Analysis of external optical feedback characteristicsof asymmetric, quarter-wave-shifted,distributed-feedback semiconductor lasers

Mohammad F. Alam, Mohammad A. Karim, and Saiful Islam

External optical feedback sensitivity is analyzed for a quarter-wave-shifted, index-coupled, distributed-feedback semiconductor laser with asymmetries in reflectivity of facets and in the position of a ly4 phaseshift. Proper asymmetric structures can withstand higher levels of external optical feedback comparedwith symmetric structures. However, the mode discrimination and yield are reduced for asymmetriclasers because of statistical variation of the corrugation phases at the reflecting facets. © 1997 OpticalSociety of America

Key words: Semiconductor laser, distributed feedback, external optical feedback, noise in opticalcommunication, coupled-wave theory, coupled cavity.

1. Introduction

Distributed-feedback ~DFB! semiconductor lasers arepotential candidates for highly stable, narrow-linewidth laser sources that are necessary for high-performance optical communication systems.Index-coupled DFB semiconductor lasers require aquarter-wave phase shift within the laser structureto attain single-longitudinal-mode operation at Braggfrequency. Such quarter-wave-shifted ~QWS! lasersare usually manufactured with antireflection ~AR!coatings on both facets to ensure high single-modeyield. The AR coatings on the facets make suchQWS DFB lasers more susceptible to external opticalfeedback than Fabry–Perot ~FP! type semiconductorlasers. Unwanted external optical feedback that isdue to reflections of the laser output beam from dis-tant surfaces usually causes high levels of intensitynoise in the operation of DFB semiconductor lasers inoptical communication systems.1 External opticalfeedback having power feedback ratios even as smallas 260 dB can cause external cavity modes to build

M. F. Alam and M. A. Karim are with the Electro-Optics Pro-gram, the University of Dayton, 300 College Park Avenue, Dayton,Ohio 45469-0245. S. Islam is with the Faculty of Electrical andElectronic Engineering, Bangladesh University of Engineeringand Technology, Dhaka 1000, Bangladesh.

Received 1 April 1996; revised manuscript received 23 August1996.

0003-6935y97y184131-07$10.00y0© 1997 Optical Society of America

up along with the internal cavity modes.2 The non-linear interactions among the internal and externalcavity modes induce mode competition and enhancefluctuations on spontaneous emission, causing excessintensity noise when external cavity modes arepresent.3 Recently, a criterion of external opticalfeedback sensitivity for DFB semiconductor lasersbased on mode competition between internal and ex-ternal cavity modes was developed.2 The externalcavity modes appear when the external optical feed-back exceeds a certain minimum power feedback ra-tio called the critical feedback ratio. The criticalfeedback ratio has been identified as a sensitive func-tion of the structural parameters such as couplingstrength, facet reflectivities, and corrugation phaseangles at the phase shift position and at the facets.2,4

Mechanisms other than those dependent on modecompetition between internal and external cavitymodes have also been proposed.5–11 These phenom-ena are, however, applicable only under a strong ex-ternal optical feedback ratio of 230 dB or more.Coherence collapse is one of the dominant mecha-nisms of intensity noise generation. The deteriora-tion of performance caused by the presence ofexternal cavity modes that appear at feedback ratiosfar less than that required for the onset of coherencecollapse in DFB semiconductor lasers has alreadybeen demonstrated for nonreturn-to-zero modula-tion.12

In this paper we analyze the effect of structuralasymmetries in QWS DFB lasers on the critical feed-back ratio and demonstrate that certain configura-

20 June 1997 y Vol. 36, No. 18 y APPLIED OPTICS 4131

tions of structural asymmetries can improve theexternal optical feedback sensitivity considerably.Practical asymmetric QWS structures have alreadybeen reported that exhibit high endurance againstexternal optical feedback.12 Our analysis clarifiesthe reason behind the improvement in such asym-metric QWS structures over symmetric structuresand also takes into account the mode selectivity andexpected yield characteristics of such structures.

2. Analytical Model

Figure 1 shows the model adopted for external opticalfeedback analysis. The output beam from the rightfacet of the semiconductor laser is reflected at thesurface of an external device at a distance lext awayfrom the right facet and fed back to the laser. Weassume G to be the ratio of the feedback power to theoutput power at the facet and h to be the couplingratio of optical feedback into the active region in thelaser cavity. Hence, hG is the effective feedback ra-tio of the semiconductor laser.

