��z�� + n2�E�z��2, �1�

here E is the inner electric field of grating, � is the grat-ng period, � is the spatial phase shift, and n0, n1, and n2enote the effective mode refractive index, the linear re-ractive index modulation amplitude, and the nonlinearefractive index coefficient, respectively.

The inner electric field E can be expressed by the sumf two terms,

E = Af exp�i��0z − �t�� + Ab exp�− i��0z + �t��, �2�

here � is the carrier angular frequency, t is the time,0=� /� is the Bragg wave number, and Af and Ab repre-ent the slowly varying amplitude of forward and back-ard waves, respectively.Substituting Eqs. (1) and (2) into the wave equations

nd assuming that the loss and material dispersion cane neglected (in this paper, the nonlinear medium ofLBG is assumed to be semiconductor InP, whose loss isegligible [14]; even though its material dispersion coeffi-ient may be larger, its total material dispersion is negli-ible owing very short length), one can obtain the follow-ng nonlinearly coupled-mode equations [19]:

�Af

�z+

1

vg

�Af

�t= i�Af + ��Af�2 + 2�Ab�2�Af + �Ab�, �3a�

�Ab

�z−

1

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�Ab

�t= − i�Ab + ��Ab�2 + 2�Af�2�Ab + �*Af�,

�3b�

here vg is the light group velocity in the grating me-ium, and , , and � account for the detuning, nonlinearoefficient, and coupling coefficient, respectively, whichan be expressed by

= � − �0 = n0

�

c− �0, =

2�n2

�0,

��z� =�n1�z�

�0exp�i��z��, �4�

here c is the light velocity in vacuum, �0=2n0� is theragg wavelength, and is the confinement factor.For LT-NLBG with no spatial chirp ���z�=0�, � can be

ritten as [13,14]

��z� = �0�1 + ��z

L� , �5a�

ith

�� =��L� − ��0�

��0�, �5b�

here L is the total length of grating, �0 is the couplingoefficient at grating front end, and �� characterizes theelative variation of coupling coefficient.

For LC-NLBG with uniform coupling coefficient, thexial distribution of Bragg wavenumber can be describedy [8]

�0 = �0� +C

L2�z −L

2 �6�

here �0� is the averaged Bragg wave number and C is thehirp coefficient.

The boundary conditions are given by

z = 0: Af�0,t� = Ai�0,t�, Ar�0,t� = Ab�0,t�, �7a�

z = L: Ab�L,t� = 0, At�L,t� = Af�L,t�, �7b�

here Ai, Ar, and At are the slowly varying amplitudes ofhe incident, reflected, and transmitted waves, respec-ively.

. Numerical Simulation Methodetting the partial derivatives with respect to t in Eqs.

3a) and (3b) equal to zero, the axial evolving equations ofhe slowly varying amplitude under steady state can beeduced. As a result, the bistable steady characteristics ofT-NLBG can be analyzed numerically by means of the

ourth-order Runge–Kutta method together with bound-ry conditions.To simulate the output dynamics for short pulse input,

he modified time-domain transfer-matrix method (TMM)an be utilized [20]. Split the length L into M equal sec-ions from input end, assume each section is uniform, andabel the localized amplitudes before and after each sec-ion as Af,i+1 �Ab,i+1� and Af,i�Ab,i� �i=1,2, . . .M�, respec-ively, then

�Af,i+1�t + �t�

Ab,i�t + �t� � = TPTC� Af,i�t�

Ab,i+1�t�� , �8a�

here the time step size �t is related to spatial step sizez as �t=�z /vg�, the matrixes TC and TP respectively de-ote the coupling and detuning terms, given by

TC = � sec h�k�z� i tanh�k�z�

i tanh�k�z� sec h�k�z� � , �8b�

Tp = �expi�z� + ��Af,j�t��2 + 2�Ab,j�t��2��� 0

0 expi�z� + ��Ab,j+1�t��2 + 2�Af,j+1�t��2���� . �8c�

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2748 J. Opt. Soc. Am. B/Vol. 24, No. 10 /October 2007 Xin-Hong Jia

Starting from the initial condition, using Eqs. (8a)–(8c),he longitudinal distributions of optical fields can be com-uted. Repeating these steps, the whole output wave-orms can be obtained. It should be stressed that the pre-ision and validity can be improved by increasing M. Ourimulation results have shown that for conventionalength ��1 cm�, if M closes to 50 or more, the outputaveform remains unchanged. In this paper, M is taken

o be 70.

. RESULTS AND DISCUSSIONShe medium of NLBG is assumed to be semiconductornP, as been mentioned before, and the data used in cal-ulations are n2=2.5�10−15 m2/W, L=1 cm, k0=5 cm−1,here the nonlinear refractive index coefficient is esti-ated based on the fact that the order of InP is 104 times

arger than in silica [14].

