τ τ VOLTERRA SERIES EXPANSION OF LASER DIODE RATE EQUATION The basic laser diode equations are: 1 τ (2) The expansion of equation (1) is: (3) )( 1

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Δανάη Γλυκύς
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2 VOLTERR ERE EXO O LER OE RTE EQUTO The i ler diode eutio re: [ ][ ] V The exio of eutio i: [ ] ddig eutio d V V The iut urret i ooed of the u of,. ooet, Î, tie vryig ooet. We thu let 6 The Volterr exio of the hoto deity d the eleo deity re: d utitutig thi exio i eutio e oti

3 Yukridki ifdei iii oererii turude yilir. u eri x x x x x x x... 9 x x x x x x x x x x x x x x x x x x 6 9 egikeler Teri yii ie!! k k ii ie;!!! k k

4 e o eute like oer of to oti or or or or [ ] utitutig the exio for d i eutio e oti V V

5 or V or V or or We o olve iulteou eutio d for the Volterr oertor d ueively. To olve for d fro eutio d e ote tht ie tt. Thu fro eutio e hve 6 d fro eutio 6 V olvig eutio6 for e hve V 6 o utitute eutio 6 ito 6 to oti

6 V V V V We o exd the rket to oti V V V V V V olletig ter i like oer of e oti the ui: V V V V V V 6 Thi ui olyoil

7 i hih the oeffiiet re: V V V V V V Eh of thee oeffiiet i olyoil i. Thu: i hih d 6

8 V V lo 9 i hih 9 V V 9 d V V 9 V

9 lo i hih V V d V V [] [] [] [] [] [] [] -.699e.6e6.99e..969e9 -.e -.666e.9e.6e e9 -.e -.6e.6e.69e9.99e.6e e.6e.6e9.e.66e e.96e.69e9.669e6.6e e.e.6e9.9e6.e e9.6e.666e9.9e6.9e e9.99e.69e9.69e6.69e e9.e.9e9.9e6.6e e9.e.69e9.9e6.969e e9.9e.9e9.9e6.69e e9.9e.e9.66e6.9e e9.e.696e9.6e6.e e.6e.6e.99e6.699e e.9e.e.99e6.9e e.e.6969e.99e6.e e.9996e.6e.99e6.66e -..9e.9996e.6999e.99e6.9e -..6e.996e.96e.99e6.66e -.

10 . x 9 []. [] -. [] - ekil. Kokleri fizikel durulri o. x.. o o 9

11 x [] o 6 [] [] o There re three root for of Eutio. or eh root, there i orreodig vlue of otied fro eutio 6. V 6 x 6 o 6 9 o for hyil olutio, e reuire > d >.We thu exet tht oly oe of thee olutio ir tify thi oditio for y vlue of. The roedure to deterie d i to ue eutio, 9 d to deterie the oeffiiet, d olyoil i d olvig eutio d 6 for the three olutio ir of d. our eutio re hyilly roer oe, there ill e oly oe ir for hih oth d re oitive d rel. Thi i the the hyil eleo deity, d hoto deity of the ler diode futio of the ijetio urret,.. Oerve tht for lrge vlue of

12 V V V or oveiee, let α. The V α α α The ui eutio for,, eutio, the i roxitely α α α or lrge vlue of oe root of thi eutio i roxitely otied α α or hih the olutio i We thu oerve tht, for lrge vlue of, i tt eul to. fro outer olutio of eutio, thi i oerved to e ue for. We thi roxitio, the hoto deity for lrge i fro eutio 6 V Thi the eutio of ight lie ho

13 th loe V V x 6 e e e9 6 9 o The loe of ight lie i igure V lo, for

14 V V... ere thi reult i oerved vlid fro ext outer olutio of the eutio for ote tht e thu here tht the threhold urret th V th. Thi orreod to the ueril vlue otied exerietlly. ote fro our theoretil olutio the threhold urret, th i roortiol to hih i teerture deedet d iree ith ireig teerture. oever, the loe of urve i V hih i eetilly ideedet of teerture. We thu exet the grh of urve to e We o deterie the firt-order Volterr oertor for d y olvig eutio d iulteouly [ ] [ ] V olletig ter i eutio Thi eutio e exreed i hih

15 x.. [].. o d x o 6 6 o Eutio e olved for

16 We o utitute eutio ito eutio V olletig ter V ultilyig y V Eutio i lier differetil eutio V hih d The Lle for of thi eutio i V or V

