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Determination of spontaneous emission rate from RIN data
Optics Communications 255 (2005) 102–113 www.elsevier.com/locate/optcom
Determination of spontaneous emission rate from RIN data W. Zheng, G.W. Taylor *, Y. Huo Electrical and Computer Engineering Department, University of Connecticut, 371 Fairfield Road, Storrs, CT 06269-2157, USA Received 18 January 2005; received in revised form 27 May 2005; accepted 30 May 2005
Abstract RIN data are analyzed to obtain values of spontaneous emission rate Rsp. The values are compared with a new theoretical prediction and agreement is obtained without the use of fitting parameters. In addition to the agreement with experiment, the new theoretical expression is distinct from existing expressions since no empirical parameters are involved. 2005 Elsevier B.V. All rights reserved. Keywords: Quantum well lasers; Spontaneous emission; RIN
1. Introduction The dynamic and modulation response of laser diodes continues to be an issue of great interest for highspeed optoelectronic circuits [1–3]. To this end, considerable work has been concerned with the basic intrinsic physical limitations due both to transport [4,5] and to nonlinear gain [6] since it provides the internal damping of the frequency response and effectively determines the bandwidth. It is well known that the relative intensity noise (RIN) [7] is another measurement that provides direct access to the intrinsic laser parameters and can avoid the de-embedding of the transport effects and the extrinsic RC time constants due to packaging or RF excitation. Also, RIN has direct significance in establishing the bit error rate (BER) of optical communication systems. Recently, a small-signal model [8] was developed for analysis of the laser dynamic response and prediction of the modulation response frequency. This analysis included the temperature dependence of the refractive index as a specific mechanism which produced a closed form expression for e, the nonlinear gain parameter on the basis of the thermal conductance of the semiconductor. Using this model, the gain, the *
Corresponding author. Tel.: +1 860 4862666; fax: +1 860 4863756. E-mail address: [email protected] (G.W. Taylor).
0030-4018/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2005.05.042
W. Zheng et al. / Optics Communications 255 (2005) 102–113
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differential gain and the K factor were derived analytically and confirmed by experiment. It is the purpose of this paper to examine RIN data using this model to obtain confirmation of the predicted value of Rsp, the spontaneous emission rate. The most important parameter in the understanding and prediction of RIN is Rsp, the rate of spontaneous emission into the laser mode. For noise, it is the driving force in the fluctuation terms for both the electron and photon populations. We present an approach to the calculation of Rsp which is expressed in closed form on the basis of the Fermi levels in the quantum well at the optical threshold condition [9]. This expression shows the dependence of Rsp on temperature, frequency, Fermi energy and photon lifetime in an analytic way. In addition, it enables correlation of the spontaneous emission to the stimulated emission through the Planck thermal radiation law. In this paper, the RIN is calculated with this approach and the standard form results but with a substantial difference in magnitude. Using the modified result, the RIN calculation agrees with several experiments without the use of fitting parameters.
2. Theoretical calculations The photon lifetime sp is expressed as 1 1 1 ; ¼ sp sp0 1 þ eF
ð1Þ
where sp0 is the photon lifetime when the photon flux density F is equal to zero, and e is the nonlinear gain compression coefficient which was derived previously [8] as aRJ WmLz hmvg ð1 gwp Þ ln R1 c ; vg ¼ ð2Þ e¼ ng ng gwp C for quantum well lasers resulting from the temperature dependence of the refractive index, where gwp is the wall plug efficiency of the laser, RJ is the thermal resistance of the GaAs microwave pn junction, W is the width of the laser active layer, m is the quantum well number, Lz is the quantum well width, vg is the group velocity, ng is the group effective index, C is the optical confinement factor, and a is the refractive index temperature coefficient. An improved estimation of the gain compression coefficient which includes the surface recombination effects is derived in Appendix A, and is used in this paper for comparison with experimental data. The rate equations for electrons and photons in the quantum wells have been developed in the previous paper [8] and are used again here to solve for the dynamic noise performance with a fixed bias current. The noise in the system is generated by the Langevin noise terms FF(t) and Fn(t) which are included as excitation terms in each differential rate equation [10] dnw J qw nw ¼ vg GF þ F n ðtÞ; dt qLz sn dF F ¼ Cvg GF þ CRsp þ F F ðtÞ; dt sp
ð3aÞ ð3bÞ
where nw is the electron density in the quantum well, Jqw is the current density injected to the quantum well, sn is the carrier recombination lifetime, Rsp is the spontaneous emission into the lasing mode, and G is the gain. The gain is expressed as G¼
1 ; vg sst ð1 þ eF Þ
where sst is the stimulated emission lifetime which is given by [11]
ð3cÞ
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s1 st ¼
Dnp ð1 þ eEFp =kT Þð1 þ eEFn =kT Þ ln hkT B . ð1 þ eðhmEFn Þ=kT Þ ð1 þ eðhmEFp Þ=kT Þ L2z
ð4Þ
The gain is established by the fact that above threshold sst = Csp. The differential rate equations are linearized about the laser operating point which is determined by the analytic expressions for EFn ; EFp ; EFT ; hm; nw0 and sst0 . For the purposes of the small signal operation and noise analysis, all of these operating parameters are fixed at the dc solution. Using the quasi-Fermi energies EFn, EFp, and their sum EFT = EFn + EFp as parametric variables and performing the linearization as in [8], the following small signal equations are derived: dEFT ðxÞ F 1 dEFT ðxÞ F edF ðxÞ 1 dF ðxÞ n dEFT ðxÞ ~ ¼ þ w0 þ F n ðxÞ; ð5aÞ 2 kT ð1 þ eF Þ sst0 kT sst0 ð1 þ eF Þ sst0 ð1 þ eF Þ sn kT CF 1 dEFT ðxÞ CF edF ðxÞ þ CdRsp ðxÞ þ F~ F ðxÞ; ð5bÞ sdF ðxÞ ¼ ð1 þ eF Þ sst0 kT sst0 ð1 þ eF Þ2
snw0
where nw0 ; sst0 are defined in Appendix B, and F~ n ðxÞ and F~ F ðxÞ are the Fourier transforms of FF(t) and Fn(t). The variables dnðxÞ and ds1 st ðxÞ are transformed into the variable dEFT(x) as derived earlier [8] and the results are summarized in Appendix B. An expression for Rsp was developed in [9] given by F iT 1 A 1 ¼ ð6Þ Dmsp bsp 1 eðhmEFTo Þ=kT ; sst B sst are the Einstein constants, EFTo is the value of EFT at the optical threshold for frequency m, where A and B and Dmsp is the linewidth of the mode at the optical threshold condition which is defined as 1 . ð7Þ Dmsp ¼ 4ps0p Rsp ¼
In the above, FiT is the photon density in the mode at the optical threshold which is defined as the condition when the modal density is just comparable to the spontaneous background, i.e. Rsp @ RSTn, the net stimulated emission rate. bsp is the spontaneous emission factor bsp ¼
k3 m ; 8p2 ng n2m Dmsp WLðmLz =CÞ
ð8Þ
where nm is the material index and L is the length of the laser cavity. This expression for bsp has been derived by Lee [12], Petermann [13] and others and appears in standard texts [10,11]. Through the parameter sst (see (4)), Rsp varies with EFT = EFn + EFp (note that EFn, EFp are measured from their respective band edges). In terms of the small signal parameter dEFT, dRsp is expressed as 1 F iT dEFT ; ð9Þ dRsp ¼ F iT d ¼ sst sst0 kT where ds1 st is derived in Appendix B. From (5a), (5b), and (9), dF(x) is found as iT F~ n ðxÞ s þ s1n þ x2r sp0 ð1 þ eF Þ F~ F ðxÞ þ x2r Csp0 ð1 þ eF Þ þ nCF s n o w0 st0 dF ðxÞ ¼ ; ð10Þ s2 þ s s1n þ x2r sp0 ð1 þ eF Þ þ g0evg þ x2r 1 þ g0 veg sn þ n s sF iTð1þeF Þ2 w0 st0 p0
where xr is the relaxation resonance frequency defined as 1 CF nw0 sst0 Fvg g0 2 xr ¼ ¼ 2 2 sst0 ð1 þ eF Þ sp0 ð1 þ eF Þ
ð11Þ
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and g0 is the differential gain [8] expressed in terms of the quasi-Fermi energies (clamped values above threshold) as 1 1 dðvg sst Þ dnw 1 ba þ ab 1 1 ¼ ¼ . ð12Þ g0 ¼ vg ab vg nw0 sst0 dEFT dEFT In general, g0 is represented by the empirical constant a. Here an analytical form is available. The laser RIN (not including the shot noise contribution) is defined as [10] dB S dF ðxÞ S dF ðxÞDf RIN or RINðdBÞ ¼ ; ¼ 2 Hz F F2 where Df is the frequency bandwidth of the measurement system, and Z 1 2 S dF ðxÞ ¼ hdP ðt þ sÞdP ðtÞi expðixtÞds ¼ hjdF ðxÞj i.
