IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL.52, NO.7, June 2016
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Abstract—In this paper we show that the dynamic per-
formance of multi-mode vertical-cavity surface-emitting lasers (VCSELs) can be modeled by single-mode rate equations developed for edge-emitters as long as the lasing modes share a common carrier reservoir. However, this assumption does not hold for ultra-high performing VCSEL devices. Due to the high photon densities inside these optimized VCSELs, the common carrier reservoir splits up as a result of the spatial hole burning (SHB) effect. This is caused by the high intensity of the multiple transverse modes. In this case, a small-signal modulation response with a different shape is expected. We derive an easy-to-apply fitting function which allows the extraction of consistently expanded figures of merit. This novel function works for all VCSELs, particularly including devices with carrier reservoir splitting. Further, we use this new model to perform a detailed analysis of our latest VCSEL generation with a modulation bandwidth of up to 32.7 GHz.
Index Terms—Direct Modulation, High-Speed, Multi-Mode, Small-Signal-Analysis, Small-Signal-Modulation Response, Transfer Function, VCSEL.
I. INTRODUCTION irectly modulated vertical-cavity surface-emitting lasers (VCSELs) are playing an important role in today’s
standardized serial data rates and in the emerging technology of optical interconnect abolishing copper lines. Consequently, the demand for higher bandwidth optical interconnects has become urgent. Interconnects based on silicon photonics either lack the cost-effective multiple-wavelength laser source [1] or the matched wave-guide if using orbital angular momentums for multiplexing [2]-[3]. Directly modulated VCSELs, however, match existing photo-detector and waveguide technologies, can be scaled up meeting future demands [4], and can therefore be readily applied as directly modulated light sources in optical interconnects. In order to reach a data-rate of 100 Gb/s, the VCSEL bandwidth has to increase by 30 to 40 % towards the 40 GHz level [5]-[6].
Intensive research to meet these requirements yielded high-speed VCSELs with direct modulation bandwidths up to 20 GHz [7]-[8] covering several wavebands. Recently, further
Manuscript received March 2, 2016, revised April 27, 2016, accepted May 25, 2016, published June 2016.
W. Hamad, S. Wanckel and W. Hofman are with the Institut für Festkörperphysik and Zentrum für Nanophotonik, Technische Universität Berlin, EW 5-2, 10623 Berlin, Germany (e-mail: [email protected]).
progress has been made boosting the modulation performance via 24 GHz [9]-[5] towards 28 GHz [10].
High-speed short-distance optical interconnects rely on multi-mode light sources transmitting via multi-mode fibers to achieve better bit error rate (BER) performance. As the latest research and experiments have shown [11]-[12], multi-mode fiber interconnection systems utilizing multi-mode VCSELs provide better immunity to intensity and modal noise compared to the single-mode VCSELs. This enhanced immunity can be explained by the fact that multi-mode VCSELs exhibit low coherent characteristics, minimizing the coherence of the reflected signal and reducing both relative intensity noise and BER.
It has been experimentally observed [13]-[14] that the spectral emission characteristics of high-speed, oxide-confined VCSELs depend mainly on the aperture size, confining the modulation current, and the geometry of the injection electrodes. In general, these optoelectronic devices tend to operate at a single longitudinal mode when driven at low modulation currents near their lasing threshold. This is especially the case in small aperture VCSELs with diameters below 3 µm where the typical emission wavelength is of the same order as the confining oxide aperture. However, devices having apertures ranging from approximately 3 to 10 µm, the emission spectra shows an increase in the number of excited higher order transverse modes at higher injection currents. On one hand, this multi-mode behavior is mainly caused by the raise in the modal gain of these higher-order modes, and on the other hand, the impact of the self-focusing effects grows, leading also to a multi-mode operation. The increase in self-focusing effects can be directly attributed to the increase in spatial hole burning (SHB) of the carrier concentrations and thermal lensing [15].
Thus, in order to establish highly performing multi-mode VCSELs in multi-mode fiber links at ultra-high-speeds, sufficient knowledge of the intrinsic laser dynamics is of great importance. Moreover, a deep understanding of the small- signal modulation response of these multi-mode VCSELs becomes indispensable for a reliable system design, modelling and characterization. While large-signal transmission experi-ments show the feasibility of a proposed system, the final performance only depends on the VCSEL device directly. This is especially true, if this VCSEL device turns out to be the bottleneck. Better large-signal performance can always be achieved through the optimization of the system by matching drivers, emitters, waveguides and receivers. Consequently,
Small-Signal Analysis of Ultra-High-Speed Multi-Mode VCSELs
Wissam Hamad, Stefan Wanckel and Werner Hofmann, Member, IEEE
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IEEE JOURNAL OF QUANTUM ELECTRONICS, VOL.52, NO.7, June 2016 2
large-signal experiments cannot reliably produce figures of merit for the isolated high-speed VCSEL. Small-signal experiments, on the other hand, can be calibrated accurately to isolate the VCSEL performance. The difficulty lies here in the proper modelling of the measured response to acquire reliable figures of merit to judge the device performance.
Despite the intensive research conducted to understand the underlying physics behind the multi-mode behavior in oxide VCSELs and their impact on the intrinsic laser dynamics, many ambiguities still exist concerning the nature of the multi-peak phenomenon occurring in the small-signal modulation response of VCSELs [16]-[17]. These multiple local maxima which appear in the transfer function signal of the multi-mode VCSELs, deviate substantially from the standard single-mode model normally applied to characterize these multimode devices. Moreover, the adapted single-mode fit model, which is based on a single-mode rate equation analysis, can reproduce only one resonance peak.