17 The V i hih hih e rereeted V igure o the Lle for of eutio i [ ] K K igure We thu hve Î V K 6

18 igure V K o lo ekil igure We o exie the tt d utitutig eutio ito eutio, e oti olletig ter e hve oti iilrly for e utitute eutio d ito eutio to oti olletig ter, V

19 [ ] [ ] [ ] We o oerve tht for lrge, Thu for lrge /o o

20 6 6 x 9 o o The ole of re the root of ± o the root re We thu here ±, o < o tht 9

21 oeuetly ± or, ote tht if, e ould hve d o tht x 9 6 o 6 o

22 x.... o /o o X X

23 igure ote tht the yte i tle. lo, re ole i roortiol the hih i roortiol to. We o deterie the eod-order Volterr oertor, d.or thi We ue eutio d [ ] [ ] [ ] olletig ter i eutio Thi eutio e exreed i hih d re give y eutio d reetively d [ ] [ ] Eutio e olved for We o utitute eutio i eutio

24 olletig ter Multilyig y Thi eutio i lier differetil eutio eutio hih d re give y eutio d reetively. lok digr thu i igure o fro eutio d fro eutio o

25 igure Whih i euivlet to igure d igure

26 o fro Eutio [ ] [ ] or lrge, o tht [ ] hih i otied Î V K [ ] igure We o deterie the third-order Volterr oertor, d. or thi, We ue eutio d [ ] [ ] [ ] olletig ter i eutio [ ] Thi eutio e exreed 6 6

27 i hih d re give y eutio d reetively d Eutio 6 e olved for 6 We o utitute eutio 6 i eutio olletig ter d ultilyig y e oti 6 Thi i lier differetil eutio 6 hih d re give y eutio d reetively. iilr to eod-order oertor, e hve igure hih i give y eutio 6 ote tht for lrge o o tht for lrge 6

28 lo, for e hve hih i otied igure hih,,, d re fit d eod-order oertor of d reetively. lo, if deired, i otied fro eutio 6. or lier feedk i hih KK, R 6 R R R R R, R, K R { } [ RK ] R R RK R R, R R R R [ [ R K ] R K ] R R e too # for lieriztio uig L feedk i our e, e t e K K K 9 ote tht feedk yte oertor, re tle oertor if R i tle oertor o R K

29 R t K e R K t e The e i tle oertor if the root of K e t re ll i the left- hlf le. or oveiee, e defie t σ jω i hih σ t σ d ω t ω. The Kt t t e ω The Kt r σ, ω σ ω t σ t e ω Kt i σ, ω σ ω t σ e iω 6 i h σ, i h ω σ, ω j σ, ω. o if oly if r d i. ro e. i σ, ω i i ω r i ω σ [ t σ ]e Kt To the eutio to e tle, thi eutio ot hve y olutio for σ. o the right ide of e. i ootoilly ireig futio of t > Kt Kt σ. Thu, e h o olutio for σ if or K < t 9 K The rge of K hih the yte i tle e lrge ie e hve ot idered tht e ut e iulteouly tified. or our ler diode reter, reult of outer iulte re t K..6. x K hih K x i the xiu vlue of K for hih the eutio i tle. We thu oerve tht K i reole loer oud of K for tility.

30 or K K, the yte defiitely i tle. or K [,K ] e th hve e j K j t t j ω ω ω ω d e j K j t t j t t j R ω ω ω ω ω ω x W i 9

31 i i VOLTERR LZ EŞTLKLER [ ] i?? i i i i i [ ] [ ]

32 6 [ ] 6 UE MY ORMUL UE W t i i i

33 i t i i i i t

34 UE W: ???????? J J J J J J J J

35 MOW UE LULTE TE X TERM i i i i i i i i i i i i i i i

36 O MOW MY ORMUL OR E i i i t

37 i i i i i i i i i i MOW U MY ORMUL O E

38 t i i i t

39 i i i i

40 i i i t WE RE MOW T 6 6

41 6 6! ! t. i t i t i t Y

42 K K J J J J R 6. i t i t i κ κ

43 t i i i i i i i i i i i i κ κ μ κ κ μ μ μ Φ i i o o J Z j Z K J Z J Z Z K j Z K j Z J Z Z K K i K

44 i. i. i i i. i i i κ κ κ κ j j j j j j j j WE RE OLY TERETE TE... RMO TERM. TEY RE i i i i. i i i

45 t i i t i i i i i i i i i

46 i i i i i t i i i i

47 [ ] i i i O RT RMO...?? [ ] i TE RM...??

48 i t i j o Z j j j j j j j j j j j j κ κ κ κ κ κ κ

49 i t i t j Z j Z j Z j Z j j j j j Z κ κ κ κ κ

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