ð13Þ
ð14Þ
1
Substituting dF(x) from (10) into (14), we obtain S dF ðxÞ ¼
ðx2 þ c2a ÞhF~ F ðxÞF~ F ðxÞi þ c2b hF~ n ðxÞF~ n ðxÞi þ 2ca cb hF~ F ðxÞF~ n ðxÞi 2
ðx2 x20 Þ þ x2 C20
;
ð15Þ
where ca ¼
1 þ x2r sp0 ð1 þ eF Þ; sn
ð16Þ
ð1 þ eF ÞF iT cb ¼ x2r Csp0 ð1 þ eF Þ 1 þ x2r Csp0 ð1 þ eF Þ; F e F iT Fvg g0 e 2 2 x0 ¼ xr 1 þ þ 1þ ; 2 g 0 vg sn g 0 vg sn F sp0 ð1 þ eF Þ 1 e 1 C0 ¼ þ x2r sp0 ð1 þ eF Þ þ ¼ þ Kf 20 sn g 0 vg sn
ð17Þ ð18Þ ð19Þ
and K¼
4p2 ðsp0 ð1 þ eF Þ þ e=g0 vg Þ . 1 þ e=g0 vg sn
ð20Þ
The previously derived relationships [10] exist for F~ n ðxÞ and F~ F ðxÞ hF~ F ðxÞF~ F ðxÞi ¼ 2Rsp F ; nw ~ ~ hF n ðxÞF n ðxÞi ¼ 2 Rsp F þ ; sn V hF~ F ðxÞF~ n ðxÞi ¼ 2Rsp F ;
ð21aÞ ð21bÞ ð21cÞ
where V is the cavity volume. It is noted that there is a substantial dependence upon Rsp. Substituting (21) into (15), we finally obtain the RIN expression as RIN ¼
~ þ Bx ~ 2 A ðx2 x20 Þ2 þ x2 C20
ð22aÞ
;
where 2 ~ ¼ 2Rsp c2 þ c2 2ca cb þ cb nw A ; a b F Rsp F sn V
~ ¼ 2Rsp . B F
ð22bÞ
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W. Zheng et al. / Optics Communications 255 (2005) 102–113
When the partition noise is included, the total measured output RIN will be " # ~ þ Bx ~ 2 hm 2g0 ðca þ C0 Þx2 ca x20 A 1þ RINtotal ¼ þ 2 2 ðx2 x20 Þ þ x2 C20 P ðx2 x20 Þ þ x2 C20
ð23Þ
as described by others [14], where P is the total output power and g0 is defined as g0 = vg ln(1/R)/(Lsp). Although the form of the result in (22a) is identical to that derived previously [7], there are significant ~ B; ~ x0 and C0 which are expressed in (23), (18) and differences in the derivations of the parameters A; (19), respectively. There are two standard forms for Rsp found in textbooks, which are [10,14] Rsp ¼ Absp n2w ;
ð24Þ
Rsp ¼ Cvg nsp aðnw ntr Þ=V ;
ð25Þ
where nsp is the population inversion factor, a is the differential gain, and ntr is the transparency carrier density, whereas in this paper we use (6). The Rsp value (24) is significantly higher than experiment or the new prediction. The Rsp value (25) is much closer to the value extracted from experiment, but is based upon the numerical values a, nth and ntr obtained from simulation in contrast to the form of (6). For example, for the experimental laser parameters considered in Section 3, (6) yields 2.2 · 1022 cm3 s1, whereas (24) yields 4.6 · 1024 cm3 s1 and (25) yields 2.7 · 1022 cm3 s1. This difference in the value of Rsp results not from the choice of parameter values used for the calculation but from a fundamentally different approach to the derivation.
3. Comparison to experiment and confirmation of Rsp Comprehensive RIN measurements have been reported in [7,15,16] for MQW lasers. The various parameters and symbols are listed in Table 1 along with the values used in the calculation for MQW laser samples I, II and III with different structures. In analyzing typical RIN data it is customary to extract the parameters x0, C0 and Rsp as fitting parameters since expressions for differential gain, nonlinear gain, n and F are not available without numerical analysis. With this fitting procedure, x0 and C0 parameters give reasonable ~ and B ~ fitted parameters do not agree with the predictions (24) and (25). results but the parameters A Sample I is a buried heterostructure MQW InGaAs–InGaAsP laser operating at 1.55 lm [7]. The active layer region consists of 16 In0.53Ga0.47As wells and 15 InGaAsP barriers both lattice-matched to the InP substrate. In Fig. 1, we show the experimental RIN data and compare it to the RIN calculated on the basis of (16)–(23) and using the measured physical material values such as the carrier lifetime, the waveguide loss and the active layer structural parameters. The parameters used are shown in Table 1, which include a spontaneous recombination coefficient A = 3.0 · 1010 cm3/s, a gain compression coefficient e = 3.7 · 1017 cm3 and a value of bsp = 0.006 as predicted by (8). To obtain this agreement, the value of Rsp must be 2.2 · 1022 cm3 s1. From (6), we also find this value, whereas from (24) and (25) we have 4.6 · 1024 and 2.7 · 1022 cm3 s1, respectively. In Fig. 2, we show the measured values of the resonance frequency f0 = x0/2p for these lasers and compare to the predicted value. In Fig. 3, we compare the measured and predicted damping factor C0. In the above two figures we find good agreement with the same set of parameters used for Fig. 1, i.e. the values in Table 1. This is confirmation of the theoretical approach. Sample II is a buried heterostructure 6-quantum-well InGaAs–InGaAsP laser operating at 1.5 lm [15]. From the relations between the resonant frequency and the square root of output power per facet reported we note that the RIN data are for the quantum well material In0.66Ga0.34As which is compressively strained to the substrate. In Fig. 4 the experimental RIN is compared with the calculation for three values of output power (per facet). A spontaneous recombination coefficient A = 2.3 · 1010 cm3/s, a gain compression coefficient e = 4.5 · 1017 cm3 and a value of bsp = 0.001 were used for this MQW laser. To obtain this
W. Zheng et al. / Optics Communications 255 (2005) 102–113
107
Table 1 MQW laser parameters for three wavelengths Sample
I
II
III
Wavelength, k (lm) Amplifier length, L (lm) Amplifier width, W (lm) Quantum well material Barrier material Quantum well number ˚) Quantum well/barrier width (A Waveguide loss, a (cm1) Effective mass for electrons, mn (m0) Effective mass for holes, mp (m0) Carrier lifetime, sn (ns) Optical confinement factor, C Gain compression factor, e (cm3) Spontaneous emission factor, bsp Rsp calculated from (6) (cm3 s1) Rsp calculated from (24) (cm3 s1) Rsp calculated from (25) (cm3 s1)
1.55 250 1.5 In0.53Ga0.47As InGaAsP 16 80/100 15 0.041 0.423 0.15 0.253 3.7 · 1017 0.006 2.2 · 1022 4.6 · 1024 2.7 · 1022
1.5 300 1.5 In0.66Ga0.34As InGaAsP 6 40/120 15 0.036 0.045 1 0.04 4.5 · 1017 0.001 3.3 · 1022 1.2 · 1024 4.4 · 1022
1.3 300 1.3 InGaAsP InGaAsP 10 40 15 0.049 0.181 0.5 0.065 1 · 1017 0.003 1.3 · 1023 5.8 · 1024 4.9 · 1022
RIN (dB/Hz)
-120
-130
-140
Simulation
-150
Experimental data for P=0.83mW Experimental data for P=8.3mW
0
2.5
5
7.5
10
12.5
15
17.5
Frequency (GHz)
Fig. 1. Simulated and measured RIN spectra [7] for the 16-quantum-well InGaAs–InGaAsP laser.
