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, N q - 0 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 gu yen Van Phu*, D inh Van H oangt, 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 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 A b stra c t I n tr o d u c tio 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 experimentally during 1980s [l]-[ ], [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 systems w ith the possibility of its applications as an optical switch (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 ents Because of m any special advantages of utilizing semiconductors as optical bistable elements, the most efforts of researchers in the field of optical bistability have been focused on developing various sem iconductor m aterials and devices [19] In this paper we propose ones of theses devices: a DFB sem iconductor laser with two sections In Section II, startin g from dynamical equations describing this laser we have received th e exhibition of optical bistability effect in the stationary sta te limit The influence of some dynam ical laser param eters on this effect are dem onstrated in Section III Section IV contains conclusions Typeset by 47 N g u y e n Van P hu, D in h Van H o a n g , C ao Long Van 48 System o f rate equations— The operating characteristics of semiconductor 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 sta rts from Maxwell s equations together with a quantum -mechanical approach for the induced polarization A DFB laser with two sections is shown schematically in figure Here, section A with injection current I\ IS an amplifying section, section B with injection current /2 much smaller than 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: dt eVi - m — n ef f g ip - W jfri - m N i (!) ~0r = ~ m - ^ - ( u - u>j)nj - JV2 dt eV2 rieff at (2) = (r ar/i + r 2T?2) - ^ - ỡ (w0 - Wj)(nj + 1) - 7nj + n ef f Here Vi v N i , N are the volumes and carrier densities of sections A , B corre­ spondingly; Ti j is photon density; e, Co in v a c u u m ; Tieff is t h e effective are refraction electric charge of electron and velocity of light in dex of m aterial, supposed to be t h e s a m e for 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] B qN i 72 B oN i e i - B 2N ’ with So B\ B2 are material coefficients, £ is saturation coefficient indicating the different relaxations of carrier densities between two sections; r i , r are confinement factors or Peterm an coefficients in sections A and B\ is coefficient which describes the photor loss O ptical b ista b ility effect in D F B la se r w ith tw o s e c tio n s 49 in section A , D and mirrors; Function g(u>0 - c j j ) describes the broadening of spectral laser line which is given in the form of Lorentzian: g(uj0 - U!j) = y — - i + ( ^ ) with r is the w idth of the gain line; A j = 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 The interaction coefficient /3 is usually taken to be unity (p — I s - [7]-[8]) In the stationary regime, we put all time derivatives in (1),(2), (3) to obtain: zero and = r t \ ~ m w y f ỡ(w0 - w > j - i W i , (4) = eV2 ~ V2w y f 9^ ~ Uj )ni ~ 7a7V2’ = (ri7?i + r 2772) - ^ - 5(^0 n ef f - Uj)(nj + 1) - Uj + Py/p~n~j (6 ) 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), ( ) th at A n ] '2 - E n ) /2 + C n 3/ - D n )/2 + p y /P ^G rij - p ự K Q n ] - ậ y f p u = 0, where the coefficients A , E , c , D, G, Q are given by: r i a ? a 2^4ổ4eV15 r 2a i a ^ V e V '25 A iB lT n + ££ 02T 22 r _ r ia ? i/3g 3eVj ABoTn r 2a ịi/3g3eV2 4£B 0T22 T i a Ị a 2iy3g 3I i B i B r 2a i a ^ V ^ - S i 2£5 02T n + 2£B gr22 r 1a Ị a 2i'3g 3B + TiOLiáịv*g2B i - 270'ia 2ỉ' 2ỡ2 B ị