T.aW^hB, r2a|i/V/2fl2 Txa\vWZ + r2a2Vg2

V N U . J O U R N A L O F S C IE N C E , M a th e m a tic s - Physics. T .X X II , Nq2 - 2 0 0 6

O P T I C A L B IS T A B IL IT Y E F F E C T IN D F B L A S E R W IT H T W O S E C T IO N S

N g u y en Van Phu*, D in h Van H oan gt, C ao Long V an #

* Faculty o f Physics, Vinh University t Faculty o f Physics, Hanoi N ational University ft In stitu te o f Physics, University o f Zielona Góra

Podgórna 50, 65-246 Zielona Góra, Poland

A b s t r a c t . In this paper the optical bistability in DFB laser with two sections has been demonstrated. Influence of some dynamical laser parameters involved in the problem (as current intensity, saturation coefficient and gain values) on this effect has been considered.

1. I n t r o d u c t io n

As known, the large num ber of DFB (distributed feedback) lasers used inside a transmitter makes the design and m aintenance of such a light wave system expensive and impractical. T he availability of sem iconductor lasers which can be tuned over a wide spectral range would solve this problem . One of these is m ulti-(tw o or more) section DFB laser, considered theoretically and experim entally during 1980s [l]-[7], [13]-[18] and were used in commercial lightwave system s by 1990.

On the other hand, optical bistability effect, discovered since the 1970s in different optical system s w ith th e possibility of its applications as an optical sw itch (or ’optical tran sisto r’), an optical differential amplifier, optical lim iter, optical clipper, optical dis­

crim inator, or an optical m em ory element, has given rise to a large num ber of different theoretical and experim ental treatm en ts. Because of m any special advantages of utilizing sem iconductors as optical bistable elements, the m ost efforts of researchers in th e field of optical bistability have been focused on developing various sem iconductor m aterials and devices [19].

In this p ap er we propose ones of theses devices: a D FB sem iconductor laser w ith two sections. In Section II, sta rtin g from dynamical equations describing this laser we have received th e exhibition of optical bistability effect in th e statio n ary sta te lim it. T he influence of some dynam ical laser param eters on this effect are d em o n strated in Section III. Section IV contains conclusions.

Typeset by 47

48 N g u y e n Van P h u , D in h Van H o a n g, C ao Long Van 2. S ystem o f rate eq u ation s—

The operating characteristics of sem iconductor lasers are described by a set of rate equations th a t govern the interaction of photons and electrons inside the active region. A rigorous derivation of the rate equations generally s ta rts from Maxwell s equations together with a quantum -m echanical approach for the induced polarization. A DFB laser w ith two sections is shown schem atically in figure 1. Here, section A w ith injection current I\ IS an amplifying section, section B with injection current /2 much smaller th an 11 takes a. role as a saturable absorber section.

F i g . l . Schematic illustration of a DFB laser with two sections.

Then we have the following system of rate equations:

- m — g i p0 - W j f r i - m N i ( ! )

dt eVi n ef f

~ 0 r = ~ m - ^ - 9 ( u 0 - u>j)nj - ^{7 2}JV² (2)

dt eV2 rieff

= ( r ar/i + r 2T?2) - ^ - ỡ(w0 - Wj)(nj + 1) - 7nj +

at n ef f

Here Vi v 2 N i , N 2 are the volumes and carrier densities of sections A , B corre­

spondingly; Ti j is photon density; e, Co are electric charge of electron and velocity of light

i n v a c u u m ; T i e f f i s t h e e f f e c t i v e r e f r a c t i o n i n d e x o f m a t e r i a l , s u p p o s e d t o b e t h e s a m e f o r

two sections; T]i is the amplification coefficient, which depends on the carrier density in the form rji = aịNị ⁺ Pi, ^where ai,Pi are material gain coefficients (i = 1,2); 71,72 ^are relaxation coefficients of carrier densities given in the form [8]

BqNi B o N i

72 e i - B 2N 2 ’

with So B\ B 2 are material coefficients, £ is saturation coefficient indicating the different relaxations of carrier densities between two sections; r i , r 2 are confinement factors or Peterm an coefficients in sections A and B\ 7 is coefficient which describes the photor. loss

O p tica l b is ta b ility e ffe ct in D F B la s e r w ith tw o s e c t i o n s ₄₉ in section A, D and mirrors; Function g(u>0 - c j j ) describes th e broadening of spectral laser line which is given in the form of Lorentzian:

g(uj0 - U!j) = ---y— - 2 i + ( ^ )

with r is the w idth of the gain line; Aj = U)Q - Uj is detuning factor; Ct»0,Uj are circular frequencies in the center of the gain line and of j th mode. The unity in factor (rij + 1) indicates the presence of spontaneous emission in laser operation and the last term p y/rijK ) deals with the interaction of signal and laser radiation. T he interaction coefficient /3 is usually taken to be unity (p — I s- 1 [7]-[8]).