agreement, the value of Rsp is determined to be 3.3 · 1022 cm3 s1. From (6), we also find this result whereas from (24) and (25) we have 1.2 · 1024 and 4.4 · 1022 cm3 s1, respectively. In Fig. 5, we show the measured values of the resonance frequency f0 = x0/2p for these lasers and compare to the predicted value. In Fig. 6 we compare the measured and predicted damping factor C0. In the above two figures we find good agreement with the same set of parameters used for Fig. 4. We have found a deviation to exist in x0 between the simulation and the experiment for the power of P = 0.5 mW (P1/2 = 0.71 mW1/2). This is indicated as a shift to high frequency in Fig. 4 and a dip in the measured frequency in Fig. 5. The RIN peak as a function of output power can be obtained from (22) by using x = x0, i.e. ~ þ Bx ~ 2 A 0 . RINpeak P ¼ x20 C20
ð26Þ
In Fig. 7, the peak RIN versus 10 log(1/P3) for samples I and II is shown with three and two experimental peak RIN values, respectively. It is evident that the peak RIN is almost linearly proportional to 1/P3 as observed in (26) through the dependence of C0 and x0 upon P, for both samples I and II.
W. Zheng et al. / Optics Communications 255 (2005) 102–113
14 12
f0 (GHz)
10 8 6 4 Simulation
2
Experimental data
0
0.5
1
2
1.5
2.5
3
Power½ (mW ½ )
Fig. 2. Simulated and measured [7] resonance frequency (f0) as a function of the square root of the facet power.
60
Γ0 (109 Rad/s)
50 40 30 20 Simulation
10
Experimental data
0
50
100
150
f02 (GHz2 )
Fig. 3. Simulated and measured [7] damping (C0) as a function of the square of the resonance frequency ðf02 Þ.
Simulation Experimental data for P=0.5mW
–110
Experimental data for P=2.2mW
RIN (dB/Hz)
108
Experimental data for P=5.4mW
–120
–130
–140
0
2.5
5
7.5
10
12.5
15
Frequency (GHz)
Fig. 4. Simulated and measured RIN spectra [15] for the 6-quantum-well InGaAs–InGaAsP laser.
W. Zheng et al. / Optics Communications 255 (2005) 102–113
109
14 12
f0 (GHz)
10 8 6 4 Simulation
2
Experimental data
0
1
2
3
4
Power½ (mW ½ )
Fig. 5. Simulated and measured [15] resonance frequency (f0) as a function of the square root of the facet power.
50
40
Γ0 (109 Rad/s)
30
20
10
Simulation Experimental data
0 0
25
50
75
100
125
150
f02 (GHz2 )
Fig. 6. Simulated and measured [15] damping (C0) as a function of the square of the resonance frequency ðf02 Þ.
Sample III is a strained 10-quantum-well InGaAsP–InGaAsP distributed feedback laser operating at 1.3 lm [16]. In Fig. 8, the simulated and measured dependence of RIN on bias current under modulation at 1.9 GHz is shown. A spontaneous recombination coefficient A = 2 · 1010 cm3/s, a gain compression coefficient e = 1 · 1017 cm3 and a value of bsp = 0.003 were used for this MQW laser. To obtain this agreement, the value of Rsp determined is 1.3 · 1023 cm3 s1. From (6), we also find this result, whereas from (24) and (25) we have 5.8 · 1024 and 4.9 · 1022 cm3 s1. From Fig. 8, we observe that when measured at a fixed frequency, RIN decreases as the bias current (output power) increases. In addition the RIN decreases more slowly as the current is decreased.
4. Summary A new formulation of the RIN spectrum for the laser diode has been described based on a Fermi level representation of both the electron and photon rate equations. This model is able to predict the RIN, the resonance frequency and the damping without resorting to fitting parameters. Specifically the model shows that the value of Rsp is most important in determining the RIN magnitude and two separate measurements
110
W. Zheng et al. / Optics Communications 255 (2005) 102–113 -100
RINpeak (dB/Hz)
-110
sample II
-120 sample I
-130 Simulation Experimental data
-140 -30
-20
-10
0
10
3
10 log(1/P ) (P in mW)
Fig. 7. Simulated and measured peak RIN as a function of 10 log(1/P3) for samples I [7] and II [15].
-120
f =1.9GHz
-125
RIN (dB/Hz)
-130 -135 -140 Simulation
-145
Experimental data
-150 -155 40
60 80 Bias current (mA )
100
120
Fig. 8. Simulated and measured [16] dependence of RIN on bias current with modulation at 1.9 GHz.
confirm the magnitude and validity of the Rsp formulation. This result is of significance because Rsp is the controlling factor in many critical laser calculations such as ASE (amplifier spontaneous emission) noise, linewidth, coherence, etc. Other advantages of the solution are the ability to predict (from the basic parameters) temperature and loss dependence of the spontaneous emission, the gain and the differential gain.