A semiconductor laser exhibits two groups of lasingmode under the effect of external feedback exceedingthe critical feedback ratio. One is a group of internalcavity modes ~p modes! whose oscillating frequenciesare determined by the laser cavity itself. Another isa group of external cavity modes ~m modes! that buildup around each internal cavity mode with a fre-quency separation characterized by the distance toreflection point lext. Figure 2 shows a typical opticalfrequency spectrum of a DFB laser under externaloptical feedback. We denote N to be a mode numbercorresponding to an internal or an external cavitymode.

We obtained the rate equations for photon number

Fig. 1. Schematic of a phase-shifted DFB semiconductor laserwith external feedback.

Fig. 2. Optical frequency spectrum of the internal and externalcavity modes of a DFB laser under optical feedback. The internalcavity modes are denoted by p. The external cavity modes aredenoted by m and m9.

4132 APPLIED OPTICS y Vol. 36, No. 18 y 20 June 1997

SN of mode N and the injected electron density h bytaking nonlinear interaction among lasing modesinto account as2,3

dSN

dt5 FAN 2 BSM 2 (

MÞN~HN~M! 1 D!SM 2 Gth~N!GSN

1ChV

ts1 ^N~t!V, (1)

dh

dt5 2

1V (

NANSN 2

h

ts1

IeV

1 ^e~t!, (2)

AN 5 ja@h 2 hg 2 b~lN 2 l0!2#, (3)

B 59j2aRcv

2\v

4ε0n2V Spin

\ D2

~hth 2 hs!, (4)

D 5 2B, (5)

HN~M! 53j2a2

4V~hth 2 hg!~1 1 ja!

3 F j~vN 2 vM! 1B~I 2 Ith!

eGth~N!

1CGth~N!Ith

I 2 IthG

4 HF j~vN 2 vM! 1B~I 2 Ith!

eGth~N!

1CGth~N!Ith

I 2 IthG

3 F j~vN 2 vM! 11ts

I 2 Ig

Ith 2 IgG

1Gth~N!

ts

I 2 Ith

Ith 2 IgJ 1 c.c., (6)

where AN is the linear gain coefficient, j is the con-finement factor of the optical field in the active re-gion, a and b are the coefficients giving the gain slopeand the wavelength dispersion relation, respectively,l0 is the wavelength at the gain peak, hg is the trans-parent level of the electron density, B and D are thegain saturation coefficients for the identical mode anddifferent mode, respectively, that are due to the burn-ing effect on energy of the laser polarization charac-terized by the intraband relaxation time tin for theelectron wave,13 Rcv is the dipole moment, n is therefractive index, and hs is an injection level that char-acterizes the gain saturation coefficient. HN~M! isanother saturation coefficient that is due to the beat-ing vibration of the injected electron density givingan asymmetric saturation profile of photon ener-gy.14,15 hth is the threshold electron density, V is thevolume of the active region, ts is the electron lifetime,Ig and Ith are the transparent current level and thethreshold current level, respectively, given by

Ig 5ehgV

ts, Ith 5

ehthVts, (7)

and a is the linewidth enhancement factor.16 Gth~N!

is the threshold gain level for mode N. C is thespontaneous emission factor17 defined as a ratio of

the spontaneous field going into a lasing field. ^N~t!and ^e~t! are fluctuation components that are dueto spontaneous emission. Correlation functionsamong these fluctuation terms are given as follows:

^^NV^MV& 5 2~SND 1 1!ChD

tsVdN,M, (8)

^^eV^eV& 5 2@1 1 C (N

SND#hD

tsV, (9)

^^NV^eV& 5SNDGth~N!

