Fstrag Z. El-Halafawy* Mal"d GI El-moly

Abd EL-Naser: A, Mob-& Mohammed S: F, Tabbow

ABSTRACT h the present paper, under thermal operating conditions, the performance characteristics

of high speed laser diode ( based on Vertical Cavity Surface Emitting Laser (VCSEL) ) is deeply and parametrically investigated. The processed characteristics are : tbe harmonic response transfer function, both the resonance frequency and 3dB band-width, and the rise time. These diodes affect the transmitted bit-rate in high-speed advanced optical communication systems. The effects of both ambient temperature (and consequently the inner temperature) and the injected current (power) tm deeply investigated, l%e pulse rise time, and the resonance frequency as well as the 34B bandwidth are the major criterions of the device speed. It is found that the device undergoes good performance: higher resonance frequency, higher bandwidth, and smaller rise time as the proposed figure of merit increases.

I. Introduction fn recent years, a new type of semiconductor laser has attracted considerable inkerest, namely, the vertical-

cavity surface-emitting laser (VCSEL). This device offers many advantages over edge-emitters, resulting in its growing popularity in thc field of optoelectronics, including single-longitudinal-mode operation, circular output beams, suitability for monolithic twdimensional integration, and compatibility with on-wafer probe testing[l]. The architecture of VCSEL can make particularly good use of the broad quantum-we11 gain-bandwidth product for the generation of ultrashort pulses, and can easily achieve gigahertz-level repetition rates[2]. Although VCSELs- have chiefly attracted interest to date as frequency-doubled blue sources, their potential as practical and inexpensive ultrashort pulse sources is equally great. It is recently demonstrated a picosecond mode-locked VCSELs operating at 1.5pm [Z]. Most characteristics of Gas-based VCSELs in the 0.8 to 1.0 pm wavelength range are now comparable than that of edge-emitters in the lower power (-lmW) regime where many short-haul data communications applications fall. For these applications, fiber loss and dispersion are generally not significant factors. These VCSELs are also proposed for many applications, ranging fkom printing to optical switching[3]. Moreover, these lasers have proven power-scaling capability[2], and there are many additional attractive Characteristics of VCSELS include their circularly shaped, low-numerical-aperture output beams for easy coupling to fibers or free-space optics; their single-axial mode spectra (although lateral modes still need to be dealt with) for potential wavelength- division multiplexing (WDM) or wavelength addressing schemes; their high power conversion efficiency in the IOW power range far reduced heating in highly integrated circuits and their natura1 vertical emission or array applications. . .

Efforts in InP-based longer wavelength VCSELs, favored for long-haul fiber -based telecommunications (1.3-i.6pm), have met with slower progress because of inherent difficulties in Cclnsttycting highly reflecting mirrors as well as high-gain active regions, Nevertheless, recent work with wafer - bonded GaAs mirrors and strained quantum well active regions suggests that viable devices may be hrthcoming. Similarly, shorter wavelength, visible VCSELs have been somewhat more difilcult to engineer, but again, recent progress suggests that viable devices will shortly emerge. These we desired for plastic-fiber data links data links (-650 nm) as well as storage and display applications [3]. VCSELs have emerged as practical contenders for applications that require high continuous wave output beams. They combine the power scalabiiity and near-diffraction-limited output of a diode pumped solid-state laser with the spectral versatility for a semiconductor quantum-well gain medium. High power VCSELs Me usually

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pumped optically, exploiting the mature diode pumplaser technology developed for SoIid-state systems. As frequehcy-doubled sources that generate blue radiation, VCSELs offer a cumput and nrgged dtemative to the argon-ion laser for the excitation of fluorophores in bio-analytical applications [Z].

II. Basic Model VCSEL possesses a complex thennal performance. F i t , 8s its temperature increases, a VCSEL's gain

spectrum broadens and its peak location shifts to longer wavelengths. The device's emission wavelengths also hcrease with temperantre, though considerably less than the gain -41. Consequently, depending on the initial hatim of the gain peak relative to the wavelength, the laser gain will either deaease or increase with temperature as thc gain peak and wavelength became more or less mismatched[4].