In addition to the historic assumption that the modulation response of “highly” index guided VCSELs show typical single mode transfer function characteristic [17]-[18], anomalies in the modulation response data are experimentally observed and reported by several research groups [13], [19], [7], [14], and [20]-[21]. Several explanations are presented for these maxima and minima ranging from optical reflections [19] to self-induced pulsation [22] and mode partition noise [18]. Some of the more promising approaches [23]-[24] are based on the interaction of the different lasing modes through their carrier reservoirs. This is due to the onset of strong modal competition behavior. The modes that are generated in the optical cavity of the VCSEL compete for the carrier density in the active region.
Investigating the majority of published VCSEL modulation data, we found strong deviations between the experimental data and the utilized modelling, especially for certain driving condition of the VCSEL devices. The small-signal modulation response of an index guided multi-mode VCSEL can typically be divided into three different driving-current regions. In the low-current region, which is characterized by the onset of strong modal competition, we frequently observe an interference-like pattern in the frequency response. This effect is caused by the large difference in the power of the different modes. In the intermediate-current region, the modulation frequency response is mainly dominated by the relaxation oscillation frequency. Surprisingly, we see the shape of this transfer function resembling a single mode laser, even though the device is clearly operating in the multi-mode regime. The most interesting region for ultra-high-speed operation, however, is the so called high current region.
Besides the well understood mechanisms which control the strength and the form of relaxation oscillation frequency (e.g. carrier diffusion, nonlinear gain suppression and carrier transport effects), the contribution of co-dominant higher order modes is still under discussion. Despite of the longitudi-nal single-mode emission and the fact that small aperture VCSELs support mainly the fundamental mode, higher order transverse modes can be detected in the emission spectra at high injection currents. In this case the transverse modes have
low spatial overlap. This causes different carrier reservoirs to establish each serving its spatially separated mode.
Even though the experiments deviated from the single-mode case, only theorists [25] considered advanced modeling. Up to now, experimentalists are modelling their multi-mode VCSEL devices exclusively with single-mode rate equations, even inconsistencies occur frequently, and some measured data can only be fitted with large error. In this paper we investigate whether in some cases this common practice might still be justified. Furthermore, in order to resolve the discrepancies mentioned above, we derive an advanced fitting model and show under which conditions it is obligatory to be used. Finally, we analyze our latest generation of ultra-high speed VCSEL devices with this novel model.
Deriving the multi-mode transfer functions, the model based on the standard single-mode rate equations was expanded to consider the coupling of two (or more) different modes through their lateral spatial distribution and overlap in the carrier reservoirs. If the modal competition for the carrier reservoir is not included explicitly in the interaction matrix as coupling factors, it is not possible to obtain more than one resonance peak in the modulation response fit model. Thus, solving the resulting interaction matrix gives rise to a single-mode transfer function even if two or more modes are included. However, if the spatial distribution of the modal fields in the carrier reservoir is taken into consideration, two or more resonance peaks can be fitted. This approach led to the development of our multi-mode transfer function. As this model was consistently expanded from the commonly used single-mode case, we also could easily expand well-known figures of merit. Our latest generation of VCSELs serve as sample-devices for detailed investigation.
II. SMALL-SIGNAL TRANSFER-FUNCTION MODELS We will begin our discussion of the different models by
going through the most fundamental model, the single-mode case, where one mode interacts with its carrier reservoir in the active region. This case is important to get familiar with the different derivation steps and the matrix representations of the linearized rate equations. The later derivation of the different multi-mode cases is going to be based on a generalized matrix presented at the end of the single-mode discussion.
A. Single-Mode Model As already known, the lasing dynamics of a single mode
laser can be, to a certain extent, fully described by a model consisting of a reservoir containing the electron carriers and a photon reservoir [26]-[27]. This model is depicted in Fig. 1. Based on particle conservation, the rates of change in both, carrier- and photon densities can be expressed as
- -inj th stimdN J J R Sdt
(1)
( - ( ))stim g i m spdS S R Jdt
(2)
where N is the carrier density and S is the photon density in
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the active region and the optical cavity, respectively. Г is the confinement factor. The first equation describes the total carrier density change. With the injected carrier density Jinj, the carrier recombination due to spontaneous emission or losses Jth, and the stimulated net-emission coefficient Rstim representing the stimulated emission rate.
The second equation states the net increase in the photon density being equal to the effective spontaneous generation rate Jsp multiplied by the confinement factor Г plus the photons generated by the stimulated emission which are not coupled out or being absorbed. Further, vg is the group velocity, αi and αm represent the internal- and mirror losses, respectively.
In order to study the dynamic behavior, or the so called modulation response of the driven laser, the derived rate equations have to be analyzed with the time derivatives included. As there is no analytical solution of the full rate equations, small-signal analysis have to be carried out. Thus linearizing and rewriting the system of equations in (1-2) yields
inj NN NSj dN dJ dN dS (3)
SN SSj dS dN dS (4)
with the following simplified rate coefficients:
=NN th g gJ v aS v aSN
(5)
=NS g g p g thv g v a S v g (6)
=SN sp g gJ v aS v aSN
(7)
= ( ) 0SS g g i m g pv g v v a S (8)
where g represents the optical material gain under all operation conditions and gth at threshold. Further, a and ap are the differential gain and the negative gain derivative after photon density, respectively. The assumptions made in (5-8) are approximations which can be made for a VCSELs operating above threshold and neglecting gain compression.