. Performance Analysis and Design for Linearlyapered Nonlinear Bragg Gratings an example, Fig. 1(a) shows the input–output charac-

eristic curves for LT-NLBG, where ��k�=30% and L=3.

he input and output powers are normalized as Pin/Pcnd Pout/Pc, Pc is the critical input power defined as Pc4�0Aeff /3�n2L, and Aeff=0.4 �m2 is the effective area ofaveguide [14]. The switching-on threshold of negative-

apered NLBG ��0.88� is larger than positive-taperedLBG ��0.61� owing to the inner diffused energy distri-ution [13]. The operation range boundary is representedy the vertical dotted lines. For input power within �0.61o 0.88, the transmittance is larger if the incident direc-ion is from left to right because the input power exceedshe switching-on threshold of positive-tapered NLBG. Onhe other hand, for incident direction from right to left,he output is always located at the down branch of bista-ility loop of negative-tapered NLBG; in other words, theight is “stopped.”

Figure 1(b) shows the contour diagram of the on–off ra-io of LT-NLBG with the variation of input power and�k�, where L=3. For fixed ��k�, the on–off ratio is de-reased with the raised input power level because of thencreased output power of down branch of bistable loopor negative-tapered NLBG, as shown in Fig. 1(a). Then–off ratio is enhanced with the increased ��k� owing tohe strengthened nonreciprocal characteristic. The ex-

ig. 1. (a) Input–output characteristic curves for LT-NLBG, where ��k�=30%, L=3; (b) contour diagram of on–off ratio of LT-NLBGith the variation of input power and ��k�, where L=3; (c) contour diagram of on–off ratio of LT-NLBG with the variation of input powernd ��k�, where L=4; (d) contour diagram of on–off ratio of LT-NLBG with the variation of input power and detuning, where ��k�30%.

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Xin-Hong Jia Vol. 24, No. 10 /October 2007 /J. Opt. Soc. Am. B 2749

ended input power range can be obtained for larger ��k�,nd the boundary almost linearly varies with the varia-ion of ��k�.

However, only the light with power within thewitching-on thresholds of positive and negative-taperedLBG shows the nonreciprocal behavior. Out of this

ange, the detuning between the Bragg frequency and in-ident light frequency has to be adjusted. Figure 1(c) re-eals the contour diagram of the on–off ratio of LT-NLBGith the variation of input power and ��k�, where L=4.ompared with Fig. 1(b), the input operation range ishifted to lower power level. For example, the operationange is �0.32 to 0.6 for ��k�=30%. This result attributeso the fact that, for smaller detuning, the input light islose to the centre of the stop band, thus the largerwitching-on threshold is necessary to start of bistability.oreover, the on–off ratio of LT-NLBG is larger for

maller detuning if the ��k� is fixed owing to the enhancednner energy provided by the larger input power.

To give an overall insight of detuning adjustment ac-ording to different input power level, Fig. 1(d) shows theontour diagram of the on–off ratio of LT-NLBG with theariation of input power and detuning, where the larger

�k� (30%) is used to achieve the larger on–off ratio and

ig. 2. (a) Input–output characteristic curves for LC-NLBG, whhe variation of input power and �C�, where L=3; (c) contour diaC� when L=4; (d) contour diagram of on–off ratio of LC-NLBG

peration range. When the input is altered from �0.25 to.2, the detuning must be changed from �4.5 to �2. Al-hough the operation range boundary is linearly shifted tohe greater power level with decreased detuning, the in-erval of the upper and lower boundaries remains almostnchanged. This diagram allows us to determine the re-uired detuning according to practical power level andhe expected on–off ratio.

. Performance Analysis and Design for Linearlyhirped Nonlinear Bragg Gratingigure 2(a) shows the input–output characteristic curves

or LC-NLBG, where �C�=3 and L=3. Similar to taperedLBG, the clear nonreciprocal characteristic can be ob-

erved in chirped NLBG. The negative-chirped NLBGhows the larger switching-on threshold compared to theositive-chirped NLBG. The reason is that, for negative-hirped NLBG, the detuning is more and more decreasedlong the grating axis; in other words, the resonance feed-ack between inner energy and detuning is suppressed,hus the larger switching-on threshold is required to ex-ite the bistability.

Figure 2(b) gives the contour diagram of the on–off ra-io of LC-NLBG with the variation of input power and �C�,

=3, L=3; (b) contour diagram of on–off ratio of LC-NLBG withf on–off ratio of LC-NLBG with the variation of input power andhe variation of input power and detuning, where �C�=3.

ere �C�gram owith t

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2750 J. Opt. Soc. Am. B/Vol. 24, No. 10 /October 2007 Xin-Hong Jia

here L=3. For the fixed input power, the operationange boundary is almost linearly increased with the in-reased chirp, owing to the intensified difference of thewitching-on threshold. However, the on–off ratio is wors-ned if the chirp value is increased to some degree. Thisan be explained as follows. For chirped NLBG, there ex-st many Bragg frequencies along the grating axis; thusor a given incident frequency, the light will only experi-nce finite resonance feedback, and thus the inner energys weakened considerably, the on–off ratio is also small,specially for larger chirp value. The above discussionsmply that the chirp value has an optimization range tobtain the wider operation range and a larger on–off ra-io.