Sccond, thermal leakage of caders out of the active region can lead to a redudion of injection efficiency, which contributes to a VCSEL's thermal roll-over. As the device temperame rises, the position of the active layers fami levels increaSes relative to the band gap. consequently, the active-layer becomes irtcreasingly incapable of confurng carriers. 'Ihe resulting leakage can be modeled as a function of carrier density and temperature[Sl. Because of the Cartier density dqendence, spatial hole burning can d t in further reduction of the injection &CienCy[5J.

For simplicity, we can neglect the effects of spatial hole W i g , so that the threshald current will be assumed to be soldy a function of temperature. A h , we choose to model the offset current using a polynomial function of temperature in the following fo1~nula[6]:

&,ff(T)=a~+alT+azT 2+a3T 3+a4T (1)

where the coefficients By introducing the offset c m t into the standard laser rate equations through an empbkd fit lo

expimental data, we will be able to model LI (light current) c w e s at different t e m w as well as take advantage of many of the desirable properties of rate equations, in particular the ability to model non& behavior such as smaIl-signal modulation. Furthermme, because the rate equations are a widely accepted taol for modeling semianductor lasers, they serve as an efficient description of the basic lasing behavior of a wide variety of VCSELs. Thus, the introduction of m empirical description of t h e d behavior into this basic fonnulation should retain much of its general applicability. The modified rate equations are[6]:

- Q can be determined during pameter extraction.

-= dN qi(I-Ioff(T)) --- N po(N-N~)s (2) dt 9 ' fn 1 + E S

- = - - ds s +- P N + g o ( N - N T ) S (3) dt =p T I l 1 + E S

where N and S are carrier and photon numbers respectively, q, is the injection efficiency, I is the injection current, q is the electronic charge, T~ is the carrier recombination life time, 7p is the photon life t h e , E is the gain compression factar, is the spontaneous emission coupling coef'ficient, Go is the gain coefficient, and NT is the carrier transpwncy number.

For high frequency modulation by small signals, the above system of equations cm be linearized. For this purpose the electron and photon densities, and the current through the laser active m e are expressed as:

N = N , +nejwt (4) s=s,+mJ* (5) I=I,+icjw (6)

where n << No, s .c< S, and i << I,,, No and So are the steady state densities of electrons and photons respectively, and o) = &$is the angular laser modulation fresuency, rad / sec. The steady state electrons and p b w densities are given in terms of &, (steady state laser bias current) putting &/dt = 0.0 and dY&=O.O into Eqns.(2,3), hence by some manipulations we can get the steady state photon density as :

- A 2 + ,/- / > s o =

2 A 1

._ w h a :

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'Ihe normalized transfer function G.(j U) is defined as : ( s l i ) , - U0 2 (19) G n ( j m ) =

( s / i ) q ) = O - m a - m 2 + jBU From Qn.( 19 ), the resonance frequency mr and the peak at resonance G,, can be Written as :

Similarly, the 3-dB bandwidth or the modulation bandwidth %a and, the gain cross over frequency 0, Can be calculated to yield :

and 2 2 2 mg = 2 ~ , - B

To complete the model, we still require expressions for the temperature, the applied voltage and, the optical output power. First, while it is certainly possible to adopt detailed numerical representations of the VCSEL temperature profile as a function of the heat dissipation throughout the device[l], a much simpler method is to describe the temperature via a themd rate equation which accounts for the transient temperature increase as a result of heat dissiptian[lj,[9]. Following this approach, the temperature equation can be written as:

d T T 5 To + ( I v - P , ) R * - fh - d t

where &, is the VCSEL's thermal impedance ( which relates the change in device temperatUte to the power dissipated as heat ), rh is a thermal time constant (which i s necessary to account for the n o m m response time of the device temperature, observed to be on -the order of I p s [6], and To is the ambient temperature. Under dc conditions, the dT/dt term disappears; thus, from the resulting equation it is clear that ( IY- Po) models the power

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dissipated in the VCSEL, where it assumed that any power not carried in the optical output is dissipated as heat in the device [4].