The above system of linearized rate equations in (3-4) can be represented in the following matrix form
=0
=
NN NS inj
SN SS
j dN dJj dS
M x i
(9)
Solving this matrix yields
1
detinj SS
SN
dJ j dNi
dS
x MM
(10)
where the det M can be already expressed in terms of figures of merit as:
2 2det R j M (11)
Introducing the relaxation oscillation frequency ωR and
damping factor γ in terms of the rate coefficients gives 2 = and =R NN SS NS SN NN SS (12) Taking the approximations made for a VCSELs operating
above threshold into consideration and neglecting gain compression (12) can be further simplified to
2 and R NS SN NN (13) Rewriting (10) in terms of the small-signal response dS to
the perturbation in the carrier density dJinj, and later replacing the carrier density by the laser power P leads to the formation of the theoretical transfer function H(ω). This transfer function describes the small-signal modulation response of the laser to a driving sinusoidal current I and is defined as
2
2 2
( ) = = SNd g th
Rd
R
dP hH v gdI e detM
he j
(14)
where ηd is the differential quantum efficiency.
Fig. 1. Single Mode Reservoir Model. This model was used for deriving therate equations in this section. A single drive current Jinj pumps a single carrierreservoir. Electron-hole pairs from this reservoir N supply a single photonreservoir S. The arrows symbolize the interaction in between and the supplyof the reservoirs. This diagram is also valid for particle densities and theirdifferential changes.
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As we are primarily interested in the relative modulation response, the constant front in (14) is scaled to one. Thus, any constant which can be factorized is set to one. This step is carried out without losing the generality of the modulation fit function and can be later compensated by inserting a variable offset value. In order to initiate the modulation fit function by zero attenuation response, this fit function is further multiplied by the inverse of H(0) which represents the modulation response at zero. In this paper we omit this last technical fit step for a better visual representation of the derived equations. Thus the modulation response is given in proportion form.
The relative modulation response function is used to characterize the performance of VCSEL devices. Extracting the different parameters from the fitted scattering parameter S21 data, yields the well-known figures of merit. Thus, until now the measured data are fitted to a relative single-mode function proportional
2 21( )
R
Hj
(15)
This intrinsic modulation response resembles a second-order filter function with two complex conjugated poles. However, for modeling the total transfer function of the VCSELs, this intrinsic response is multiplied by a third pole which models external parasitics. This gives the well-known 3-Pole transfer function in (16). An elaborate discussion on the modelling of the parasitics as a single-pole low-pass filter is given in chapter III of this work.
2 2
( ) = ( ) ( )
1 1 1 ( / )
Intrinsic Extrinsic
R P
H H H
j j
(16)
B. Multi-Mode Model The matrix representation (9) of the single mode rate
equations can be expanded to include various effects. These expanded models can be used to analyze the laser dynamics and derive the modulation transfer function in almost all operation and configuration cases.
N N N N N S N S1 1 1 n 1 1 1 m 1 inj1
N N N N N S N Sn 1 n n n 1 n m n in
1S N S N S S S S1 1 1 n 1 1 1 m
mS N S N S S S Sm 1 m n m 1 m m
j ... ... dN dJ
... j ... dN dJ=
dS... j ...
dS... ... j
j(n k)
0
0
= M x i (17)
Thus, the developed multi-mode interaction Matrix
presented above in (17), can model the interaction and coupling of the different lasing modes among each other, and with their underlying carrier reservoirs. More importantly, this model also supports the formation of multiple carrier reservoirs, one for each lasing mode distinctly. Furthermore, the interaction between these different carrier reservoirs among one another and themselves is supported.
The different matrix entries in (17) represent the rate coefficients of the linearized multi-mode rate equations analogous to the case of the single-mode matrix representation. However, expanding the number of carrier- and photon reservoirs to n and m reservoirs, respectively, induces the introduction of a second subscript character to the already existing ones. This step is essential in order to be able to distinguish the impact resulting from the different reservoirs on themselves and on one another. For example, μN1S2 defines the impact on the first carrier reservoir N1 caused by the changes in the second photon reservoir S2. In obvious cases, this second subscript is omitted as in the single-mode, single-carrier reservoir case discussed previously.
It is important to mention at this stage that the rate coefficients (entries) of the multi-mode matrix for some of the upcoming multi-mode rate equation analysis are still consistent with the single mode coefficients. However, particle conservation for carrier- and photon reservoir densities should be taken into consideration.
Finally, this general interaction matrix can be sized and adapted to model the different dynamics in small-signal analysis resulting from different considerations and device configurations. In the following, coupling and interaction effects are analyzed by tailoring the interaction matrix to satisfy these boundary conditions and considerations with the final aim to derive an analytical transfer function model. This analytical function is utilized to fit the experimental S21 data allowing us to extract reliable information about the dynamic properties of the real-world device. Furthermore, this analytical model can be used to categorize the different operation regions as discussed in the introduction.
Fig. 2. Shared Reservoir Multi-Mode Model. A single drive current Jinj
pumps a single carrier reservoir. Electron-hole pairs from this reservoir Nsupply several photon reservoirs Sm. Our calculation shows that this case fallsback to a single-mode-like transfer function.