V2 2 2~SND 1 1!ChD

tsV. (10)

Here, ^NV and ^eV are the frequency components offluctuation terms ^N~t! and ^e~t!, respectively. SNDand hD are dc components of photon number andelectron density, respectively. The nonlinear inter-actions among lasing modes are described in terms ofD and HN~M! in Eq. ~1! and are called the mode com-petition phenomena. The mode competition en-hances fluctuations on spontaneous emission causingexcess intensity noise when more than one mode ispresent. The relative intensity noise ~RIN! can becalculated from

RIN 5

KS(N

SNVD2LS(

NSNDD2 , (11)

where SNV is the fluctuation component of the photonnumber at angular frequency V. Calculation of RINas a function of the effective feedback ratio hG revealsthat, when the feedback ratio is small enough, thelaser operates without the generation of excess noiserevealing the intrinsic quantum noise. On the otherhand, when the ratio exceeds a minimum criticallevel hGc, the RIN starts to increase abruptly.2 Thecritical feedback ratio hGc above which RIN increasesabruptly to a high value also corresponds to thatparticular feedback ratio below which only a singleinternal cavity mode around the Bragg wavelengthcan exist but above which external cavity modes startto appear ~in addition to the internal cavity modes!.Thus, the critical feedback ratio hGc is a key param-eter for feedback noise sensitivity in semiconductorlasers.

In the model shown in Fig. 1, the DFB laser isassumed to have two sections with lengths l1 and l2.The sections 2l1 , z , 0 and 0 , z , l2 are, respec-tively, denoted section 1 and section 2 for conve-nience. Amplitude reflectivities of the left and rightfacets are r1 and r2, and the corresponding powerreflectivities of the facets are R1 and R2, respectively.Structural asymmetry can be introduced by setting l1Þ l2 and r1 Þ r2 in the DFB structure. For any onesection, the index of refraction n~z! is assumed to

vary along the z direction as

n~z! 5 n# 1 FDn2

exp~ j2bBz 1 jf! 1 c.c.G , (12)

where n# is the average refractive index over the zdirection and Dn is its amplitude variation. The cor-rugation is assumed to have a first-order Bragg grat-ing with a spatial period L. Thus, the Bragg wavenumber is given by bB 5 pyL. The initial phase ofthe corrugation at z 5 0 is assumed to be f, and thespecific values of f are f1 and f2 in sections 1 and 2,respectively. The DFB laser having a quarter-waveor ly4 phase shift has f1 5 f2 5 0.5p. Equation ~12!was used along with Maxwell’s equations to arrive ata pair of coupled-wave equations for a DFB laser.18

We assume N to be a mode corresponding to either aninternal or an external cavity mode. Solutions of thecoupled-wave equations for the Nth mode can be writ-ten as

AN~z! 5 a1 exp~gNz! 1 a2 exp~2gNz!, (13)

BN~z! 5 b1 exp~gNz! 1 b2 exp~2gNz!, (14)

where AN~z! and BN~z! are the forward and backwardcomponents of the propagating wave of mode N.The propagation constant gN is given by

gN2 5 Sg 2 a

22 jdbND2

1 ki2, (15)

where g and a are, respectively, the gain and losscoefficients of the traveling wave. bN satisfies thewave equation for a DFB laser in an unperturbedmedium without any loss, corrugation, or externalfeedback, given by

~¹2 1 mε0n#2vN

2!FN~x, y!exp~2jbNz! 5 0, (16)

where m is the magnetic permeability, ε0 is the per-mittivity of the free space, and FN~x, y! is the nor-malized transverse component of the fielddistribution of mode N. The deviation of the wavevector from the Bragg wave vector is given by dbN >bN 2 bB. The index coupling coefficient ki is givenby

ki 5mε0vN

2

2bB *2`

`

*2`

`

n# DnuFN~x, y!u2dxdy, (17)

and the angular lasing frequency vN of mode N isgiven by

vN 5 2pfN 5bN

n# Îε0m5

bB 1 dbN

n# Îε0m. (18)

Next, the boundary conditions for the two sectionsof the laser are utilized to find the different solutions:AN,1~z! and BN,1~z! in section 1 and AN,2~z! andBN,2~z! in section 2. Because of the external opticalfeedback, the modified reflectivity of the right facet

20 June 1997 y Vol. 36, No. 18 y APPLIED OPTICS 4133

becomes19

r29 5 ÎR2 1 ~1 2 R2!ÎhG exp~2juext!, (19)

where uext 5 4plextylext is the phase delay of thefeedback light in the external cavity having wave-length lext. Multiple reflection of the feedback lightbetween the laser right facet and the surface of theexternal device is neglected assuming that hG issmall enough. The boundary conditions at z 5 2l1,z 5 0, and z 5 1l2 are given by

r29AN,2~l2!exp~2jbBl2! 5 BN,2~l2!exp~ jbBl2!, (20)

r1AN,1~2l1!exp~ jbBl1! 5 BN,1~2l1!exp~2jbBl1!, (21)

AN,2~0! 5 BN,1~0!, (22)

AN,1~0! 5 BN,2~0!. (23)

Using the boundary conditions in Eqs. ~20!–~23!, weget the condition for oscillation of a phase-shiftedindex-coupled DFB semiconductor laser as follows:

H2gNr1 exp~2j2bBl1!cosh~2gNl1! 2 Fr1 exp~2j2bBl1!

