8
.11.2
499.84 1997!
499
Tim e (ns) C a rrie r N um ber (lunit= 10® )
2170 2163 2160 2133 2130 2133 2130 — !>»'.»• ‘ ' ' 4»‘.ay ' ' ■ w.K' ' ' ' i »‘.w— — w. %" "w.w— ssr Tim e (ns) 8.11.3
In o u r F M m ode-locking experim ent w ith InG aA sP -LiN bO g la se r we observed both solutions excited sim ultaneously (Fig. 5.8). We are re lu c ta n t to use o u r e x p e rim e n ta l re s u lt in th is discussion because o f the pronounced com posite-cavity effects o f our h y b rid ca vity. R ecently, an experim ental in ve stig a tio n o f pure FM m ode-locking o f an exte rn a l ca vity sem iconductor laser was r e p o r t e d T h e gain elem ent was a tw o-section o p tica l a m p lifie r and the ca vity was form ed using an e xte rn a l g ra tin g . A s u rp ris in g re s u lt o f th is paper, w hich received no com ment, was th a t tw o d iffe re n t regim es o f FM m ode-locking were observed in tw o experim ents u tilis in g d iffe re n t laser chips. In both cases th e same type o f e xte rn a l ca vity was used. In the fir s t case, a single tra in o f pulses was generated, w h ile in the second case a double tra in o f pulses (s im ila r to our trace in Fig. 5.8) was produced.
I t is ce rta in th a t fu rth e r experim ental evidence is needed to ju s tify the v a lid ity o f our theoretical re sult. In the case o f long e xte rn a l cavities the effects o f the lin e w id th enhancement factor are reduced. I f the level o f the o p tica l feedback is sm all then the combined effect o f com posite ca vity, asym m etry o f feedback, and spontaneous em ission noise m ay become the dom inant control mechanism o f the mode-locked output.
8.3.2 N um erical S im u la tio n E xcluding the L in e w id th Enhancem ent Factor
We have also perform ed n u m e rica l sim u la tio n s o f F M m ode-locking assum ing a zero value fo r the lin e w id th enhancem ent factor. O ur re sults are presented in Fig. 8.12. The m ain ch a ra cte ristic o f F M m ode-locking w ith pQ=0 is double pulsing. T his sim ultaneous e x c ita tio n o f tw o mode- lo ckin g solutions is shown in the calculated w aveform s o f the photon num ber (Fig. 8.12.2) and ca rrie r num ber (Fig. 8.12.3). The co n trib u tio n s o f the tw o pulse tra in s are resolved clearly in the overall spectrum shown in Fig. 8.12.1.
Spectrum (arbitrary units)
— « 8—
Frequency (G H z)
8
.
12.1F ig . 8.12 FM m ode-locking when the lin e w id th enhancem ent factor is set to zero (Pc=0); averaged spectrum over 50 pulses w ith center
frequency a t 130GHz (8.1 2.1), steady-state photon num ber (8.12.2) and
c a rrie r num ber (8.12.3) versus tim e
Photon Number (lunit=10^)
49ë.ëô ’ ’ ' ^ .é2 4^ 4 g à . t i 4 b s . K 4^9 499 8.
12.2 Tim e (ns) C a rrie r N um ber (lu n it= 1 0 ® )2170 2169 2160 2199 2190 % % — — w » — srf Tim e (ns) 8.12.3
The second characteristic o f our re s u lt is th a t the pulses obtained fo r pg=0 have a d iffe re n t shape than the equivalent pulses achieved fo r Pg=5. I f the lin e w id th enhancem ent factor is set to zero, the pulses become s lig h tly
s h o rte r (1 2.2ps) and th e tim e -b a n d w id th p ro d u c t in cre ase s to
a p p ro xim a te ly 0.63. According to K uizenga and Siegm an's tim e-dom ain
th e o ry^ based on the self-consistent pulse analysis, the tim e -b a n d w id th
product o f a Gaussian pulse produced by FM m ode-locking is 0.626.
F u rth e r in v e s tig a tio n is needed in order to prove th e v a lid ity o f our num erical re sults produced fo r zero detuning. The fir s t step w ould be to include the coupling coefficient due to the c a rrie r num ber v a ria tio n Aq in th e program . H ow ever, i f the program is k e p t in its o rig in a l fo rm , a d d itio n a l in s ig h t can be obtained i f detuning o f the m o du la tin g frequency is included.