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As already mentioned, higher order transverse modes evolve in real-world VCSEL devices under certain operation currents or geometrical layouts. Thus, in order to include the impact of the different interacting lasing modes on the transfer function, we start our investigation by the simplest multi-mode case. Here, we assume one carrier density reservoir N serving as a carrier source for the two different lasing mode densities S1 and S2 as depicted in Fig. 2. Consequently, the interaction matrix becomes a 3 x 3 matrix
NN NS NS1 2 inj
S N S S S S 11 1 1 1 2
2S N S S S S2 2 1 2 2
j dN dJj dS = 0
dS 0j
(18)
To avoid modelling of any mode beating effects which are
not supported by our VCSEL geometrical configuration, the two rate coefficients μS1S2 and μS2S1 are set to zero. Thus, any interaction between these two lasing modes through their photon reservoirs is inhibited and just the interaction of both modes with their common carrier reservoir is permitted.
Before proceeding with the discussion, it is important to point out that the impact of the interaction of the two modes among each other through their photon reservoirs, results in many interesting phenomena in the modulation response. Many mechanisms are employed to realize this mode coupling ranging from injection-locking techniques [28] to schemes that are based on coupled cavities and modulator integration [29]. These so called optical feed-back mechanisms are a result of the photon–photon resonance between two transverse oscillating modes under strong slow-light feedback.
In this paper, the focus is exclusively on the modulation response effects resulting from the interaction between the different carrier reservoirs, each belonging to a different resonating mode, among one another and with their own- and other photon density reservoirs.
Going back to the interaction matrix (18), and solve for -1x = M i , gives the following system of equations:
1
111 1
1 211
2 31
= 0 =0
inj
inj
dN dJ mdS dJ mdS m
M (19)
1
11m to 1
31m represent the first column entries of the inverse
matrix M-1 with
121 1 2 2
1 ( )det S N S Sm j
M (20)
and
131 2 1 1
1 ( )det S N S Sm j
M (21)
As in the single mode case μS1S1 and μS2S2 can be neglected
and are therefore set to zero. Rewriting (19) in terms of the small-signal response dP to the perturbation in the carrier density dJinj, and taking into consideration that the total output power is now the sum over all mode powers (P = PS1+PS2), results in the following intrinsic modulation response:
1 2
1 121 31
1 22
1 1 2 2
( ) = ( )
( )
S N S N
NS S N NS S N NN
H c dS dS
c m m
cj
(22)
where cη is a constant. Again here we are just interested in the relative small-signal frequency response. Rewriting and comparing this derived intrinsic transfer function to that of the single mode case, reveals in a very interesting finding. This model, consisting of one carrier reservoir and two lasing modes, yields also a second-order filter function with two complex conjugated poles as a modulation response. Thus, the intrinsic modulation response can be expressed as
2
1 1 2 22
'
2 2'
1( )
1
NS S N NS S N NN
R
R
Hj
j
(23)
with 2
' 1 1 2 2R NS S N NS S N and NN (24)
The analysis above for one carrier- and two photon
reservoirs can be generalized as long as all lasing modes share the same carrier reservoir. Thus, expanding the interaction matrix to model m different photon density reservoirs, and calculating the intrinsic modulation response results in
11( ) m
m ModesH m (25)
and can be also expressed as
2
2''
2 2''
1( )
1
m NS S N NNm m
R
R
Hj
j
(26)
which also represents a two complex conjugated pole transfer function just as a single-mode laser would have. Thus, single-mode rat equations can be used to model these cases.
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Until now the standard applied technique to characterize the performance of both, single- and multi-mode oxide confined VCSELs, is fitting the measured S21 data with a two complex conjugated pole transfer function. This approach originated from experimental observations which, in many cases, show a very similar profile to that of the single mode.
Equation (26) above, confirms the methodology to extract important device parameters of multi-mode VCSELs from the measured response by single-mode fitting. This behavior in modulation response becomes obvious when taking a closer look at the crucial parameters, damping factors and relaxation resonance frequencies, in the derived transfer functions.
The damping factor γ, as defined in (5) for the single mode case, remains unchanged throughout the derivation of the modulation responses for the two- and multi-mode cases. However, the photon density S in the two- or more mode cases, is redefined to be the sum over all photon densities. Despite this fact, the value of the rate coefficient μNN stays almost the same in all cases except for some negligible factors added for each further lasing mode. Until now, the developed models and the derived transfer functions can simulate a modulation response with only one maximum. These models represent a classical single-resonance frequency response. However, under certain operation conditions, a S21 measured curve which is rich in maxima and minima is observed. In this case the deviation from the classical single mode model becomes significant. The assumptions matching this case are presented in Fig. 3. Here, the extraction of figures of merits using the classical fit yields unreliable performance parameters. In order to develop a more reliable transfer function which can model the S21 data of high-speed VCSELs and give a deeper understanding of the laser mode dynamics of multi-mode oxide VCSELs, the interaction matrix is expanded to include more carrier reservoirs. In the following we are going to derive a transfer function from an interaction matrix which includes two photon reservoir densities S1 and S2 along with two carrier reservoir densities N1 and N2. The interaction of these reservoirs is ensured
through the sixteen different matrix entries as can be seen in (27). Fig. 3 shows a schematic representation of these interactions which are based on the governing rate equations.
By implementing the approximations and boundary conditions discussed in the single mode cases, the matrix entries in general matrix (17) are then reduced to fourteen having twelve different interaction coefficients.
In order to further simplify the interaction matrix, only the cross interactions between carrier and photon density reservoirs which are shown in Fig. 3 are modelled in (27). Thus, the interaction of the carrier reservoirs with the photon reservoirs, μS1N2 and μS2N1, are set to zero in agreement with the established device physics.
