Mode partition noise in multi-transverse mode
vertical-cavity surface-emitting lasers
A. Valle and L. Pesquera
Instituto de Fsica de Cantabria (CSIC-Universidad de Cantabria)
Facultad de Ciencias, Avda. Los Castros s/n
E-39005 Santander Spain.
ABSTRACT
Relative intensity noise spectra of weakly index guided VCSELs in a multi{transverse mode regime are analyzed
by using a model that takes into account all the transverse modes supported by the waveguide. Several resonance
peaks are obtained in the noise spectra that correspond to the relaxation oscillation frequencies of the transverse
modes. In is shown that for low spatially overlapping transverse modes, low RIN operation can be maintained.
However, the excitation of transverse modes with a signicant mode overlap leads to a clear enhancement of the
RIN at low frequencies.
Keywords: Semiconductor Lasers, Vertical cavity lasers, Transverse modes, Relative Intensity Noise
1. INTRODUCTION
Vertical-cavity surface-emitting lasers (VCSELs) are highly attractive light sources for applications in interconnections and optical communications, because of their useful characteristics such as single longitudinal mode
operation, low threshold current, circular output beam, and wafer scale integration. An important consideration for such applications of VCSELs is the understanding of their noise characteristics. Several experimental
studies of the relative intensity noise (RIN) in VCSELs have been performed. In general the noise performance of VCSELs operating in the fundamental mode is comparable to that of edge emitting lasers. However,
VCSELs exhibit dynamic and static characteristics signicantly dierent from those of edge emitting lasers. It is
known that VCSELs can emit multiple high order transverse modes due to spatial hole burning (SHB).
The
existence of higher order transverse modes can cause degradation of the noise performance, particularly at kinks
in the ligth{current characteristics where localised increases in RIN have been experimentally observed. RIN
measurements have also shown that the spectrum exhibits several peaks. Similar behavior was demonstrated in
the small{signal modulation response. The peaks in the frequency response correspond to distinct resonance
frequencies of the transverse modes.
Theoretical modelling of VCSEL noise have used a model which includes the spatial dependence of both
the optical eld and the carrier density. The results show that the mode partition noise for two transverse modes
without azimuthal dependence is signicantly reduced when their spatial overlap is small, because of reduced
carrier competititon. Multitransverse mode regime have been recently analyzed by using a model that includes
both the azimuthal and radial dependences of electric eld and charge carriers.
In this work this model is
used to study the RIN spectra of VCSELs operating in the multitransverse mode regime. It is considered that
the the dominant mechanism for mode selection is the modal gain. The eects of dierent transverse modes
on the RIN spectra corresponding to each transverse mode and to the total intensity are investigated. The
meaning of the dierent peaks appearing in the noise spectrum is claried. The relationship of the low frequency
enhancement of RIN with the spatial overlap of transverse modes is also studied.
The paper is organised as follows. In section II the two{dimensional model is summarised. Multi{transverse
mode Light-current characteristics are then presented in section III. In section IV the RIN spectrum in the
multi{transverse mode regime is analysed. Finally, section V is devoted to summarise the obtained results.
1,2
3{9
10{14
3
9
13
15
16
15
17,18
Other author information: (Send correspondence to A.V.)
A.V: Email: vallea@besaya.unican.es; Telephone: 34-942-201465; Fax: 34-942-201459.
2. TWO-DIMENSIONAL MODEL
The model utilised in the present work incorporates both spatial dependence of carrier and optical eld proles.
The cylindrical weakly{index guided VCSEL structure considered in the following analysis is illustrated
schematically in Fig. 1. of Ref. 16. The refraction index at the core region, ncore, is slightly greater than the one
at the cladding region, ncladd, in such a way that the carrier induced changes in the refractive index are smaller
than the built{in refractive index dierence. Then, waveguiding properties of the laser are not signicantly
aected by temporal changes in the carrier density. It is appropriate to utilize a cylindrical coordinate system,
(r; ) to describe the spatial distributions of charge carriers and optical elds in the structure. The active region
of the device is taken to be of radius a, equal to radius of the waveguide. The active region consists of three
GaAs-Al : Ga : As quantum wells with a well width of 100 amstrongs and barrier thickness of 150 amstrongs.