8.3.3 FM -Laser O peration
A w e ll know n re s u lt o f general laser theory is th a t s u ffic ie n t detu n in g o f the m odulating frequency in FM m ode-locking leads to a frequency swept mode o f operation called FM -laser operation. We have discussed FM -laser operation in C hapter 5.1.2 and considered the use o f a composite c a v ity InG aAsP -LiN bO g laser in order to dem onstrate it.
In our num erical model we have observed tw o types o f behaviour fo r FM - laser operation, dependent on the inclusion o f the lin e w id th enhancem ent factor. Two spectra obtained by averaging over 500 pulses are shown in Fig. 8.13. B oth spectra are obtained fo r positive d e tu n in g o f lOOMHz and our standard d riv in g conditions. For P^=5, the spectrum has a shape o f a d isto rte d FM spectrum (Fig. 8.13.1). A proper FM spectrum is obtained w hen the lin e w id th enhancement factor is set to zero (Fig. 8.13.2). As in the case o f detuned operation o f m ode-locking by c u rre n t m o du la tio n ,
unstable pulse sequences are achieved both fo r Pq=0 and p^=5. F o r the
same am ount o f detuning, the ra tio o f the pulse he ig h t to the DC pedestal is la rg e r when the lin e w id th enhancement factor is included. In the
Spectrum (arbitrary units)
566... ÿi6’....
8.13.1
Frequency (GHz) Spectrum (a rb itra ry u n its)
8.13.2
W «6...556----«6----*0 Frequency (GHz) F ig . 8.13 FM -laser operation obtained for +100M Hz detuning; averaged spectrum over SOOpulses w ith center frequency a t 130GHz.
L in e w id th enhancement factor Pg=5 (8.13.1) and Pq=0 (8.13.2)
example presented in Fig. 8.13 th is ra tio is 2.7 fo r Pc=5, and 0.25 fo r Pg=0. T h is in fe rs th a t the sem iconductor laser is s t ill s tro n g ly p u ls in g fo r lOOMHz detuning i f the lin e w id th enhancement factor is included. F or the same am ount o f d e tu n in g, the flu c tu a tio n s in th e o u tp u t pow er are re la tiv e ly sm all i f the lin e w id th enhancement factor is set to zero.
8.4 A M M ode-Locking
F in a lly , to conclude th is chapter, we present tw o s im u la tio n s o f A M m ode-locking. Equations 7.57, 58 and 39 have been used in conjunction
w ith the param eter values lis te d in A ppendix 1. As in the case o f FM
m ode-locking, we have o n ly u tilis e d the coupling c o n trib u tio n o f th e
c a rrie r num ber v a ria tio n represented by Eq. 7.39 and set to zero. The
loss m odulator section was assumed to be short i f compared to the le n g th o f the m o n o lith ic cavity. As in our previous sim u la tio ns o f active mode- locking, we have assumed a parabolic loss dependence. In the case o f A M m ode-locking, we have introduced a parabolic d is trib u tio n o f coupling
coefficients (both and A^^^g) to com pensate fo r th e d iffe re n t
m odulation depths o f in d iv id u a l modes.
8.4.1 N um erical S im ulation In clu d in g the L in e w id th Enhancem ent Factor
In our num erical sim u la tio n o f A M m ode-locking we have used tw e n ty five modes, a DC current o f 30mA and the am plitude o f RF v a ria tio n o f the m irro r re fle c tiv ity o f approxim ately 10%. Fig. 8.14. shows the re su lts o f th is sim ulation. The firs t p lo t (Fig. 8.14.1) represents the spectrum w hich is calculated from a sequence o f fifty pulses. A lth o u g h we ob ta in a stable sequence o f mode-locked pulses, the spectrum shows a sm all am ount o f broadening w hich is probably caused by the m odal phase va ria tio n s. Due to the in c lu s io n o f the lin e w id th enhancem ent fa cto r, the spectrum is asym m etric. The pulse shape and the e q u iva le n t c a rrie r v a ria tio n are presented in Figs. 8.14.2 and 8.14.3. The pulses are 17.2ps long and the tim e-bandw idth product is 0.62.