Again, photon-photon interaction is not supported by our device structure, and therefore μS1S2 and μS2S1 are omitted as well. These assumptions lead to the following system matrix:
1 1 1 2 1 1 1 2 1 1
2 1 2 2 2 1 2 2 2 2
11 1
22 2
=0 0 0
00 0
N N N N N S N Sinj
N N N N N S N S inj
S N
S N
j dN dJj dN dJ
dSjdSj
(27)
This matrix representation of the linearized rat equations is
used for the derivation of the multi-mode transfer function. This matrix has ten different interaction coefficients of which six are already introduced in the single mode model. Two new entries represent mainly the interaction or communication between the two different carrier reservoir densities N1 and N2, and can expressed by
1 2 2 2 12 2 2 2 12/ /N N g N Nv aS V V V V (28)
2 1 1 1 12 1 1 1 12/ /N N g N Nv aS V V V V (29)
and the following interaction coefficients are needed to model cross reabsorption:
1 2 (mod 2) 2 12 2 2 2 12/ /N S g th e N Sv g V V V V (30)
2 1 (mod 1) 1 12 1 1 1 12/ /N S g th e N Sv g V V V V (31)
In the derivation of (28) and (29), diffusion terms are neglected. On one hand, diffusion effects are already considered as first-order effects in the definition of the steady-state quantities, S and N. They are also included in the photon-density-ratio q which is going to be represented later. On the other hand, this approximation does not affect the shape of the modulation response for high-speed VCSELs [17]-[25].
We define V1 / V12 = s1 and V2 / V12 = s2 representing the spatial dependency of the two interacting carrier reservoirs with respect to the two different mode volumes [30]. Furthermore, Vi has a volume dimension and represents the effective volume of the ith mode. The degree of spatial overlap between the two lasing modes, mode 1 and mode 2, is given by s = s1 s2 = V1V2 / (V12)2. If this factor s is small, which means the overlap between the two modes is weak, the two modes can barely interact through their carrier reservoirs. This leads
Fig. 3. Multiple Reservoir Multi-Mode Model. In this case the modes confinetheir independent carrier reservoirs. This behavior corresponds to spatial holeburning of multiple transverse mode ensembles of a VCSEL. A single drivecurrent Jinj also splits up proportional to a power ratio defined by the twotransverse modes. This model can explain the dynamic behavior of real-worldhigh-speed VCSEL devices well.
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to the so called decoupled model where the total transfer function of this system is given by the sum of the individual single mode function of the different modes. The transfer function belonging to this case is represented at the end of this section. For a VCSEL with an oxide aperture diameter ranging from 3 to 10 µm, the typical two lasing super-modes have the shape of LP01 and LP11 modes. In a circular oxide aperture, the LP01 has its main optical intensity in the center of this aperture, whereas the LP11 mode power is mainly in the outer region. Typical values of s1 and s2 for this type of modes are 0.67 and 0.94 for the two super-modes LP01 and LP11, respectively [30].
The last two parameters which are still to be defined in the above matrix representation in (27) are the driving current densities dJinj1 and dJinj2. As shown in Fig. 3, the VCSEL is uniformly injected with the carrier density Jinj. Before reaching the carrier reservoirs, this injected carrier density Jinj is split among the two reservoir in proportion to the photon densities of the respective modes. We define p1 and p2 as two pumping currents which are proportional to the respective current densities and the supplied photon reservoir densities. Thus, p1 ~ dJinj1 ~ S1 and p2 ~ dJinj2 ~ S2. As we are just interested in the mode power ratio, we introduce the parameter q. This q represents the photon-density-ratio or the mode-power-ratio between the two modes and is defined as q = (Г2 vg2 a2 S2)/ (Г1 vg1 a1 S1) ≈ S2 / S1. The lower case numbers 1 and 2 distinguish between the two set of parameters belonging to the two different lasing modes. These parameters are discussed in the single mode case.
Finally, solving the matrix in (27) and summing over all mode photon densities, the modulation response can be expressed as
with cκ being a constant. Again, the lower case numbers 1 and 2 distinguish the relaxation oscillation frequencies ωRi and damping factor γi of each lasing mode and are defined as
2 and Ri NiSi SiNi i NiNi . If the cross interactions between the carrier- and photon
density reservoirs are modeled as discussed above, then (32) can be expanded to
Even though (33) appears to be bulky, it consists of four well know figures of merit, two damping factors along with two relaxation oscillation frequencies. Moreover, if the geometrical configurations of the device are known, s1 and s2 can be calculated or extracted and set as constants.
For the decoupled case, where the degree of spatial overlap is negligible, s1 and s2 can be set to zero. Thus the total modulation response in (33) reduces to a power-weighted sum of the individual single mode responses
2 2 2 22 2 1 1
1 /( )R R
q qHj j
(34)
Finally, if one of the mode powers dominates over the other
mode powers, the modulation response collapses again to the single-mode model. For example, if the power of mode one becomes very dominant compared to the second mode, S1 >> S2, the modulation response reduces consistently to the single-mode case
2 21 1
1( )R
Hj
(35)
III. VCSEL DEVICE STRUCTURE AND PERFORMANCE The device architecture is an optimized version of our very
successful high-speed, temperature-stable 980 nm VCSEL design [5]. It shares the very short half-lambda cavity and a binary bottom-mirror with 32 pairs. Doping levels were further optimized minimizing internal loss. Like the previous design two oxide apertures and highly conducting current-spreading materials are used for low RC-parasitic. For higher differential gain we used as active region a 5x InGaAs multiple quantum
well structure with strain compensated GaAsP barriers. Instead of the 22 -pair Al12Ga88As/Al90Ga10As top-mirror we utilize an improved 18 -pair GaAs/Al90Ga10As mirror with lower photon lifetime, better confinement and better heat extraction properties. The structure was grown by IQE Europe ensuring the commercialization potential. Like the previous structure the VCSEL was processed in our two-mesa design with Cyclotene™ passivation utilizing inductively coupled plasma (ICP) dry etching with a coplanar ground-signal-ground (GSG) pad layout.