Appendix A. Derivation of gain compression coefficient The effects of surface recombination on the cleaved facets have a significant impact on the current flow pattern in edge emitting lasers. This in turn establishes thermal gradients in the structure which are effective in determining the nonlinear gain compression coefficient e. Previously [8], e was calculated based upon the temperature increase in the laser expressed by (2), (which did not include the facet recombination) and its amplitude was about 10 times smaller than the experimentally determined value. The result is modified here to include surface recombination at the facets. The continuity equation for the laser is expressed in terms of the electron (the equation for holes is identical) as
W. Zheng et al. / Optics Communications 255 (2005) 102–113
on J 0 n n0 d2 n ¼ þ Dn 2 ðRST RAB þ Rsp Þ; ot qd dx sn
111
ðA1Þ
where d is the active region including the SCH regions, the quantum wells and the barriers, Dn is the electron diffusion coefficient, and RST, RAB and Rsp are the per unit volume rates of stimulated recombination, stimulated absorption and spontaneous recombination, respectively, which are determined generally for the laser as average values throughout the cavity. Over most of the length of the laser, dn/dx 0 and assuming n n0, then (A1) reduces to 0¼
J0 n F ; qd sn s0p
ðA2Þ
which is the standard laser equation for above threshold operation. When above threshold, the solution of (A2) provides n = nth where nth is the clamped carrier density value. However, the facets are characterized by a high density of interface states which means the carrier density drops dramatically as the facet is approached. Therefore near the facet, (A1) is expressed as 0¼
J 0 n n0 d2 n þ Dn 2 ; dx qd sn
ðA3Þ
where the F =s0p term is missing because the reduced carrier densities render negligible the processes RST and Rsp (the process of RAB provides a modified sn). Our interest is to determine the length of the region affected by surface recombination. To do this we solve (A3) for the entire laser in the absence of lasing and subject 0 sn j ¼ qSnð0Þ (S is the surface recombination velocity) and nð1Þ ¼ Jqd . to the boundary conditions qDn dn dx x¼0 Then we determine the position when the solution of (A3) is equal to clamped carrier density value nth. To represent the region of recombination, the standard solution of (A3) is [17] sn J 0 sn S expðx=Ln Þ sn J 0 ð1 expðx=Ln ÞÞ ; ðA4Þ nðxÞ ¼ n0 þ 1 Ln þ s n S qd qd where Ln is the diffusion length from the cleaved facets towards the center, n0 (intrinsic) is taken to be negligible and snS Ln. The position for n = nth is obtained as J0 r Ln ln ðA5Þ Ln J th =J 0 J 0 J th for large input current density. The approximation J0 Jth represents the range of interest of J0. Within the region of r there is a diffusion current flowing toward the surface given by J s ¼ qDn
dnðxÞ Ln J 0 expðx=Ln Þ. dx d
The average current density in the recombination area is Rr Z J s dx Ln J 0 r Ln J 20 ð1 expðJ th =J 0 ÞÞ ¼ expðx=Ln Þdx ¼ . J sav ¼ 0 r dJ th dr 0
ðA6Þ
ðA7Þ
Then the average current in the recombination area is Is ¼
wLn J 20 ð1 expðJ th =J 0 ÞÞ ; J th
ðA8Þ
where w is the width of the laser. The current in the central area (away from the facets) is approximately I b ¼ wLJ 0 ;
ðA9Þ
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W. Zheng et al. / Optics Communications 255 (2005) 102–113
where L is the length of the laser. Then the ratio of the temperature changes in the diffusion area and the central area is given by 2
cT ¼
2
DT s I 2s Rs Z T s L2n ðLJ 0 2Ln J th Þ ð1 expðJ th =J 0 ÞÞ ¼ ¼ . DT b I 2b Rb Z T b d 2 L2 J 2th
ðA10Þ
The temperature changes at the cleaved facets of the laser lead to the changes of the refractive index of the active layer material which affect the reflection coefficient of the facet. Assuming the active layer refractive index at the facet is n1 = n10(1 + aDT2), where n10 is the refractive index without temperature change, the reflection coefficient can be calculated as n2 pffiffiffi n1 n2 1 n10 ð1þaDT n10 n2 2an10 n2 DT s sÞ R¼ ¼ þ ; n2 n1 þ n2 1 þ n10 ð1þaDT n10 þ n2 ðn10 þ n2 Þ2 sÞ
where n2 is the refractive index of air. Thus R can be approximated as !2 n10 n2 2an10 n2 DT s þ R0 ð1 þ dÞ; R¼ n10 þ n2 ðn10 þ n2 Þ2
ðA11Þ
ðA12Þ
where R0 = ((n10 n2)/(n10 + n2))2 is the reflection coefficient without temperature changes and d¼
4an10 n2 DT s . ðn10 þ n2 Þðn10 n2 Þ
Since the photon lifetime can be expressed as 1 c 1 1 ln þ a ; ¼ sp ng þ aDT b L R then by using small signal expansions of the ln term, we obtain 1 c 1 1 lnð1 þ dÞ ln ¼ 1 þa sp ng ð1 þ ða=ng ÞDT b Þ L R0 lnð1=R0 Þ c 1 1 1 ln þ a . 4ng n10 n2 cT ng 1 þ a DT b L R0 ng
ðA13Þ
ðA14Þ
ðA15Þ
ðn10 þn2 Þðn10 n2 Þ lnð1=R0 Þ
From the relation between power and photon density for the laser, we obtain [8] DT 1 ffi eng F =a
ðA16Þ
and substituting into (A14) we obtain 1 1 1 . ¼ sp sp0 1 þ eF
ðA17Þ
From (A15) and (A17), we identify the gain compression coefficient as e¼
aRJ WmQW Lz hvvg0 ð1 gwp Þ lnð1=R0 Þ 4ng n10 n2 cT . ng gwp C ðn10 þ n2 Þðn10 n2 Þ lnð1=R0 Þ
ðA18Þ
Previously e was derived without recombination effects to produce the first term. The second term is required to include facet recombination. It is noted that cT is dependent upon the input current and therefore the value of nonlinear gain coefficient e obtained will also be current dependent. However, the only values of interest (listed in Table 1) are those for the highest currents (powers).
W. Zheng et al. / Optics Communications 255 (2005) 102–113
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Appendix B. Relation between dn(x), ds1 st (x) and dE FT (x) When performing small signal analysis, we need to express dnw and dsst in terms of dEFT. The results were derived previously [8] and are summarized here. dnw is given by dnw ¼
ab dEFT dEFT ¼ nw0 ; a þ b kT kT
ðB1Þ
where a¼
Dn kT EFn0 =kT 1 þ 2eEFn0 =kT þ eDEn =kT ; e Lz ð1 þ eEFn0 =kT ÞðeEFn0 =kT þ eDEn =kT Þ
ðB2aÞ
b¼
Dp kT Epn0 =kT 1 þ 2eEFp0 =kT þ eDEp =kT e ; Lz ð1 þ eEFp0 =kT ÞðeEFp0 =kT þ eDEp =kT Þ
ðB2bÞ
dsst is given by 1 dðs1 st Þ ¼ sst0
dEFT ; kT
s1 st0 ¼
ba þ ab ; aþb
ðB3Þ
where a and b are given by a¼u
eðEFn0 þhmÞ=kT eEFn0 =kT ; ð1 þ eEFn0 =kT ÞðeEFn0 =kT þ ehm=kT Þ
ðB4aÞ
b¼u
eðEFp0 þhmÞ=kT eEFp0 =kT . ð1 þ eEFp0 =kT ÞðeEFp0 =kT þ ehm=kT Þ
ðB4bÞ
It is noted that a, b, a and b are derived from the steady-state solution in terms of the quasi-Fermi levels for electrons and holes.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17]
I. Suemune, L.A. Coldren, M. Yamanishi, Y. Yan, Appl. Phys. Lett. 53 (1988) 1378. R. Olshansky, P. Hill, V. Lanzisera, W. Powazinik, IEEE J. Quantum Electron. 23 (1987) 1410. C.B. Su, J. Eom, C.H. Lange, C.B. Kim, R.D. Lauer, W.C. Rideout, J.S. LaCourse, IEEE J. Quantum Electron. 28 (1992) 118. R. Nagarajan, T. Fukushima, M. Ishikawa, J.E. Bowers, R.S. Geels, L.A. Coldren, IEEE Photon. Technol. Lett. 4 (1992) 121. R. Nagarajan, T. Fukushima, M. Ishikawa, J.E. Bowers, R.S. Geels, IEEE J. Quantum Electron. 28 (1992) 1990. R. Nagarajan, T. Fukushima, J.E. Bowers, R.S. Geels, L.A. Coldren, Electron Lett. 27 (1991) 1058. M.C. Tatham, I.F. Lealman, C.P. Seltzer, L.D. Westbrook, D.M. Cooper, IEEE J. Quantum Electron. 28 (1992) 408. G.W. Taylor, Y. Huo, J. Shao, Opt. Commun. 189 (2001) 345. G.W. Taylor, Prog. Quantum Electron. 16 (1992) 73. G.P. Agrawal, N.K. Dutta, Semiconductor Lasers, second ed., Van Nostrand-Reinhold, New York, 1993. G.W. Taylor, J. Appl. Phys. 70 (1991) 2508. T.P. Lee, C. Burrus, J. Copeland, A. Dentai, D. Marcuse, IEEE J. Quantum Electron. 18 (1982) 1101. K. Petermann, IEEE J. Quantum Electron. 15 (1979) 566. L.A. Coldren, S.W. Corzine, Diode Lasers and Photonic Integrated Circuits, Wiley, New York, 1995. H. Yasaka, R. Iga, Y. Noguchi, Y. Yoshikuni, IEEE J. Quantum Electron. 29 (1993) 1098. H. Watanabe, T. Aoyagi, A. Takemoto, T. Takiguchi, E. Omura, IEEE J. Quantum Electron. 32 (1996) 1015. S.M. Sze, Physics of Semiconductor Devices, second ed., Wiley, New York, 1981.