For the last equation to be solved, we have to get an expression for the Po as a function of the laser voltage Y. A h , we have to obtain another expression for the laser voltage to convert Eqn. (24) to be a function of the injected curcent I and the temperature of the source. Then by giving the program the value of the injected current we can get easily the temperature value. IJI. ResulQ and Discussion

The investigated device is 3.1-pm diameter thinaide-apatured VCSEL reported by T h i k u l t et d [IO]. Thc lsscr is coinposed of an Ab9GaolAsGaAs ptype DBR three In0~~Gao~3As-GaAs quantum wells, an AbgGe& cavity, and BII AlAssraAs n-type DBR. 3 y using the data that is included in the w e that authors mtd and fading it to fitting program for polynomial curye fittin the faded data, we get :

Po=2.44233 x 10'- 1.05522 x 10' Vi- 8.90924 x 10- V2- 1.31399 V3f4.21374x 10' V4 +I .50434 x 10" v5 (25)

(26) V=2.18755+ 1 .13597~ 10-'1-3.62664x 10312+4.81684x lO"l' with a root mean square errors of 0.47859Ox10-' and 0.458252~10-~ for the two equations respectively. Then, using the provided dat461: q, E 0.821, 2.68 X IO* r,, s 1.201 ns, Go = 8,486 X lo5 SI, No = 1.286 X 106, z, = 2.884 ps, E = 3.888 X IO", a3 = 0 &', aa = 0 n~4.K'. By providing the mentioned data CO the software program, we get the shown results. As shown in Fig 1, the 3-dB bandwidth increases as the injection current increases, and it decreases linearly with the increase of temperature.

Using the curve fitting p'Dcedure to get the corresponding polynomial equations of the 3- bandwidth from the curves of Fig. 2. we et :

fm= 2.1054 x 10 + 3.1237 I - 1.2132 x IO' l2 t 1.404 x lW3 1' at To = 285 "K

at To = 290 "K

= 0.896 WmW, a, = 2.213 X IW3 mA, a, = -1.719 X lo4 mAM, = 3.355 X lo4 "IC',

(27)

(28)

(29)

(30)

(31)

f

f 3 a = 2.0538 x 10' + 3.0893 I - 1.1828 x 10" l2 + 1.248 x I'

f 3 4 = 2.0078 x 10' + 3.0239 I - 1.1285 x 10" I2 + 1.019 x l o 3 I' at To= 295 "K

fm = 1.9730 x 10' + 2.9092 I - 1.0336 x IO-' I' f 6.68 x lo4 P To=3oo"K

f& = 1.9617 x lo'+ 2.7066 I - 8.632 x

f3* = a0 + a1 I + az I' + a3 l3

l2 + 9.6 x l o 5 I3 at To = 305 "K which can be arranged in the form :

Then, by using the curve fitting p d u r e to get the polynomial equations of all the ow&cients against the cmesponding ambient temperature, we get the following results :

81, a2,and a3

a. =1.0083xId -1.0499x1dTo +3.7319~16~T~ - 4 . 4 3 4 9 ~ ~ 1 0 ~ ~ ~

a1=-2.2416x102+2.435 To - 8 . 6 x W 3 TZ +1.0043 K ~ O - ' T ~

a2=1.7707 x ld-1.8962 x llr' '6 +ti654 xlQ4 3 - 7.7068 x

a3 =-5.49L1b1 +5.92 x W 3 % -20958 xlr5$ +24416 xlfl'$

(32) (33) (34) (35)

Figures 3 and 4 show the variations of the rise time of the device agdnst variations of the ambient t e m p e w at different values for the injection current, and against variations of the injection current at different ambient tempagtures respectively.