B kẻl T.aW^hB,r2a|i/V/2fl2 Txa\vWZ + r2a2V g2 B 0T n 2£B0T22 2£ B T l a 1a 2v 2g2B rr T 2a ia ^ 2g2B l rr 2^5^eVi 11 + 2ZB2 eV2 22 YiOtiGt2V2g2h B \B T20i\0i2V2g2h B \B ^ B le V x 2ỊB Ịẽ V r i _ T i a i ug r 2a 2vg T iC tiyg liB i 2B 0eVi 11 2Ì B ữeV2 22 2B 0eVi 'yaii'gBiZ + ^5 ^ £B 0’ Y2a 2V9 h B i 2£B 0eV2 7’ (7) N g u y e n Van P h u , D in h Van H o a n g , C a o L on g Van 50 Or — ctịvgB i Bo 0C2vgB2 n _ OL\a.2V2g2B \B (B o - „ -1 -; V — Co u = r — ; = 9{v - Wj), n e // Tn = ^ / / flo e V i + ĩịẼị] r 22 = y/^I2B0eV2 + / f B f W hen the external optical signal disappears (Pw = 0), we obtain from (7) ( 8) A rP j - EVij 4- C rij - D = For the most of sem iconductor lasers used in practice we have also [7] V\ = V2 — V, B \ = B — 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( uo- Wj ) = and obtain finally: (9) Hj — 'Hrij + Crij —M = 0, where: n = c= Bo D (F1T22 + r2r„) + ^(ri/iT22+ r2T2Tu) + TllTẠ-(riaỉy+r2au - 27Ổ) a u eV B o t e T ^ T n + T2I2T n ) + B °T" T™m B T uT aa eK = B ỰiTi + /2r2) + BqT\\T22 a 2u2eV av +1} + r2) + (IY T n + r2T22) / 7I n IT g^(fr,rn + r2r22) - jLfc/.r, + J3r2) - e§° /T T = a u { T 1T22 + T2T n ) i The Eq (9) represents the catastrophe manifold of the Riemann-Hugoniot (or ‘cusp’) catastrophe A s in the M ather-Thom classification [20] T his catastrophe is given by the following potential function V(x\ a, b): V( x \ a, b) — - X + - a x + bx ( 10 ) The physical system described by this potential function has evolution generated by variations in the control param eters a = L - V.2/2),b = ;H C/ - M - 27i3/27 The O ptical b is ta b ility effe ct in D F B la se r w ith tw o se c tio n s 51 system, in accordance with the general principle of minimization of potential energy, will tend to dwell on the catastrophe surface M given by M = { ( x , a , b) : X + a x + b = } 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 , ) : X3 + a x + b = ,3 x 4- a = 0} (11) The X variable may be eliminated from the system of equations defining the set E Then we obtain the bifurcation set z?3 given by: B = {(a, b) : 4a 4- 2762 = 0} This set determ ines the param eters range involved in the 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 ) not change the form of the hysteresis curves as we will see in the next Section Influence o f som e dynam ical param eters on optical bistab ility effect T h e a p p e a n ce o f o p tic a l b ista b ility e ffe c t We can now solve numerically the equations (7), ( 8) The values of the param eters involved in the problem are taken from the experimental d a ta for a concrete semiconduc­ tor laser on InG aA sP given by K inoshita [7] and Yong-Zhen Huang [8]: Co = 3.10 10cm s_1; e = 1,6.1(T19C; Vi = 84.10- 12cm3; v = 84.10“ 12cm3; B = 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; £ — ; r = ; r2= 0.2 ; = , n.io^s1; Pi = 0; = s ” 1; p„ = 1022cm"3 110° InjecUoncurrenl I^(A) F ig Hysteresis curve of optical bistability effect in DFB laser with two sections N g u y e n Van P h u , D in h Van H o a n g , C ao Long Van 52 In the MATLAB language, we have received a hysteresis curve of optical bistability effect shown in Fig Here injection current I\ is control param eter and distance X1X2 indicates the width of bistability (BSW) 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 : X 10“ 5Ẩ,2.5 X 10- A ,2.8 X 10- 5i4 The corresponding curves are presented in Fig 3: The dotted line corresponds to the value of /2 = X 10-5 j4, the dashed and solid lines correspond to the values of /2 = 2.5 X 10~5A and /2 = 2.8 X 10” 5A T he results are given in the Table I T ab le I h(A) BSW(A) 0.1543 X 2.5 10~5 X 10~5 2.8 0.4423 X 10~5 1.3486 F ig Influence of injection current /2 on