Determination of spontaneous emission rate from RIN data W. Zheng, G.W. Taylor *, Y. Huo Electrical and Computer Engineering Department, University of Connecticut, 371 Fairfield Road, Storrs, CT 06269-2157, USA Received 18 January 2005; received in revised form 27 May 2005; accepted 30 May 2005
Abstract RIN data are analyzed to obtain values of spontaneous emission rate Rsp. The values are compared with a new theoretical prediction and agreement is obtained without the use of fitting parameters. In addition to the agreement with experiment, the new theoretical expression is distinct from existing expressions since no empirical parameters are involved. 2005 Elsevier B.V. All rights reserved. Keywords: Quantum well lasers; Spontaneous emission; RIN
1. Introduction The dynamic and modulation response of laser diodes continues to be an issue of great interest for highspeed optoelectronic circuits [1–3]. To this end, considerable work has been concerned with the basic intrinsic physical limitations due both to transport [4,5] and to nonlinear gain [6] since it provides the internal damping of the frequency response and effectively determines the bandwidth. It is well known that the relative intensity noise (RIN) [7] is another measurement that provides direct access to the intrinsic laser parameters and can avoid the de-embedding of the transport effects and the extrinsic RC time constants due to packaging or RF excitation. Also, RIN has direct significance in establishing the bit error rate (BER) of optical communication systems. Recently, a small-signal model [8] was developed for analysis of the laser dynamic response and prediction of the modulation response frequency. This analysis included the temperature dependence of the refractive index as a specific mechanism which produced a closed form expression for e, the nonlinear gain parameter on the basis of the thermal conductance of the semiconductor. Using this model, the gain, the *
Corresponding author. Tel.: +1 860 4862666; fax: +1 860 4863756. E-mail address: [email protected] (G.W. Taylor).
0030-4018/$ - see front matter 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2005.05.042
W. Zheng et al. / Optics Communications 255 (2005) 102–113
103
differential gain and the K factor were derived analytically and confirmed by experiment. It is the purpose of this paper to examine RIN data using this model to obtain confirmation of the predicted value of Rsp, the spontaneous emission rate. The most important parameter in the understanding and prediction of RIN is Rsp, the rate of spontaneous emission into the laser mode. For noise, it is the driving force in the fluctuation terms for both the electron and photon populations. We present an approach to the calculation of Rsp which is expressed in closed form on the basis of the Fermi levels in the quantum well at the optical threshold condition [9]. This expression shows the dependence of Rsp on temperature, frequency, Fermi energy and photon lifetime in an analytic way. In addition, it enables correlation of the spontaneous emission to the stimulated emission through the Planck thermal radiation law. In this paper, the RIN is calculated with this approach and the standard form results but with a substantial difference in magnitude. Using the modified result, the RIN calculation agrees with several experiments without the use of fitting parameters.
2. Theoretical calculations The photon lifetime sp is expressed as 1 1 1 ; ¼ sp sp0 1 þ eF
ð1Þ
where sp0 is the photon lifetime when the photon flux density F is equal to zero, and e is the nonlinear gain compression coefficient which was derived previously [8] as aRJ WmLz hmvg ð1 gwp Þ ln R1 c ; vg ¼ ð2Þ e¼ ng ng gwp C for quantum well lasers resulting from the temperature dependence of the refractive index, where gwp is the wall plug efficiency of the laser, RJ is the thermal resistance of the GaAs microwave pn junction, W is the width of the laser active layer, m is the quantum well number, Lz is the quantum well width, vg is the group velocity, ng is the group effective index, C is the optical confinement factor, and a is the refractive index temperature coefficient. An improved estimation of the gain compression coefficient which includes the surface recombination effects is derived in Appendix A, and is used in this paper for comparison with experimental data. The rate equations for electrons and photons in the quantum wells have been developed in the previous paper [8] and are used again here to solve for the dynamic noise performance with a fixed bias current. The noise in the system is generated by the Langevin noise terms FF(t) and Fn(t) which are included as excitation terms in each differential rate equation [10] dnw J qw nw ¼ vg GF þ F n ðtÞ; dt qLz sn dF F ¼ Cvg GF þ CRsp þ F F ðtÞ; dt sp
ð3aÞ ð3bÞ
where nw is the electron density in the quantum well, Jqw is the current density injected to the quantum well, sn is the carrier recombination lifetime, Rsp is the spontaneous emission into the lasing mode, and G is the gain. The gain is expressed as G¼
1 ; vg sst ð1 þ eF Þ
where sst is the stimulated emission lifetime which is given by [11]
ð3cÞ
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W. Zheng et al. / Optics Communications 255 (2005) 102–113
s1 st ¼
Dnp ð1 þ eEFp =kT Þð1 þ eEFn =kT Þ ln hkT B . ð1 þ eðhmEFn Þ=kT Þ ð1 þ eðhmEFp Þ=kT Þ L2z
ð4Þ