3 S2jdbN 1g 2 a

2 D 2 jki exp~ jf1!Gsinh~gNl1!J3 $2gNr29 exp~2j2bBl2!cosh~2gNl2!

2 Fr29 exp~2j2bBl2!S2jdbN 1g 2 a

2 D2 jki exp~ jf2!Gsinh~gNl2!%

5 H2gN cosh~gNl1! 1 F2jkir1 exp~2j2bBl1 2 jf1!

1 S2jdbN 1g 2 a

2 DGsinh~gNl1!J3 H2gN cosh~gNl2! 1 F2jkir29 exp~2j2bBl2 2 jf2!

1 S2jdbN 1g 2 a

2 DGsinh~gNl2!J . (24)

Equation ~24! involves complex quantities. By set-ting both the real and imaginary parts of Eq. ~24!equal to zero, solutions for the quantities g and dbNcan be obtained when all the other parameters ~in-cluding the feedback ratio hG! are known. Gain gthat satisfies Eq. ~24! represents the threshold gainlevel gth~N! for mode N. Equation ~24! is solved withdifferent values of the feedback ratio to find all pos-sible ~internal and external cavity! modes of oscilla-tion for each value of feedback ratio. Withoutexternal feedback, only internal modes are present,and each internal cavity mode has its unique oscil-lating wave number bN and net threshold gain @gth~N!

2 a#. The mode having the lowest threshold gain–length product ~g 2 a!L will lase with highest power

4134 APPLIED OPTICS y Vol. 36, No. 18 y 20 June 1997

and is the dominant mode when no feedback ispresent. The threshold gain–length product differ-ence D~g 2 a!L between the dominant mode and themode having the next higher ~g 2 a!L is an indica-tion of mode selectivity of a DFB laser ~without ex-ternal feedback!. Usually a side-mode suppressionratio of better than 30 dB ~which is often a criterionfor single-mode nomenclature! can be obtained withDFB lasers. Also, the single-mode behavior of aDFB laser is well maintained under fast modula-tion.20 Thus, a DFB laser is usually dynamicallystable, unlike Fabry–Perot type lasers that tend toproduce transient multimode oscillations when mod-ulated because, when the laser current is modulatedfrom below threshold, the gain transiently exceedsthe threshold gain for other modes over a certainwavelength range in Fabry–Perot lasers. Thethreshold gain level Gth~N!, which should be substi-tuted into Eq. ~1!, is obtained from gth~N! as

Gth~N! 5

*0

L

gth~N!@uEN~1!~z!u2 1 uEN

~2!~z!u2#uFN~x, y!u2dz

n# Îε0m *0

L

@uEN~1!~z!u2 1 uEN

~2!~z!u2#uFN~x, y!u2dz

.

(25)

When external feedback exceeding the critical feed-back ratio is present, external cavity modes start toappear. Since the external cavity modes oscillate atfrequencies close to each of the internal cavity modesand their threshold gain levels are close to that of theinternal cavity modes, excessive intensity noise isgenerated due to mode competition.3,21 One can findthe critical feedback ratio by increasing incremen-tally the feedback ratio to determine that amount offeedback beyond which the number of solutions to Eq.~24! in the vicinity of the Bragg wavelength changefrom a single solution ~representing single-mode op-eration at the Bragg wavelength! to multiple solu-tions ~representing both internal and external cavitymodes!. The higher the critical feedback ratio of aDFB laser, the more it is resistant to external cavitymode induced noise.

3. Results

To investigate the effects of asymmetries in reflectiv-ity and in the position of a phase shift, we take a QWSDFB laser with kiL 5 2.0 and L 5 500 mm and varyR1, R2 and the ratio l1yL. The distance to externalreflector lext is taken as 0.1 m.