The VCSEL device performance was characterized by a calibrated 40-GHz vector
network analyzer (HP8722C). To suppress large-signal artefacts, the modulation amplitude was set as low as -15 dBm. Calibration was done until the wafer-plane of a Cascade GSG150 probe with the matched calibration substrate. Optical coupling was done with a cleaved multimode-fiber of 1 m length. To minimize distortion of the signal by optical feedback, the coupling was reduced to a reasonable level of 0 dBm in the fiber. Remaining small
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reflections caused by the straight FC/PC connectors could be extracted from the whole dataset allowing the systematic error to be de-embedded with no influence on the measured bandwidth. The high-speed photo-detector was calibrated by a femtosecond laser-pulse allowing its transmission behavior to be subtracted from the measurement yielding correct data even beyond 30 GHz.
L-I-V characteristics of the device are presented in Fig. 4. The 980-nm VCSEL features an aperture for current confinement of about 7 µm in diameter with a threshold current below 0.8 mA, and a differential slope of ~ 0.64 W/A at 25°C (equaling a differential quantum efficiency of 50 %). The maximum optical output power is above 7 mW, and the differential series resistance is 75 Ω.
An elaborate small-signal analysis was carried out. The original data of the modulation responses at the indicated bias level are calibrated and plotted in Fig. 5. These measured S21 data are later fitted to various rate-equation based models and several figure of merit could be extracted as depicted in Fig. 6-8. The maximum 3 dB bandwidth of the device including chip-parasitics was found to exceed 32.7 GHz at 14 mA. The VCSEL reached thermal-rollover at 17 mA. For better visibility, the corresponding S21 curve at this current is not depicted. It is worth to mention that the small response offset appearing in Fig. 5 at the first two S21 data sets is not artificially introduced to the plot. It is probably caused by low power levels at low driving currents.
In order to demonstrate the potential of the multi-mode model presented in this work (33), the S21 data in Fig. 5 is analyzed. For comparison, we used the established single-mode model (16) as well, and showing its pitfalls. In analo-gous to (16), (33) is further multiplied by a parasitic pole in order to model the device parasitics. It is important to men-tion, that the frequency response corresponding to the electri-cal parasitic, can be well approximated by a single-pole low-pass filter. This simple first order approximation is consistent with an elaborate analysis by equivalent circuit modelling of the S11 signal measured simultaneously with our S21 data as presented in Fig. 6. The method used here requires a high level of understanding of the device layout to reduce the degrees of freedom created by introducing several equivalent circuit elements. Even though this method delivers slightly higher
precision in theory, the remaining uncertainties might well cancel out this advantage by larger error-bars. Consequently we suggest using the single parasitic pole method if the device under investigation features a low-parasitic chip layout similar to the devices presented here. The parasitic small-signal response of such a device is dominated by an electrical equivalent-circuit including a series resistance Rm of both Bragg mirrors as well as an current crowding resistance Ra parallel to a capacitance Ca caused by the aperture. This simple electrical equivalent-circuit sufficiently models the parasitic behavior well as long as the capacitance of the bonding pads Cp is kept small pushing their effects to be in the range of 50 GHz. Therefore, the parasitic response of this kind of devices can be well modeled by a simple single-pole low-pass filter [27] in the given frequency range.
Fig. 4. L-I-V Characteristics of the oxide-confined 980-nm VCSEL at roomtemperature. The characterized VCSEL has an aperture diameter of ~7 µm.
Fig. 5. Calibrated small-signal modulation response of a 980-nm multi-modeand oxide-confined VCSEL with an aperture diameter of ~7 µm. The curvesdepict the relative response data S21 for various driving currents at roomtemperature. The modulation current is increased gradually up to 14 mAwhere thermal rollover was reached. At this current the measured 3-dBbandwidth was found to approach 33 GHz.
Fig. 6. Parasitic response of the same device, measured at 14 mA bias point.The solid line represents the single-pole approximation as in (16) with aparasitic cut-off frequency of fP
MM. The scatters represent the extrinsicresponse extracted from device impedance measurement and fitting with thegiven equivalent circuit. The measurement data, the fit and the parasiticequivalent circuit with extracted device values are inset.
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The figures of merit extracted using (33) can then be directly compared to that of (16) with the detailed definitions of the single-mode model derived from (12) as follows:
2
2
0
4π
2
1( )
( )
o
R
NN SS g g p th sp
pg g i m
g i m
K
R 0
Nv aS v a S J J
S
a av aS v
v
f K
(36)
defining the K-factor K, a fitting parameter and damping offset γ0 in laser parameters as defined in (5) – (8). For the multi-mode case, we can define multiple K-factors Ki for each independent mode ensemble using the same definition. The relaxation oscillation frequency depends on the biasing conditions. Therefore, we can define the D-factor as follows:
R thf D I I where 12π
g i
res
v aD
eV
(37)
with Vres as the volume of the optical resonator [27]. Again, for the multi-mode case, we can define multiple D-factors Di for each independent mode ensemble using the same definition. A similar interrelationship is also observed for the laser bandwidth f3dB. Here, we speak from a modulation current efficiency factor MCEF with
3dB thf MCEF I I (38)
A comparison between the extracted figures of merit from both models is depicted in Fig. 7-9 and the values of these parameters are stated in Table I.