The gain is established by the fact that above threshold sst = Csp. The differential rate equations are linearized about the laser operating point which is determined by the analytic expressions for EFn ; EFp ; EFT ; hm; nw0 and sst0 . For the purposes of the small signal operation and noise analysis, all of these operating parameters are fixed at the dc solution. Using the quasi-Fermi energies EFn, EFp, and their sum EFT = EFn + EFp as parametric variables and performing the linearization as in [8], the following small signal equations are derived: dEFT ðxÞ F 1 dEFT ðxÞ F edF ðxÞ 1 dF ðxÞ n dEFT ðxÞ ~ ¼ þ w0 þ F n ðxÞ; ð5aÞ 2 kT ð1 þ eF Þ sst0 kT sst0 ð1 þ eF Þ sst0 ð1 þ eF Þ sn kT CF 1 dEFT ðxÞ CF edF ðxÞ þ CdRsp ðxÞ þ F~ F ðxÞ; ð5bÞ sdF ðxÞ ¼ ð1 þ eF Þ sst0 kT sst0 ð1 þ eF Þ2
snw0
where nw0 ; sst0 are defined in Appendix B, and F~ n ðxÞ and F~ F ðxÞ are the Fourier transforms of FF(t) and Fn(t). The variables dnðxÞ and ds1 st ðxÞ are transformed into the variable dEFT(x) as derived earlier [8] and the results are summarized in Appendix B. An expression for Rsp was developed in [9] given by F iT 1 A 1 ¼ ð6Þ Dmsp bsp 1 eðhmEFTo Þ=kT ; sst B sst are the Einstein constants, EFTo is the value of EFT at the optical threshold for frequency m, where A and B and Dmsp is the linewidth of the mode at the optical threshold condition which is defined as 1 . ð7Þ Dmsp ¼ 4ps0p Rsp ¼
In the above, FiT is the photon density in the mode at the optical threshold which is defined as the condition when the modal density is just comparable to the spontaneous background, i.e. Rsp @ RSTn, the net stimulated emission rate. bsp is the spontaneous emission factor bsp ¼
k3 m ; 8p2 ng n2m Dmsp WLðmLz =CÞ
ð8Þ
where nm is the material index and L is the length of the laser cavity. This expression for bsp has been derived by Lee [12], Petermann [13] and others and appears in standard texts [10,11]. Through the parameter sst (see (4)), Rsp varies with EFT = EFn + EFp (note that EFn, EFp are measured from their respective band edges). In terms of the small signal parameter dEFT, dRsp is expressed as 1 F iT dEFT ; ð9Þ dRsp ¼ F iT d ¼ sst sst0 kT where ds1 st is derived in Appendix B. From (5a), (5b), and (9), dF(x) is found as iT F~ n ðxÞ s þ s1n þ x2r sp0 ð1 þ eF Þ F~ F ðxÞ þ x2r Csp0 ð1 þ eF Þ þ nCF s n o w0 st0 dF ðxÞ ¼ ; ð10Þ s2 þ s s1n þ x2r sp0 ð1 þ eF Þ þ g0evg þ x2r 1 þ g0 veg sn þ n s sF iTð1þeF Þ2 w0 st0 p0
where xr is the relaxation resonance frequency defined as 1 CF nw0 sst0 Fvg g0 2 xr ¼ ¼ 2 2 sst0 ð1 þ eF Þ sp0 ð1 þ eF Þ
ð11Þ
W. Zheng et al. / Optics Communications 255 (2005) 102–113
105
and g0 is the differential gain [8] expressed in terms of the quasi-Fermi energies (clamped values above threshold) as 1 1 dðvg sst Þ dnw 1 ba þ ab 1 1 ¼ ¼ . ð12Þ g0 ¼ vg ab vg nw0 sst0 dEFT dEFT In general, g0 is represented by the empirical constant a. Here an analytical form is available. The laser RIN (not including the shot noise contribution) is defined as [10] dB S dF ðxÞ S dF ðxÞDf RIN or RINðdBÞ ¼ ; ¼ 2 Hz F F2 where Df is the frequency bandwidth of the measurement system, and Z 1 2 S dF ðxÞ ¼ hdP ðt þ sÞdP ðtÞi expðixtÞds ¼ hjdF ðxÞj i.
ð13Þ
ð14Þ
1
Substituting dF(x) from (10) into (14), we obtain S dF ðxÞ ¼
ðx2 þ c2a ÞhF~ F ðxÞF~ F ðxÞi þ c2b hF~ n ðxÞF~ n ðxÞi þ 2ca cb hF~ F ðxÞF~ n ðxÞi 2
ðx2 x20 Þ þ x2 C20
;
ð15Þ
where ca ¼
1 þ x2r sp0 ð1 þ eF Þ; sn
ð16Þ
ð1 þ eF ÞF iT cb ¼ x2r Csp0 ð1 þ eF Þ 1 þ x2r Csp0 ð1 þ eF Þ; F e F iT Fvg g0 e 2 2 x0 ¼ xr 1 þ þ 1þ ; 2 g 0 vg sn g 0 vg sn F sp0 ð1 þ eF Þ 1 e 1 C0 ¼ þ x2r sp0 ð1 þ eF Þ þ ¼ þ Kf 20 sn g 0 vg sn
ð17Þ ð18Þ ð19Þ
and K¼
4p2 ðsp0 ð1 þ eF Þ þ e=g0 vg Þ . 1 þ e=g0 vg sn
ð20Þ
The previously derived relationships [10] exist for F~ n ðxÞ and F~ F ðxÞ hF~ F ðxÞF~ F ðxÞi ¼ 2Rsp F ; nw ~ ~ hF n ðxÞF n ðxÞi ¼ 2 Rsp F þ ; sn V hF~ F ðxÞF~ n ðxÞi ¼ 2Rsp F ;
ð21aÞ ð21bÞ ð21cÞ
where V is the cavity volume. It is noted that there is a substantial dependence upon Rsp. Substituting (21) into (15), we finally obtain the RIN expression as RIN ¼
~ þ Bx ~ 2 A ðx2 x20 Þ2 þ x2 C20
ð22aÞ
;
where 2 ~ ¼ 2Rsp c2 þ c2 2ca cb þ cb nw A ; a b F Rsp F sn V
~ ¼ 2Rsp . B F
ð22bÞ
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W. Zheng et al. / Optics Communications 255 (2005) 102–113
When the partition noise is included, the total measured output RIN will be " # ~ þ Bx ~ 2 hm 2g0 ðca þ C0 Þx2 ca x20 A 1þ RINtotal ¼ þ 2 2 ðx2 x20 Þ þ x2 C20 P ðx2 x20 Þ þ x2 C20
ð23Þ
as described by others [14], where P is the total output power and g0 is defined as g0 = vg ln(1/R)/(Lsp). Although the form of the result in (22a) is identical to that derived previously [7], there are significant ~ B; ~ x0 and C0 which are expressed in (23), (18) and differences in the derivations of the parameters A; (19), respectively. There are two standard forms for Rsp found in textbooks, which are [10,14] Rsp ¼ Absp n2w ;
ð24Þ
Rsp ¼ Cvg nsp aðnw ntr Þ=V ;
ð25Þ
where nsp is the population inversion factor, a is the differential gain, and ntr is the transparency carrier density, whereas in this paper we use (6). The Rsp value (24) is significantly higher than experiment or the new prediction. The Rsp value (25) is much closer to the value extracted from experiment, but is based upon the numerical values a, nth and ntr obtained from simulation in contrast to the form of (6). For example, for the experimental laser parameters considered in Section 3, (6) yields 2.2 · 1022 cm3 s1, whereas (24) yields 4.6 · 1024 cm3 s1 and (25) yields 2.7 · 1022 cm3 s1. This difference in the value of Rsp results not from the choice of parameter values used for the calculation but from a fundamentally different approach to the derivation.