In the statio n ary regime, we put all time derivatives in (1), (2), (3) to zero and obtain:

0 = r t \ ~ m w y f ỡ(w0 - w > j - 7 i W i , (4)

0 = eV2 ~ V2w y f 9 ^ 0 ~ Uj )ni ~ 7a7V2’

0 = (ri7?i + r2772) - ^ -5(^0

- Uj

Uj + Py/p~n~j.

We suppose also th a t (3i = 0, which is usually valid for the most of semiconductor lasers used in practice (e.g. InGaAsP, see [7], [8]) and we also suppose to ignore the presence of spontaneous emission. It follows from (4), (5), (6) th a t

A n ] '2 - E n ) /2 + C n 3/ 2 - D n )/2 + p y /P ^ G r ij - p ự K Q n ] - ậ y f p u = 0, (7) where the coefficients A, E , c , D, G, Q are given by:

r i a ? a 2^4ổ4eV15 2 r 2a i a ^ V e V '25 1 A iB l T n + 4 £ £02T22

r _ r ia ? i/3g 3eVj r 2a ị i/ 3g3eV2 T i a Ị a 2iy3g 3I i B i B 2 r 2a i a ^ V ^ 2 - S i 5 2

A B oTn 4£B 0T 22 2 £ 502T n + 2 £B gr22

r1a Ị a 2i'3g 3B 2 + TiOLiáịv*g2B i - 270'ia2ỉ'2ỡ2B ị B2 k ẻ l

T.aW^hB, r2a|i/V/2fl2 Txa\vWZ _{+ r2a2Vg2}

2 B 0T n 2£B0T22 2 £ B 0

T l a 1a 2v 2g2B 2 rr T 2a ia 2 ^ 2g2B l rr 2^5^eVi 11 + 2ZB20eV2 22

YiOtiGt2V2g2h B \ B 2 T20i\0i2V2g2h B \B 2 'yaii'gBiZ + 7 0 2^5 ^ 2

^ B l e V x 2ỊB Ịẽ V 2 £B 0 ’

r i _ T i a i ug r 2a 2vg T iC tiy g liB i Y 2a2V9h B i

2B 0eVi 11 2Ì B ữeV2 22 2B 0eVi 2£B 0eV2 7 ’

50 N g u y e n Van P h u, D in h Van H o a n g , C a o L o n g Van

ctịvgB i 0C2vgB2 n _ OL\a.2V2g2B \B 2

Or — --- „ ---1--- ; V —

Bo

Co

(B o u = r — ; 9 = 9 {v0 - Wj),

n e //

Tn =

ĩịẼị]

y/^I2B0eV2

W hen the external optical signal disappears (Pw = 0), we obtain from (7)

A rP j - EVij 4- C rij - D = 0. ⁽8⁾

For the most of sem iconductor lasers used in practice we have also [7] V\ = V2 — V , B \ = B2 — B C*1 = c*2 = Ct. We consider for simplicity the resonance case in which the generating mode frequency coincides with Uo, then we have

g ( u o - Wj ) = ^1.

and obtain finally:

Hj — 'Hrij + Crij — M = 0, (9)

where:

n =

c =

Bo

D

(F1T22 + r2r„) + ^(ri/iT22 + r2T2Tu) +

TllTẠ-(riaỉy + r2au _{- 27Ổ)}

2 B o t e T ^ T n + T2I 2T n ) + B °T" T™m

+ r2) +

+ r2T22)

a u e V B T u T aa

=

eK

2BqT\\T22

B

ỰiTi + /2r2) + +1}

/7I

n

a v I T

a 2u2eV

T = a u { T 1T22 + T2T n ) i

g^(fr,rn + r2r22) - jLfc/.r, + J3r2) - e§°

The Eq. (9) represents th e catastrophe manifold of th e Riem ann-H ugoniot (or

‘cusp’) catastrophe A s in the M ather-T hom classification [20]. T his catastrophe is given by the following potential function V( x\ a, b) :

V( x \ a , b ) — - X 4+ - a x 2 + bx. ⁽10)

T he physical system described by this potential function has evolution generated by variations in the control param eters a = L - V.2/2),b = ;HC/ 3 - M - 27i3/27. The

O ptical b is t a b il i ty e ffe c t i n D F B la s e r w ith tw o s e c tio n s 51

system, in accordance with the general principle of minimization of potential energy, will tend to dwell on the catastro p h e surface M3 given by

M 3 = { ( x , a , b) ^{: X}3 + a x ⁺ b = 0 } .

The set of degenerate critical points £3, defined by the condition of having multiple roots by the polynomial w( x ) = X3 4- a x 4- 6, is expressed by

£ 3 = { ( x , a ,6) : X3 + a x + b = 0 ,3 x2 4- a = 0} . (11) The X variable m ay be elim inated from the system of equations defining the set E3. Then we obtain th e bifurcation set z?3 given by:

B 3 = {(a, b) : 4a34- 2762 = 0}.

This set determ ines th e param eters range involved in th e problem for which the bistablity

e f f e c t o c c u r s . T he left side of (9) is an universal unfolding of the function f (rij) = nJ which is structurally stable: th e small change of the control param eters (physical param eters involved in the problem ) do not change the form of the hysteresis curves as we will see in the next Section.