In the same way, we can get the corresponding polpornid ~wtions of rise time from the curves of Fig.4. as : T= 1.2319 x 10' - 7.5559 x 1 0 ' I + 3.441 1 x lo-' I - 5.3341 x lo4 l3

~ 1 . 2 5 6 0 ~ 10'-7.8906x 10'1+3.6028x 1O"I2-5.4685x 1041'

T= 1.2589 x 10'- 7.6304 x 1 0 ' I + 3.3198 x 1021* - 4.5529 x lo4 I3

(34)

(37)

(38)

at T0=285"K

at To=29O0K

at To = 295 *K ! t

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T= 1 . 2 7 2 7 ~ 10'-74029x 10"1+3.2151 x 1O~*I2-4.0142x IO4 i3

T = 1.2901 x 10' - 7.4367 x 10 ' I f 2.8565 x iO-z12 - 2.5326 x IO4 I3

~ = b ~ + b ~ 1 + t + d + b 3 1 ~

bo =-2w74+ CM862 To - O.1663 1

(39)

(40) at To=300nK

at To = 305 O K

which cm be arranged in the form :

Then, by using the curve fitting procpdure to get the poIynornial equations of all the ooeficients bp b,, h,md b against the corresponding ambient temperature, we get the following results :

2 + 0.~1412xlO=-~ $ (41)

$ =42613-0,4542T0 +0.157!W~lO-~ T'-O.I826~t0-~$ (42)

b;! =-5.0191+0.05381;, - 0 . 1 8 9 6 ~ 1 6 ~ $ +0221l2x1O4'$ - 143)

3 =~1653~~00177-4, + a m i 7 X 1 c + 2 -a72szxi1Pd (44) Figures 5 and 6 show the variations of the raonance frequency of the device against variations of the ambient temperature at different values for the injection current, and against variations of the injection current at different ambient temperatures respectively. Similarly, the resonance fresuency f , can be represented from the curves of Fig.6. by the following equdons:

at To = 285 O K

at T, = 290 O K

at To = 295 O K

at T0=3000K

at T0=305QK which can be arranged in the form :

f,= 1.2199 x 10' + 1.7395 I - 6.5898 x 10" I* + 7.3969 x lo4 l3

f,= 1.1898 x 10' + 1.7278 1-6.4573 x 10212 + 6.5552 x IO4 I3

f,= 1.1626~ 10' + 1.6985 1-6.1912 x 10212 + 5.3034 x IO' I3

f, = 1.1420 x 10' + 1.6396 I - 5.6853 x I2 + 3.3402 x 10'' I3

f,= 1.2060 x 10' + 1.3350 I - 3.2SS3 x I O 2 l2 - 2.9168 x IO' I3

f , = ~ + c1 I +Q I~ +

(45)

(46) .

. (47)

(48)

(49)

Then, by using the m e fitting procedure to get the polynomial equations of all the coefficients CO, CI,

c0 = 1268.3 - 13.346 To + 0.047199 To2 - 0.55584~10 q,and q against the corresponding ambient temperature, we get the following resdts :

(50)

(5 1)

(52)

(53)

- 4 3 TO

CI= -378.5 + 4.0652 To - 0.01439 To2 + O.16876~10-~ T i 9=29.315-031427 '&, +0.1112&10-2 2- 0 . 1 3 0 4 8 ~ 1 0 - ~ ~

5 =469686+ 0.00747% -026441 xl(r4 $ +a30945 xlO-'$ At pulse of duration z, = 40 psec, Figs.7, and 8 show the time responses of the nomalhd output power for the

*mentioned device against the time of operation at the assumed values of the ambient temperature, and at different values for the injection currentg.