3. Comparison to experiment and confirmation of Rsp Comprehensive RIN measurements have been reported in [7,15,16] for MQW lasers. The various parameters and symbols are listed in Table 1 along with the values used in the calculation for MQW laser samples I, II and III with different structures. In analyzing typical RIN data it is customary to extract the parameters x0, C0 and Rsp as fitting parameters since expressions for differential gain, nonlinear gain, n and F are not available without numerical analysis. With this fitting procedure, x0 and C0 parameters give reasonable ~ and B ~ fitted parameters do not agree with the predictions (24) and (25). results but the parameters A Sample I is a buried heterostructure MQW InGaAs–InGaAsP laser operating at 1.55 lm [7]. The active layer region consists of 16 In0.53Ga0.47As wells and 15 InGaAsP barriers both lattice-matched to the InP substrate. In Fig. 1, we show the experimental RIN data and compare it to the RIN calculated on the basis of (16)–(23) and using the measured physical material values such as the carrier lifetime, the waveguide loss and the active layer structural parameters. The parameters used are shown in Table 1, which include a spontaneous recombination coefficient A = 3.0 · 1010 cm3/s, a gain compression coefficient e = 3.7 · 1017 cm3 and a value of bsp = 0.006 as predicted by (8). To obtain this agreement, the value of Rsp must be 2.2 · 1022 cm3 s1. From (6), we also find this value, whereas from (24) and (25) we have 4.6 · 1024 and 2.7 · 1022 cm3 s1, respectively. In Fig. 2, we show the measured values of the resonance frequency f0 = x0/2p for these lasers and compare to the predicted value. In Fig. 3, we compare the measured and predicted damping factor C0. In the above two figures we find good agreement with the same set of parameters used for Fig. 1, i.e. the values in Table 1. This is confirmation of the theoretical approach. Sample II is a buried heterostructure 6-quantum-well InGaAs–InGaAsP laser operating at 1.5 lm [15]. From the relations between the resonant frequency and the square root of output power per facet reported we note that the RIN data are for the quantum well material In0.66Ga0.34As which is compressively strained to the substrate. In Fig. 4 the experimental RIN is compared with the calculation for three values of output power (per facet). A spontaneous recombination coefficient A = 2.3 · 1010 cm3/s, a gain compression coefficient e = 4.5 · 1017 cm3 and a value of bsp = 0.001 were used for this MQW laser. To obtain this
W. Zheng et al. / Optics Communications 255 (2005) 102–113
107
Table 1 MQW laser parameters for three wavelengths Sample
I
II
III
Wavelength, k (lm) Amplifier length, L (lm) Amplifier width, W (lm) Quantum well material Barrier material Quantum well number ˚) Quantum well/barrier width (A Waveguide loss, a (cm1) Effective mass for electrons, mn (m0) Effective mass for holes, mp (m0) Carrier lifetime, sn (ns) Optical confinement factor, C Gain compression factor, e (cm3) Spontaneous emission factor, bsp Rsp calculated from (6) (cm3 s1) Rsp calculated from (24) (cm3 s1) Rsp calculated from (25) (cm3 s1)
1.55 250 1.5 In0.53Ga0.47As InGaAsP 16 80/100 15 0.041 0.423 0.15 0.253 3.7 · 1017 0.006 2.2 · 1022 4.6 · 1024 2.7 · 1022
1.5 300 1.5 In0.66Ga0.34As InGaAsP 6 40/120 15 0.036 0.045 1 0.04 4.5 · 1017 0.001 3.3 · 1022 1.2 · 1024 4.4 · 1022
1.3 300 1.3 InGaAsP InGaAsP 10 40 15 0.049 0.181 0.5 0.065 1 · 1017 0.003 1.3 · 1023 5.8 · 1024 4.9 · 1022
RIN (dB/Hz)
-120
-130
-140
Simulation
-150
Experimental data for P=0.83mW Experimental data for P=8.3mW
0
2.5
5
7.5
10
12.5
15
17.5
Frequency (GHz)
Fig. 1. Simulated and measured RIN spectra [7] for the 16-quantum-well InGaAs–InGaAsP laser.
agreement, the value of Rsp is determined to be 3.3 · 1022 cm3 s1. From (6), we also find this result whereas from (24) and (25) we have 1.2 · 1024 and 4.4 · 1022 cm3 s1, respectively. In Fig. 5, we show the measured values of the resonance frequency f0 = x0/2p for these lasers and compare to the predicted value. In Fig. 6 we compare the measured and predicted damping factor C0. In the above two figures we find good agreement with the same set of parameters used for Fig. 4. We have found a deviation to exist in x0 between the simulation and the experiment for the power of P = 0.5 mW (P1/2 = 0.71 mW1/2). This is indicated as a shift to high frequency in Fig. 4 and a dip in the measured frequency in Fig. 5. The RIN peak as a function of output power can be obtained from (22) by using x = x0, i.e. ~ þ Bx ~ 2 A 0 . RINpeak P ¼ x20 C20
ð26Þ
In Fig. 7, the peak RIN versus 10 log(1/P3) for samples I and II is shown with three and two experimental peak RIN values, respectively. It is evident that the peak RIN is almost linearly proportional to 1/P3 as observed in (26) through the dependence of C0 and x0 upon P, for both samples I and II.
W. Zheng et al. / Optics Communications 255 (2005) 102–113
14 12
f0 (GHz)
10 8 6 4 Simulation
2
Experimental data
0
0.5
1
2
1.5
2.5
3
Power½ (mW ½ )
Fig. 2. Simulated and measured [7] resonance frequency (f0) as a function of the square root of the facet power.
60
Γ0 (109 Rad/s)
50 40 30 20 Simulation
10
Experimental data
0
50
100
150
f02 (GHz2 )
Fig. 3. Simulated and measured [7] damping (C0) as a function of the square of the resonance frequency ðf02 Þ.
Simulation Experimental data for P=0.5mW
–110
Experimental data for P=2.2mW
RIN (dB/Hz)
108
Experimental data for P=5.4mW
–120
–130
–140
0
2.5
5
7.5
10
12.5
15
Frequency (GHz)
Fig. 4. Simulated and measured RIN spectra [15] for the 6-quantum-well InGaAs–InGaAsP laser.
W. Zheng et al. / Optics Communications 255 (2005) 102–113
109
14 12
f0 (GHz)
10 8 6 4 Simulation
2
Experimental data
0
1
2
3
4
Power½ (mW ½ )
Fig. 5. Simulated and measured [15] resonance frequency (f0) as a function of the square root of the facet power.
50
40
Γ0 (109 Rad/s)
30
20
10
Simulation Experimental data
0 0
25
50
75
100
125
150
f02 (GHz2 )
Fig. 6. Simulated and measured [15] damping (C0) as a function of the square of the resonance frequency ðf02 Þ.
Sample III is a strained 10-quantum-well InGaAsP–InGaAsP distributed feedback laser operating at 1.3 lm [16]. In Fig. 8, the simulated and measured dependence of RIN on bias current under modulation at 1.9 GHz is shown. A spontaneous recombination coefficient A = 2 · 1010 cm3/s, a gain compression coefficient e = 1 · 1017 cm3 and a value of bsp = 0.003 were used for this MQW laser. To obtain this agreement, the value of Rsp determined is 1.3 · 1023 cm3 s1. From (6), we also find this result, whereas from (24) and (25) we have 5.8 · 1024 and 4.9 · 1022 cm3 s1. From Fig. 8, we observe that when measured at a fixed frequency, RIN decreases as the bias current (output power) increases. In addition the RIN decreases more slowly as the current is decreased.
4. Summary A new formulation of the RIN spectrum for the laser diode has been described based on a Fermi level representation of both the electron and photon rate equations. This model is able to predict the RIN, the resonance frequency and the damping without resorting to fitting parameters. Specifically the model shows that the value of Rsp is most important in determining the RIN magnitude and two separate measurements
110
W. Zheng et al. / Optics Communications 255 (2005) 102–113 -100
RINpeak (dB/Hz)
-110
sample II
-120 sample I
-130 Simulation Experimental data
-140 -30
-20
-10
0
10
3
10 log(1/P ) (P in mW)
Fig. 7. Simulated and measured peak RIN as a function of 10 log(1/P3) for samples I [7] and II [15].
-120
f =1.9GHz
-125
RIN (dB/Hz)
-130 -135 -140 Simulation
-145
Experimental data
-150 -155 40
60 80 Bias current (mA )
100
120
Fig. 8. Simulated and measured [16] dependence of RIN on bias current with modulation at 1.9 GHz.
confirm the magnitude and validity of the Rsp formulation. This result is of significance because Rsp is the controlling factor in many critical laser calculations such as ASE (amplifier spontaneous emission) noise, linewidth, coherence, etc. Other advantages of the solution are the ability to predict (from the basic parameters) temperature and loss dependence of the spontaneous emission, the gain and the differential gain.