A symmetric QWS DFB laser having l1 5 l2 5 0.5Lis usually AR coated with AR coatings on both facetsso that the corrugation phases at the facets do notaffect the lasing characteristics of the laser. Forsuch a symmetric AR–AR structure, the dominantmode is at the Bragg wavelength. Our computa-tions indicate that, for an AR–AR QWS structure, thely4 phase shift should be exactly at the center of the

laser ~l1yL 5 0.5! for the highest single-mode selec-tivity. When the phase shift position is moved awayfrom the center of the structure, the mode at theBragg wavelength experiences an increase in re-quired threshold gain, which causes D~g 2 a!L toreduce drastically. When the phase shift is movedfar away from the center, i.e., the ratio l1yL or l2yLbecomes small, two side modes having equal thresh-old gains become the dominant modes and the single-mode behavior of the QWS laser is lost.

Figure 3 shows the effect of varying the power re-flectivity of the left facet ~R1! on the critical feedbackratio of a QWS laser with l1 5 l2 and an AR-coatedright facet. The phase of the corrugation at the leftfacet is assumed to be zero for simplicity. There isalmost no variation of the critical feedback ratiowhen R1 is varied. For an AR-coated left facet ~R1 50!, the critical feedback ratio does not depend on thecorrugation phase of the left facet. For nonzero R1,however, the critical feedback ratio depends on thefacet corrugation phase, and the values shown in Fig.3 can be taken as indications of average values of thecritical feedback ratio. Thus, for a QWS laser with aphase shift at the center of the structure, increasingthe reflectivity of the left facet ~while keeping thereflectivity of the right facet equal to zero! does notnecessarily improve external optical feedback sensi-tivity. On the other hand, the mode selectivity var-ies statistically because of the uncertainty in the facetcorrugation phase during fabrication and the yieldcan be expected to be reduced for such a structure.

Next, we discuss the effect of varying the phaseshift position on the mode selectivity parameterD~g 2 a!L and on the critical feedback ratio for aQWS laser having a cleaved ~CL! left facet ~R1 5 0.32!and an AR-coated right facet. Figure 4 shows thevariations of D~g 2 a!L and the critical feedback ratioas functions of the ratio l1yL for a CL–AR structure.The corrugation phase at the left facet is assumed tobe zero, so the values for D~g 2 a!L and the critical

Fig. 3. Variation of the critical feedback ratio with power reflec-tion coefficient of the left facet R1. The structural parameters areR2 5 0, l1 5 l2 5 250 mm, 2bBl1 5 2bBl2 5 0, kiL 5 2, lext 5 0.1 m,and phase shift at the center f1 5 f2 5 90°.

feedback ratio are indicative of average values ratherthan exact values. When the phase shift position ismoved away from the center toward the reflectingfacet, an increase in the critical feedback ratio isobserved. The mode selectivity parameter D~g 2a!L shows its highest value around l1yL 5 0.35. InFig. 5, the variations of D~g 2 a!L and the criticalfeedback ratio are plotted as functions of the phaseshift on reflection at the cleaved ~left! facet 2bBl1 forthe best mode selectivity case l1yL 5 0.35. Thephase shift associated with reflection at the left facetvaries because of statistical variation of the corruga-tion phase at the left facet. The vertical axes of theplots in Fig. 5 are proportional to the probabilitydistribution functions of D~g 2 a!L and the criticalfeedback ratio, respectively, and the horizontal axis isproportional to the number of devices fabricated.The threshold gain difference between the main modeand the dominant side mode remains almost constantat around 1.0 except when 2bBl1 approaches 180°.For comparison, the value of D~g 2 a!L obtained inan AR–AR QWS DFB laser with the phase shift atthe center of the structure is 1.45. Thus the thresh-old gain difference between the main and dominantside modes, D~g 2 a!L, is somewhat reduced in theasymmetric structure. Approximately 70% of thedevices show a value of D~g 2 a!L around 1.0. Thus,devices having lower D~g 2 a!L values around 2bBl15 180° can be expected to reduce the yield by at least30% from AR–AR symmetric structures. The criti-cal feedback ratio also varies because of variations in2bBl1. The critical feedback ratio for an AR–ARQWS DFB laser with the phase shift at the center is234 dB. In comparison, the critical feedback ratioin Fig. 5 remained better than 234 dB for all values

Fig. 4. Variation of the threshold gain difference between themain mode and the dominant side mode, D~g 2 a!L, and thecritical feedback ratio as functions of l1yL for a CL–AR structure.The structural parameters are R1 5 0.32, R2 5 0, 2bBl1 5 2bBl2 50, kiL 5 2, lext 5 0.1 m, and phase shift f1 5 f2 5 90°. The circlesrepresent the corresponding values for a symmetric AR–AR struc-ture.