These figures indicate that as either To at I or both increase, the overshoot increases as well as, the time response shrinks. A figureof merit FM isdesignedas:

The variationS of the set (fr , f 3 - d ~ , T) against the variations of FM are displayed in Figs.9,10, and 11. It is clear that the device possesses good perfonnance (higher resonance frequency, higher bandwidth, smaller rise rime) at higher value of FM according to the higher technological limits. The forlowing nonlinear correlations are cast :

FM=VTo. (54)

fm= 1.5486~ loL+ 1.4056~ 103FM-2.7714~ lo4 FM2 +2.7841 x 105FM3'

r - 1.2625 x 106w + 2.4799 x IO5 FM'i 1.2839~ 1W2 (55)

= 1.5043 x 10' - 5.0306 x 10' FM + 1,4355 x 10' FM2 - 2.2375 x lo5 FM3

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+1.7780 x 106F& - 5.4507 x 1O5FMS * 1.0813 x I O 2

105~M4 + 6.8814 x io5 F M ~ f 1.283% 10 '~

(56)

(57) fr = 8.9386 + 7,9216 x IO2 FM - 1,5886 x IO' FM2 +1.6&84 x lo5 FM3 - 8.5592 x

w b : 0.01 5 FMs0.09 APK 5.0 5 1 5 25.0 mA 285.0 5 To 5 305.0 OK.

These hmIin&ar correlations (positive or negative) facilitak the best operating conditions. Figures 12 and 13 show the variations of the product of resonance frequency - in GHz - and, rise time - in psec - of the device against variations of the injection current at different ambient temperatures and, against variations of the ambient t e m p " at different values for the injection current respectively. Using the curve fitting procedure to get the corresponding polynomial equations of the product, P, of the resonance fkquency and the rise time from the curves of Fig. I L we get :

Pr =1.7148 x IO' + 4.3147 I - 5.683 x lo-' 1' + 3.6413 x IO-* I3 - 1.2089 x I O 3 I' + 1.5747 x I O 5 Is (58 )

6.1 103 x 10" Is (59)

4.2606 lo4 is (60)

i .4223 x 10" 1' (61)

4.8586 x IO4 I5 (62)

at To = 285 OK Pr= 1.7279 x 10' + 3.0990 I - 3.5262 x 10-'12 + 1,8620 x I' - 5.3190 x lo4 I' +

at To = 290 OK Pr= 1.7268 x lo2 i- 2.6865 I - 2.9303 x IO-' I2 + 1.4302 x IO9 l3 -3.8586 x IO4 I4 +

at To = 295 'K Pr 1.6933 x IO2 -t 3.8619 1 - 5.1507 x IQ-' 1' -I- 3.2663 x f - 1.0843 x IO-' I' +

at To = 300 "K Pr= 1.7107 x IQ2 + 2.4986 I - 2.8549 x 10*'l2 + 1.4420 x 1W2 i3 -4.0936 x IO4 I' +

at To = 305 "K

Iv. Conclusions In this paper, we. clarified that the introducing of a thermal o f k t current into the standard rate equations

provides m effective mean to account for thermal effects. We studied the effects of taking the heat dissipation due to the injection current on the mentioned device. These effects are analyzed and discussed based on the 3-dB bandwidth, rise time, and the r e m " frequency. The time response is investigated also where it is found that the overshoat increases as well as the response shrinks as either To or I or both in-. A designed figure of merit is proposed to control the good performance of the device. The product of resonance. frequency and the rise time are also processed for the device at different operating conditions (injected current and ambient temperahe).

-

References 111 K. fga, F. Koyama, and S. Kinoshita, a Surface - Emitting Semiconductor Lasen ", IEEE J. Quantum

-[21 A. Tropper. and S. Hoogland, WCSELs Achieve High Power and Spectral Versatility", SPIE's OE magazine,

131 L. A. Coldren, B. J. Thibeault, "Vertical Cavity Surface-Emitting Lasers" Ch. 6, pp. 200-266,In Optical Fiber Tekcanmunications I I B , Ed. I. P. Kaminowand, T. L. Koch, AP, USA, 1997.