Appendix A. Derivation of gain compression coefficient The effects of surface recombination on the cleaved facets have a significant impact on the current flow pattern in edge emitting lasers. This in turn establishes thermal gradients in the structure which are effective in determining the nonlinear gain compression coefficient e. Previously [8], e was calculated based upon the temperature increase in the laser expressed by (2), (which did not include the facet recombination) and its amplitude was about 10 times smaller than the experimentally determined value. The result is modified here to include surface recombination at the facets. The continuity equation for the laser is expressed in terms of the electron (the equation for holes is identical) as
W. Zheng et al. / Optics Communications 255 (2005) 102–113
on J 0 n n0 d2 n ¼ þ Dn 2 ðRST RAB þ Rsp Þ; ot qd dx sn
111
ðA1Þ
where d is the active region including the SCH regions, the quantum wells and the barriers, Dn is the electron diffusion coefficient, and RST, RAB and Rsp are the per unit volume rates of stimulated recombination, stimulated absorption and spontaneous recombination, respectively, which are determined generally for the laser as average values throughout the cavity. Over most of the length of the laser, dn/dx 0 and assuming n n0, then (A1) reduces to 0¼
J0 n F ; qd sn s0p
ðA2Þ
which is the standard laser equation for above threshold operation. When above threshold, the solution of (A2) provides n = nth where nth is the clamped carrier density value. However, the facets are characterized by a high density of interface states which means the carrier density drops dramatically as the facet is approached. Therefore near the facet, (A1) is expressed as 0¼
J 0 n n0 d2 n þ Dn 2 ; dx qd sn
ðA3Þ
where the F =s0p term is missing because the reduced carrier densities render negligible the processes RST and Rsp (the process of RAB provides a modified sn). Our interest is to determine the length of the region affected by surface recombination. To do this we solve (A3) for the entire laser in the absence of lasing and subject 0 sn j ¼ qSnð0Þ (S is the surface recombination velocity) and nð1Þ ¼ Jqd . to the boundary conditions qDn dn dx x¼0 Then we determine the position when the solution of (A3) is equal to clamped carrier density value nth. To represent the region of recombination, the standard solution of (A3) is [17] sn J 0 sn S expðx=Ln Þ sn J 0 ð1 expðx=Ln ÞÞ ; ðA4Þ nðxÞ ¼ n0 þ 1 Ln þ s n S qd qd where Ln is the diffusion length from the cleaved facets towards the center, n0 (intrinsic) is taken to be negligible and snS Ln. The position for n = nth is obtained as J0 r Ln ln ðA5Þ Ln J th =J 0 J 0 J th for large input current density. The approximation J0 Jth represents the range of interest of J0. Within the region of r there is a diffusion current flowing toward the surface given by J s ¼ qDn
dnðxÞ Ln J 0 expðx=Ln Þ. dx d
The average current density in the recombination area is Rr Z J s dx Ln J 0 r Ln J 20 ð1 expðJ th =J 0 ÞÞ ¼ expðx=Ln Þdx ¼ . J sav ¼ 0 r dJ th dr 0
ðA6Þ
ðA7Þ
Then the average current in the recombination area is Is ¼
wLn J 20 ð1 expðJ th =J 0 ÞÞ ; J th
ðA8Þ
where w is the width of the laser. The current in the central area (away from the facets) is approximately I b ¼ wLJ 0 ;
ðA9Þ
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W. Zheng et al. / Optics Communications 255 (2005) 102–113
where L is the length of the laser. Then the ratio of the temperature changes in the diffusion area and the central area is given by 2
cT ¼
2
DT s I 2s Rs Z T s L2n ðLJ 0 2Ln J th Þ ð1 expðJ th =J 0 ÞÞ ¼ ¼ . DT b I 2b Rb Z T b d 2 L2 J 2th
ðA10Þ
The temperature changes at the cleaved facets of the laser lead to the changes of the refractive index of the active layer material which affect the reflection coefficient of the facet. Assuming the active layer refractive index at the facet is n1 = n10(1 + aDT2), where n10 is the refractive index without temperature change, the reflection coefficient can be calculated as n2 pffiffiffi n1 n2 1 n10 ð1þaDT n10 n2 2an10 n2 DT s sÞ R¼ ¼ þ ; n2 n1 þ n2 1 þ n10 ð1þaDT n10 þ n2 ðn10 þ n2 Þ2 sÞ
where n2 is the refractive index of air. Thus R can be approximated as !2 n10 n2 2an10 n2 DT s þ R0 ð1 þ dÞ; R¼ n10 þ n2 ðn10 þ n2 Þ2
ðA11Þ
ðA12Þ
where R0 = ((n10 n2)/(n10 + n2))2 is the reflection coefficient without temperature changes and d¼
4an10 n2 DT s . ðn10 þ n2 Þðn10 n2 Þ
Since the photon lifetime can be expressed as 1 c 1 1 ln þ a ; ¼ sp ng þ aDT b L R then by using small signal expansions of the ln term, we obtain 1 c 1 1 lnð1 þ dÞ ln ¼ 1 þa sp ng ð1 þ ða=ng ÞDT b Þ L R0 lnð1=R0 Þ c 1 1 1 ln þ a . 4ng n10 n2 cT ng 1 þ a DT b L R0 ng
ðA13Þ
ðA14Þ
ðA15Þ
ðn10 þn2 Þðn10 n2 Þ lnð1=R0 Þ
From the relation between power and photon density for the laser, we obtain [8] DT 1 ffi eng F =a
ðA16Þ
and substituting into (A14) we obtain 1 1 1 . ¼ sp sp0 1 þ eF
ðA17Þ
From (A15) and (A17), we identify the gain compression coefficient as e¼
aRJ WmQW Lz hvvg0 ð1 gwp Þ lnð1=R0 Þ 4ng n10 n2 cT . ng gwp C ðn10 þ n2 Þðn10 n2 Þ lnð1=R0 Þ
ðA18Þ
Previously e was derived without recombination effects to produce the first term. The second term is required to include facet recombination. It is noted that cT is dependent upon the input current and therefore the value of nonlinear gain coefficient e obtained will also be current dependent. However, the only values of interest (listed in Table 1) are those for the highest currents (powers).
W. Zheng et al. / Optics Communications 255 (2005) 102–113
113
Appendix B. Relation between dn(x), ds1 st (x) and dE FT (x) When performing small signal analysis, we need to express dnw and dsst in terms of dEFT. The results were derived previously [8] and are summarized here. dnw is given by dnw ¼
ab dEFT dEFT ¼ nw0 ; a þ b kT kT
ðB1Þ
where a¼
Dn kT EFn0 =kT 1 þ 2eEFn0 =kT þ eDEn =kT ; e Lz ð1 þ eEFn0 =kT ÞðeEFn0 =kT þ eDEn =kT Þ
ðB2aÞ
b¼
Dp kT Epn0 =kT 1 þ 2eEFp0 =kT þ eDEp =kT e ; Lz ð1 þ eEFp0 =kT ÞðeEFp0 =kT þ eDEp =kT Þ
ðB2bÞ
dsst is given by 1 dðs1 st Þ ¼ sst0
dEFT ; kT
s1 st0 ¼
ba þ ab ; aþb
ðB3Þ
where a and b are given by a¼u
eðEFn0 þhmÞ=kT eEFn0 =kT ; ð1 þ eEFn0 =kT ÞðeEFn0 =kT þ ehm=kT Þ
ðB4aÞ
b¼u
eðEFp0 þhmÞ=kT eEFp0 =kT . ð1 þ eEFp0 =kT ÞðeEFp0 =kT þ ehm=kT Þ
ðB4bÞ
It is noted that a, b, a and b are derived from the steady-state solution in terms of the quasi-Fermi levels for electrons and holes.
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