3. Influence o f som e d yn am ical param eters on op tical b ista b ility effect

3 .1 . T h e a p p e a ra n c e o f o p tic a l b is ta b ility e ffe c t

We can now solve num erically the equations (7), (8). T he values of the param eters involved in the problem are taken from the experim ental d a ta for a concrete semiconduc­

tor laser on InG aA sP given by K inoshita [7] and Yong-Zhen H uang [8]: Co = 3.1010cm .s_1;

e = 1,6.1(T 19C; Vi = 84.10- 12cm3; v 2 = 84.10“ 12cm3; B 0 = 10~10cm 3.s’ 1]

D \ = 5.10~ 19cm3; £?2 = 5.10“ 19cm3; n ef f = 3.4; C*1 = 4.10“ 16cm2; c*2 = 4.10~ 16cm2;

£ — 0.1;

r

r2

n.io^s1;

1022cm"3

110°

InjecUon currenl I^(A)

F ig .2. Hysteresis curve of optical bistability effect in DFB laser w ith two sections.

52 N g u y e n Van P h u, D in h Van H o a n g, C a o L ong Van In the MATLAB language, we have received a hysteresis curve of optical bistability effect shown in Fig. 2. Here injection current I\ is control param eter and distance X1X2 indicates the width of bistability (BSW).

3 .2 . T h e c h a n g e o f th e in je c tio n c u r r e n t /2

It follows from Fig. 3, th a t when /2 increases, the bistability w idth (BSW) increases too. For clearness we take three values of /2 : 2 X 10“ 5Ẩ ,2.5 X 10- 5 A ,2.8 X 10- 5i4. The corresponding curves are presented in Fig. 3: The dotted line corresponds to th e value of

/2 = 2 X 10- 5 j4, the dashed and solid lines correspond to the values of /2 = 2.5 X 1 0 ~ 5A

and /2 = 2.8 X 10” 5A T he results are given in the Table I.

T a b le I

h ( A ) 2 ^X 10~5 2.5 X 10~5 2.8 X 10~5

B S W ( A ) 0.1543 0.4423 1.3486

F ig . 3. Influence of injection current /2 on hysteresis curves of optical bistability effect.

O ther values of I'2 are /2 ⁼ 2 X 10“5j4, Ì2— 2.5 X 1 0 “ 5 i 4 ,/2 = 2.8 X 10“ 5A 3 .3 . I n flu e n c e o f th e s a tu r a tio n c o e ffic ie n t £

Choosing three values of £ we also obtain the hysteresis curves and optical bistabilty effect is dem onstrated in Fig. 4. W hen £ rises the BSW diminishes. T he results are given in Table II.

T a b le I I

£ ^0.1 0.15 0.2

B S W ( A ) 0.4354 0.2194 0.1343

O p tica l b is ta b ility e ffe c t in D F B la s e r w ith tw o s e c tio n s ₅₃

*10”

F ig . 4. Influence of saturation coefficient £ on BSW of hysteresis curves 3.4. Influence o f the g ain value a

In this case the curves of optical bistability are presented in Fig. 5. Prom this Fig.

we ^{s e e} th a t when the gain value a increases the BSW increases too. T he numerical results

are given in Table III.

T a b le III

a ( c m 2 ) 3 ^X 10- 16 4 ^X 1 (T16 5 ^X 10- 16

B S W ( A ) 0.4354 0.6857 1 . 1 2 1 1

I i o ’3

F ig . 5. Influence of gain values a on hysteresis curves of optical bistability effect.

54 N g u y e n Van P h u, D in h Van H o a n g, C a o L o n g Van 5. C onclusions

From above o b tain ed results we derive the following conclusions:

1. Optical b istab ility effect appeared like in the case of lasers containing saturable absorber (LSA) [16]. Here, th e decisive condition for having hysteresis curves of OB effect is the current Ỉ2 in section B m ust be much smaller than current I \ in section A.

2. Laser parameters as gain, saturation coefficients, etc ... will be control parameters for hysteresis curves. T h e change of dynam ical param eters involved in the problem clearly influences on characteristics of optical bistability effect as th e bistability w idth or the optical bistability height. D eterm ination of the .values of these param eters, which give the large bistability w idth for DFB laser is very im portant from experim ental and practical point of view.

In fact, the change values of laser param eters as gain, satu ratio n coefficients, etc... can be realized by changing pro p o rtio n of X or y in structure I n \ - xG axA s y P \- y of m aterial.

5. R eferences

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16. P.Q. Bao, D .v . H oang and Luc, J. o f Russian Laser Research, Kluwer A cadem ic/Plen Publishers, Vol. 20, No. 4, 297 (1999).

17. G. P. Agrawal, Fiber-Optic Communication System s, 3rd ed., John Wiley & Sons, inc., New York 2002.

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Ltd, Singapore 1999, C hapter 12.

20. G. Gilmore, Catastrophe Theory fo r Scientists and Engineers, New York 1981.