141 W. Nakwaski, Thermal Aspects of Efficient Operation of Vertical-Cavity Surface Emitting Lasers", Optical Quantum Electron, Vol. 28, pp. 335-352,1996.

[Si J. W. SmU, R. S. Geels, S. W. Corzine, and L. A. Coldren, "Modeling Temperature Effects and Spatial Hole Burning to Optimize Vert id4avi ty Surface-Emitting Laser Performance", IEEE J.Quanium Electron, Vol. 29, pp. 1295-1308, May 1993.

161 P+V. Men% J.J. Morikuni, S. M. Kang, A. V.Harton, and K. W. Wyatt, " A Simple RftbEquatiokBased Thermal VCSEL Model", J. Lightwave TechnoL,VoI. 17, NOS, pp.865-872, May 1999.

17) M. Osinski, and W. Nakwaski, '' Thennal Effects in Vertical-Cavity Surface Emitting Lasers 'I. Internat -ionaI J. of High Speed Electronics and Systems, Vol. 5, No. 4, pp. 667-730,1994.

Elecwn, Vol. 24, No. 9, pp.1845-1855, Sept.1988,

pp. 21-23, MU. 2004.

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181 S. F, Yu, W. N. Wong, P. Shwn, and E. N. Li, Theoretical Analysis of Modulati& Rtspollse and Second - order Harmonic Distortion in Vertical Cavity Surface-Emitting lase”’, IEEE J. Quantum Electron., Vol.

[9] N. Bewtra, D. A. Suda, G. L. Tan, F. Chatenoud, and !.M. Xu,“ Modeling of quantum-well Lasers with El-Thermd I n t ” ”, IEEE J.Selected Topics in Quantum Electron, Vol. 1, No. 2, pp. 331- 340,1995.

[lo] B. J . Thibeault, K. Bertilsson, E. R Hegblom, E. Stnelecka, P. D. Floyd, R. Naone, and A. Y. Cho, High- speed Characteristics of Low-Optical Loss Oxide-Apertured in Vertical Cavity Lasera ,, , IEEE Photonics

32, NO. 12, pp. 21394147,1996.

Techfiolog~ Le#ers, Vol. 9, NO. I , pp. 11-13, 1997.

Fig. 1. Variations of f3- against variations of To at the assumed set of parameters.

Injection Current., I, mA

Pig.2. Variations of f s a against variations of 1 at the assumed set of parameters. (where To in *K).

2

I .8

f .6

1.4

1.2

1

0.8

0.6

0.4

0.2

0 0 7n M 60

Time, t, psec

Y 27 1

0.01 0.03 0.05 0.07 0.09

Quality Factor, FM, A I "K

Fig.9. Variations of f3- against variations of Quality factor, FM.

2 , I

8 0.6

0.4

0.2

0

8 z

0 3n An 60 Time, t, psec

Fig.%. Variations of Po agair" *-A&--" of t at the assumed set of par AIGaAs re To in OK) when I = 25mA, T, = 4, ,,-.

9 -

8.5 - 8 -

7.5 -

7 -

6 .5 j . . ' . . . ' ' . . ' . . . . 0.01 0.03 0.05 0.07 0.09

Quality Factor, FM, APK

Fig.10. Variations of T against variations of Quality factor, FM.

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27 ,

c.

K

0.01 0.03 0.05 0.07 0.09

Quality Factor, FM, A / O K

Fig. 1 1. Variations of f, against variations of Quality factor, FM.

80

a4

82

80

70

70

74

72

70

68

+i=IPmA -0- la1 6mA -X- 1=2 Om A

172 ~

70 ~

68 -. * * a ' - - - : a * * - * ' - - - *

285 290 295 300 305

Ambient Temperature, To , *K L I " I

AI

Fig.12. Variations of C ,.T against variations of To at the assumed set of parameters.

4 6 8 10 12 14 16 10 20 22 24 Injection Current, I, mA

Fig. 13. Variations off, . t against variations of I at the assumed set of parameters. (where To in OK)