The first interesting contrasts between both models is graphically illustrated in Fig. 7 through the plotting and fitting of three different modulation response data sets at various driving currents; 2.5 mA, 5.0 mA, and 14 mA. These S21 curves are selected data sets taken from Fig. 5. For clarity, the selected curves are shifted along the y-axes. The dashed lines represent the fits which are based on the single mode model (SM), whereas, the solid lines show fits based on the advanced
Fig. 7. Three different small-signal modulation response data at variousdriving currents 2.5 mA, 5.0 mA and 14 mA are presented. These S21 curvesare selected from the calibrated data in Fig. 5. For clarity, the selected curvesare shifted along the y-axes. The dashed lines are fits to the data and they arebased on the single mode model (SM). The solid line fits are based on theadvanced multi-mode multi-reservoir model (MM). The position of theextracted relaxation oscillation frequencies fR, fR1 and fR2 along with theparasitic cutoff frequencies fP from both models are also depicted. Twostraight lines show the propagation direction of the parasitic cutofffrequencies as the current increases for the two models.
(a) (b) Fig. 8. Extracted figures of merit from the small-signal modulation responsedata using the single-mode model. Plotting the resonance frequency fR andthe bandwidth f-3dB versus the square-root of the driving current abovethreshold allow us to estimate the MCEF and D-factor (a). In (b), the K-factor is obtained by plotting the gradient of the damping-rate γ versus thesquared resonance frequency. Above a certain driving current, the parameterextraction reliability of K and D of the single-mode model is questioned.This region is shaded.
(a) (b) Fig. 9. Extracted figures of merit from the small-signal modulation responsedata of Fig. 5 using the advanced multi-mode model (33). Plotting theresonance frequencies fR1 and fR2 versus the square-root of the driving currentabove the threshold of each mode, allow us to estimate the D-factors D1 andD2 (a). In (b), the K-factors K1 and K2 are obtained by plotting the gradientsof the damping-rates γ1 and γ2 versus the squared resonance frequencies ofthe corresponding modes.
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multi-mode multi-carrier reservoir model (MM). The position of the extracted relaxation oscillation frequencies fR, fR1, and fR2 along with the parasitic cut-off frequencies fP from both models are also depicted. The tendency in the propagation direction of the parasitic cutoff frequencies, which varies as the current increases, is also indicated by straight line.
Based on this new evaluation, we can make several findings. Firstly, the novel fits match all curves much better. In particular, the fitting of the 2.5 mA-data, where no one would doubt the validity of the single-mode fit at the first place, are well fitted. For the two other curves, it is much more obvious that the old model, represented by the dashed lines, can neither explain the form of the measured data, nor the physics behind. The figure from 0 to 10 GHz of the VCSEL biased at 5 mA cannot be replicated at all by the single-mode fit. The fit attempts to follow the front notch and the resonance peak. Even though unrealistic device parasitics are assumed, the first 10 GHz are still way off. Other fitting strategies led to even worse or even less consistent results. At 14 mA, the single-mode fit only coincides with the measured data from 20 GHz onwards. To achieve this level of matching in both single-mode fits, the parasitic pole had to be allowed to range around 14 GHz for the small drive current and dropping from there towards 6 GHz, whilst the differential resistance of the devices drops with current as can be observed in the declining slope in Fig. 4. This is contra intuitive and unphysical, however, common practice. This might be the reason why the validity of small-signal experiments is questioned from time to time in the community. The fitting according to (33) does not have this problem. The parasitic cutoff frequency starts from 22 GHz raising to 27 GHz with lower differential resistance. These values of electrical parasitics coincide with those extracted from the reflected S11 measurements. Li Hui et al. also observed similar tendency in the parasitic cutoff frequency when extracting data from the reflected S11 originating from a similar VCSEL architecture [31].
Even though (33), with its 5 fitting parameters, could give the impression of being able to fit arbitrary transfer-functions, this is not the case. Basically, this equation can be seen as the sum of two well-defined single-mode transfer functions with a coupling term determined by the device geometry. However, effective fitting strategies are helpful for the extraction of reliable parameters.
All the measured data presented in Fig. 5 was fit accordingly, and the results are plotted in Fig. 8 and Fig. 9 for extraction of figures of merit like K- and D-factors. Whilst the modulation current efficiency MCEF is derived from the 3-dB bandwidth and is therefore independent of curve fitting, the extraction of K- and D-factors relies on correct fitting. The quality of the fits can no longer be verified in these values. Therefore, we shaded the areas of fitting with high error and and unphysical parasitics in Fig. 8, cancelling out most of the plot. While the D-factor seems to be still usable as it is derived from data with better fits, the K-factor can no longer be trusted by using the simple model for evaluation.
The multi-mode fit using (33), does not only give better fits and physical parasitics, but also more information about the VCSEL device. We can extract two D-factors, one for each of the two dominant modes forming their individual
carrier reservoirs. Interestingly, these two super-modes have different dynamics. This can explain experimental findings in large-signal experiments, where the achievable bit-rates and error-flooring strongly depends on coupling strategies of multi-mode devices. In Fig. 9(b) we show two K-factors being extracted corresponding to different damping characteristics for the different modes competing with each other for material gain. From the fits we found the mode-power-ratio q to be around 0.25 … 2 raising with the bias current. We also see no damping offset for the second mode ensemble, which is expected, as the VCSEL is already lasing while these higher-order modes evolve. The differences in K- and D-factor could be interpreted by assuming the differential gain to be six times higher for the first mode ensemble which is also suffering from twice as high gain compression. More detailed studies are planned for the future.