20 June 1997 y Vol. 36, No. 18 y APPLIED OPTICS 4135

of 2bBl1. Thus, the asymmetric CL–AR structurewith l1yL 5 0.35 exhibits better endurance againstexternal optical feedback compared with the symmet-ric AR–AR structure.

Figure 6 shows the variation of D~g 2 a!L and thecritical feedback ratio as functions of the ratio l1yL fora structure with a highly reflecting ~HR! left facet andan AR-coated right facet when 2bBl1 5 0. For such

Fig. 5. Variation of the threshold gain difference between themain mode and the dominant side mode, D~g 2 a!L, and thecritical feedback ratio as functions of reflection phase shift at theleft facet, 2bBl1, for a CL–AR structure. The structural parame-ters are l1yL 5 0.35, R1 5 0.32, R2 5 0, 2bBl2 5 0, kiL 5 2, lext 50.1 m, and phase shift f1 5 f2 5 90°. The horizontal line repre-sents the corresponding value for a symmetric AR–AR structure.

Fig. 6. Variation of the threshold gain difference between themain mode and the dominant side mode, D~g 2 a!L, and thecritical feedback ratio as functions of l1yL for a HR–AR structure.The structural parameters are R1 5 1, R2 5 0, 2bBl1 5 2bBl2 5 0,kiL 5 2, lext 5 0.1 m, and phase shift f1 5 f2 5 90°. The circlesrepresent the corresponding values for a symmetric AR–AR struc-ture.

4136 APPLIED OPTICS y Vol. 36, No. 18 y 20 June 1997

HR–AR DFB structures, D~g 2 a!L has a high valueof more than 1.0 for l1yL , 0.4. The critical feed-back ratio is seen to increase when the position of thephase shift is moved toward the HR facet. Figure 7shows the variation of D~g 2 a!L and the criticalfeedback ratio as functions of the phase shift associ-ated with reflection at the HR facet 2bBl1 for aHR–AR QWS DFB laser with l1yL 5 0.25. Becauseof the lower values of D~g 2 a!L for the deviceshaving 90° , 2bBl1 , 270°, yield lower than 50% maybe expected. The critical feedback ratio, on the otherhand, shows significant increase compared with the234-dB value for AR–AR symmetric QWS DFB la-sers. Thus, the HR–AR structure with l1yL 5 0.25exhibits yet better resistance to external optical feed-back compared with a CL–AR structure. This struc-ture, however, suffers from lower mode selectivityand lower expected yield.

4. Conclusion

External optical feedback sensitivity for a QWSindex-coupled DFB semiconductor laser can be im-proved by introducing asymmetries in reflectivity ofthe facets and in the location of the ly4 phase shift.A QWS laser with asymmetry only in reflectivity offacets or only in the position of the phase shift aloneis not effective in reducing the external optical feed-back sensitivity. CL–AR structures show better en-durance against external optical feedback thanAR–AR symmetric structures as well as adequatemode selectivity and yield when the position of thephase shift is approximately one third of the length ofthe laser away from the cleaved facet. The modeselectivity and yield is somewhat lower comparedwith symmetric AR–AR structures. HR–AR struc-

Fig. 7. Variation of the threshold gain difference between themain mode and the dominant side mode, D~g 2 a!L, and thecritical feedback ratio as functions of reflection phase shift at theleft facet, 2bBl1, for a HR–AR structure. The structural param-eters are l1yL 5 0.25, R1 5 1, R2 5 0, 2bBl2 5 0, kiL 5 2, lext 50.1 m, and phase shift f1 5 f2 5 90°. The horizontal line repre-sents the corresponding value for a symmetric AR–AR structure.

tures show the best endurance against external op-tical feedback when the phase shift is moved towardthe HR facet, but suffer from reduced mode selectivityand lower yield characteristics.

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