hysteresis curves of optical bistability effect Other values of I '2 are /2 = X 10 “ 5j4 , Ì — 2.5 X “ i , /2 = 2.8 X 10“ 5A 3 In 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 W hen £ rises the BSW diminishes The results are given in Table II T ab le I I £ 0.1 0.15 0.2 BSW(A) 0.4354 0.2194 0.1343 O ptical b ista b ility effect in D F B la se r w ith tw o s e c tio n s 53 *10” F ig Influence of saturation coefficient £ on BSW of hysteresis curves Influence o f the gain value a In this case the curves of optical bistability are presented in Fig Prom this Fig we s e e th a t when the gain value a increases the BSW increases too The numerical results are given in Table III T a b le III a ( c m 2) BSW(A) 0.4354 X 10- 16 X 1(T 16 0.6857 X 10- 16 1.12 11 I io ’3 F ig Influence of gain values a on hysteresis curves of optical bistability effect N g u y e n Van P h u , D in h Van H o a n g , C ao L ong Van 54 C onclusions From above obtained results we derive the following conclusions: Optical bistability effect appeared like in the case of lasers containing saturable absorber (LSA) [16] Here, the decisive condition for having hysteresis curves of OB effect is the current Ỉ in section B m ust be much smaller than current I \ in section A Laser parameters as gain, saturation coefficients, etc will be control parameters for hysteresis curves T he change of dynamical param eters involved in the problem clearly influences on characteristics of optical bistability effect as the bistability width 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, saturation coefficients, etc can be realized by changing proportion of X or y in structure I n \ - xG axA s y P \-y of m aterial References Dinh Van Hoang et al., M odem problems in Optics and Spectroscopy, II, 406 (2000) G M orthier and p Vankwikelberge, Handbook of Distributed Feedback Laser Diodes, Artech House, Norwood, MA 1999, p 1792 A Lugiato, L M Narducci, Phys Rev A 32, 1576 (1985) D Dangoisse et al., Phys Rev A 42, (1990) 1551 H Wenzel et al., IE E E J Quantum Elec 32, 69 (1996) B Sartorius et al., IE E E J Quantum Elec 33, 211 (1997) J Kinoshita, IE E E J Quantum Elec 30, 928 (1990) Yong-Zhen H uang,IE E E Photonics Tech Lett 7, 977 (1995) H Onaka et al., in: Optical Fiber Communication (O F C ’96) Post deadline papers, P art B, San Jose, 25 Feb.-l Mar., pp PD 19-1/5 10 G P Makino et al., in: Optical Fiber Communication (O F C ’96) Technical Digest, Vol 2, San Josfe, 25 Feb.-l Mar., pp PD 298-142 11 I Jiondot and I L Beylat, Electron Lett 29, 604 (1993) 12 S M Sze, Physics o f semiconductor devices, 2nd ed., Wiley, New York 1981 13 K Seeger, Sem iconductor Physics, Springer-Verlag, Berlin 1985 14 M Asada et al., IE E E J Quantum Elec 17, 947 (1981) 15 K Iga and s K inoshita, Process technology fo r semiconductor laser, Springer series in m aterials science, New York 1996 16 P.Q Bao, D v Hoang and Luc, J of 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 18 M Ohstu, Frequency Control of Semiconductor Lasers, Wiley, New York 1996 19 G s He, S.H Liu, Physics of nonlinear optics, World Scientific Publishing Co Pte Ltd, Singapore 1999, C hapter 12 20 G Gilmore, Catastrophe Theory fo r Scientists and Engineers, New York 1981  ... 110° InjecUoncurrenl I^(A) F ig Hysteresis curve of optical bistability effect in DFB laser with two sections N g u y e n Van P h u , D in h Van H o a n g , C ao Long Van 52 In the MATLAB... 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... following conclusions: Optical bistability effect appeared like in the case of lasers containing saturable absorber (LSA) [16] Here, the decisive condition for having hysteresis curves of OB effect