The device architecture optimization towards short photon life time results in a moderate, but still sufficiently high modulation current efficiency factor and D-factor of 12.5 GHz/mA0.5 and 9.1 GHz/mA0.5, respectively. The higher MCEF can be explained by the very small K-factor as low as 0.12 ns, making the VCSEL no longer damping limited. The second mode ensemble seems to suffer from slightly lower dynamics. By filtering out the slower mode, e.g. by the implementation of high-contrast-gratings (HCG), large-signal modulation characteristics are expected to be improved. Even though the device design is clearly optimized for heat extraction, the bandwidth is still limited by thermal effects as can be seen in Fig. 9(a) with the performance still saturating for higher currents.
All the three different current regions, as discussed in the introduction, can be accurately modelled by our multi-mode theory (33).
TABLE I EXTRACTED FIGURE OF MERIT FROM BOTH MODELS
Fit Model K1 [ns]
K2 [ns]
D1 [GHz/√mA] D2 [GHz/√mA]
SM-Model 0,07 - 8,8 -
MM-Model 0,12 0,36 9,1 3,5
The extracted VCSELs performance parameters the K-factors and D-factors from both models, are depicted for a direct comparison. The first row shows the data which are extracted using the single-mode model. The proposed multi-mode model was applied to extract the data which are presented in the second row.
IV. CONCLUSION In this paper, we have consistently expanded the established
single-mode laser rate equations towards a multi-mode high-speed VCSEL rate-equation model. We could show that the commonly used practice of modelling multi-mode VCSELs by single-mode rate-equations is justified as long as the carrier reservoirs do not split up. For the highest performing VCSELs, however, this assumption does not hold up, and our more comprehensive model has to be used. Even though this model includes many effects such as spatial hole burning or
HAMAD et al.: SMALL-SIGNAL ANALYSIS OF ULTRA-HIGH-SPEED VCSELs
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cross carrier reabsorption, only as little as five free fitting parameters are needed. This allows reliable small-signal analysis and the extraction of figures of merit by means of curve-fitting. Therefore, we believe this method should be the technique of choice in judging the dynamic performance of VCSEL devices.
Further, we used this model to investigate our latest generation of ultra-high-speed VCSEL devices with modulation bandwidths of up to 32.7 GHz and a modulation current efficiency of 12.5 GHz/mA0.5. The K-factor for the dominant set of modes was found to be 0.12 ns, whilst the K‐factor for the higher order transverse modes was 0.36 ns indicating a higher damping of these modes, however, without a damping offset as these modes evolve competing with other lasing modes rather than spontaneous emission below threshold. The D-factor of the dominant modes was as high as 9.1 GHz/mA0.5 while the higher-order transverse modes being pumped at high drive currents show lower efficiency of 3.5 GHz/mA0.5 only. Modelling this VCSEL device with the former theory would have required to set the parasitic response to unphysical low values. This would cause the intrinsic damping to be underestimated yielding unphysical values for the K-factor as well.
Analyzing this VCSEL with the newly proposed model not only gives much better fits, but also lets us derive very physical figures of merit allowing us to understand our device in detail. This gives us the knowledge for further optimization which is the basis of the next device generation.
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Wissam Hamad has studied B.Eng. and M.Sc. in Engineering Physics at the University of Oldenburg, Germany. After his graduation in 2012 with the specialization “Microrobotics and Micro Systems Engineering” he worked as a researcher in the division of Prof. S. Fatikow in the field of handling and characterization of nanomaterials and nanoobjects. In 2014 he joined Prof. Hofmann´s research group at TU
Berlin were his research focus is the high-speed nanostructured VCSELs. Stefan Wanckel is a graduate student in Physics at the Technical University of Berlin (TU Berlin), Germany. In 2015 he started his master-program in Engineering Physics at the same place. In 2015 he joined Prof. Hofmann’s group focusing on the modelling of VCSEL-dynamics.
Werner H. E. Hofmann (M’06) received his Dipl-Ing (M.S.) degree in electrical engineering and information technology in 2003 and the Dr.-Ing. (Ph.D.) degree in 2009, both from the Technical University of Munich, Germany. From 2003 to 2008, he was with the the group of Prof. Amann at the Walter Schottky Institute, where he was
engaged in research on long-wavelength vertical-cavity surface-emitting lasers (VCSELs). Subsequently, he joined Prof. Chang-Hasnain´s Group, University of California, Berkeley (UCB), where he worked on the incorporation of high-contrast gratings into VCSEL devices. In 2010, he joined the Technical University of Berlin (TU Berlin), Germany, as principal scientist in the group of Prof. Bimberg. Since 2013 Prof. Hofmann is leading his own group focusing on ultra-high-speed nanophotonic devices. Prof. Hofmann is also the Chief Technical Officer at the Centre of Nanophotonics at TU Berlin. Since 2014 he is also with the Xiamen University, P. R. China as guest professor and highly qualified foreign expert.
He received the E.ON Future Award in 2009 and the SPIE Green Photonics Award in 2012 for his research on high-speed, energy efficient VCSELs. Prof. Hofmann authored over 100 papers cited over 500 times and has presented over 10 invited talks and tutorials on international conferences. He is a member of the Association of the German Engineers (VDI), the German Association of University Professors (DHV) and the IEEE Photonics Society. His main research interests are high-speed surface emitters and their advance by nano-photonics.