As one may expect, the influence of detuning an on–offatio for LC-NLBG is similar as for LT-NLBG. This ideaan be confirmed by Fig. 2(c), which depicts the contouriagram of the on–off ratio of LC-NLBG with the varia-ion of input power and �C� when L=4. The worsened on–ff ratio and downshifted operation range is observed forarger detuning.

ig. 3. Input and output pulse shapes for LT-NLBG, where ��k�=.75.

ig. 4. Input and output pulse shapes for LC-NLBG, where �C�=

Finally we plot the contour diagram of the on–off ratiof LC-NLBG with the variation of input power and detun-ng in Fig. 2(d), where �C�=3. The variation tendency isimilar to LT-NLBG by comparison with Fig. 1(d). It cane found that the magnitude order of the on–off ratio forhirped and tapered has no evident difference if the de-uning is smaller because of stronger resonance feedback.ut for larger detuning, the on–off ratio of the chirpedrating is far smaller than that of tapered grating. We canherefore conclude that the tapered NLBG is more prefer-ble to obtain the larger on–off ratio and the wider opera-ion range simultaneously.

. Switching Characteristic to Short-Pulse Input forinearly Tapered and Chirped Nonlinear Bragg Gratinghe above steady-state analyses give the operation rangef optical isolator using NLBG. In this subsection, we willurther investigate the dynamic switching characteristicsor short pulse input.

Figure 3 shows the input and output pulse shapes forT-NLBG, where ��k�=30%, L=3, and the normalized

L=3; the normalized input peak power Pin /Pc is (a) 0.65 and (b)

3; the normalized input peak power P /P is (a) 0.5 and (b) 0.6.

30%,

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Xin-Hong Jia Vol. 24, No. 10 /October 2007 /J. Opt. Soc. Am. B 2751

nput peak power Pin/Pc is (a) 0.65 and (b) 0.75. The inputeak power is selected within its operation range accord-ng to Fig. 1(a). The input pulse is assumed to be Gauss-an shape with 100 ps width. It can be seen that, whenin/Pc=0.65, the normalized peak power and width of the

ransmitted pulse for positive-tapered NLBG are �0.93nd �25 ps, respectively, which implies that the inputower exceeds the switching-on threshold [see curve b inig. 3(a)]. In contrast, for negative-tapered NLBG, theutput power is negligible because the input power ismaller than the switching-off threshold [see curve c inig. 3(a)]. In fact, the narrowed transmitted pulse forositive-tapered NLBG is a Bragg soliton [15–19]. Fromhe physical point of view, the Bragg soliton comes fromhe balance of anomalous group velocity dispersion nearBG and self-phase modulation (SPM). For input pulseith larger peak power [see Fig. 3(b)], the normalizedeak power of transmitted Bragg soliton is further in-reased up to �2.4, and its width is narrowed up to10 ps owing to the enhanced SPM.Figure 4 shows the input and output pulse shapes for

C-NLBG, where �C � =3, L=3, and the normalized inputeak power Pin/Pc is (a) 0.5 and (b) 0.6. The input peakower is selected within its operation range according toig. 2(a). The Bragg soliton shape is consistent with thexperimental observation in [15]. From the comparison ofigs. 3 and 4 it can be found that the pulse switchingharacteristic of chirped NLBG is similar to taperedLBG. It should be noted that the magnitude order of theormalized peak power of the transmitted pulse for theegative chirped NLBG is �10−3, which is far smallerhan the negative tapered NLBG ��10−5�. This meanshat the tapered NLBG is more preferable for obtaininghe larger on–off ratio.

. CONCLUSIONSn summary, by using the nonlinearly coupled-mode equa-ions, we have analyzed the performance of optical isola-ors based on the nonreciprocal property of optical bista-ility in tapered and chirped nonlinear Bragg grating.he design method for detuning adjustment according tohe input power level and the expected on–off ratio is alsorovided. The numerical simulation shows that, for ta-ered NLBG, the broad operation range and greater on–ff ratio can be achieved by using a larger taper slope;here exists an optimization range of chirp coefficient forhirped NLBG; the operation range boundary is shiftedoward lower input power with the increased detuning;nd the tapered NLBG is more preferable to obtain thearger on–off ratio and wider operation range simulta-eously compared with chirped NLBG.

CKNOWLEDGMENTSacknowledge the reviewers for their helpful comments.

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