The laser cavity is dened by two highly reecting mirrors separated by a distance, L, along the longitudinal
axis. We consider that the injected current is distributed with a prole given by j (r; ; t). The modes supported
by the assumed waveguide are conventionally denoted as the LPmn modes. Several modes (LP , LP , LP ,
LP , LP , and LP ) are supported by the waveguide structure. We take into account the azimuthal degree of
c ) and sinus (LP s )
freedom by considering for each LPmn mode the corresponding modes with cosinus (LPmn
mn
c and LP s modes are, respectively,
azimuthal dependence. Then, electric eld spatial distributions of LPmn
mn
c
s
mn (r; ) = mn (r) cos(m) and mn (r; ) = mn (r) sin(m). The radial variation of the electric eld is given
by
Km (wmn r=a) otherwise:
Jm (umn r=a) if r a;
(1)
mn (r) = K (w )
mn (r) = J (u )
16{20
0 2
0 8
01
12
21
02
11
31
m mn
? ) =
m mn
; wmn = a( ? (ncladd mn ) ) = ; and Jm and Km are Bessel functions of
the rst and second kind. In the present model a cavity resonance condition is imposed in the form L = q
where q is an integer (here, q = 8). The wavevector mn is obtained from eigenvalue equations. Normalized
intensity proles j mn (r)j are illustrated in Fig. 1.
Here umn = a((ncore mn )
2
2 1 2
2
2 1 2
21
2
LP01
LP02
LP11
Normalised Power
1.0
0.5
0.0
0
Figure 1.
1
2
Radial coordinate, r (µm)
3
4
Normalized transverse mode intensity proles.
The two{dimensional dependence of the intensity of the modes LP , LP c , LP s and LP is shown in Fig. 2.
01
11
11
21
Figure 2.
Near eld distributions of LP , LP c , LP s and LP modes.
01
11
21
11
Assuming that the VCSEL can operate in several transverse modes simultaneously the electric eld is expressed as
X
i
i
?j! t
E (r; ; t) = 21
(2)
mn (r; )Emn (t)e mn + cc;
m;n;i
i (t) (i = c; s) and !mn are, respectively, the eld amplitudes and frequencies of LP i modes. We
where Emn
mn
have assumed the same polarization direction for all transverse modes. This situation can be achieved by using
some external mechanism of polarization control.
Rate equations for the complex amplitudes of the previous modes are
22,23
s
R R
h
i
i
dNw a N (r; ; t)rdrd i
1
dEmn
1
?
j
i
i
Emn (t) +
mn (t)
i
dt = 2 vg ?gmn ? p;mn
2n
2
0
0
(3)
i is the modal gain of LP i mode. Similar modal losses ( i
where gmn
mn
p;mn = p ) and material gain for all modes
have been assumed. Therefore the model describes VCSELs in which the modal gain is the dominant mechanism
i mode is given by
for the selection of high order transverse modes.
Modal gain of LPmn
16,20
i =
gmn
R 1 R 2
0
0
j
i
mn (r; )j
R 1 R 2
0
0
A(N (r; ; t) ? N )rdrd
i (r; )j rdrd
j mn
2
0
2
(4)
i (r; ) is the electric eld prole of LP i
where A is the dierential gain, N (r; ; t) is the carrier density, and mn
mn
mode. Modal gain then represents the two{dimensional degree of overlapping between the mode intensity prole
and the carrier distribution. The numerical value and meaning of the other parameters can be found in Table 1.
Spontaneous emission noise is taken into account by including Gaussian white noise terms with zero mean and
a time correlation given by
i (t) j? (t0 )i = 2mm nn ij (t ? t0 ):
hmn
(5)
mn
0
0
0
0
In order to determine the temporal evolution of the modal gain and power it is necessary to calculate the
time development of the spatial distribution of the carrier density in the laser active region:
@N (r; ; t) = Dh 1 @ r @N i + D @ N ? N + j (r; ; t)
@t
r @r @r
r @ n
ed
X X
X
i (r; )j g i jE i j
2amn j mn
a n j n (r)j g n jE n j +
?
mn mn
2
2
0
n
2
0
0
2
0
2
2
m ; i c;s
2
(6)
=1 3 =
where amn = vg ?(2d 1 j mn (r)j rdr)? . We have split the stimulated recombination term in two sums: one
for azimuthally independent modes (m = 0) and another one for modes with m 6= 0. Superindices i are dropped
for m = 0 since they are no longer needed for azimuthally independent modes.
R
2
1
0
Table 1.
SYMBOL
a
L
D
ncore
ncladd
N
d
vg
0
?
p
n
Device and material parameters
VALUE
3 m
1 m
5 cm s?
3.5
3.49
1:33 10 cm?
0:03 m
0:86 10 cms?
0.06
1:5 ps
2 ns
3
2 10?
2
1
18
3
10
1
5
MEANING OF THE SYMBOL
radius of the core
length of the cavity
diusion constant
refraction index of the core region
refraction index of the cladding region
carrier density at transparency
active layer thickness
group velocity
longitudinal connement factor
photon lifetime
carrier lifetime
linewidth-enhacement factor
spontaneous emission factor
This equation can be solved by a two dimensional discretization of the space with its corresponding long
computational time. When the injected current has a simple azimuthal symmetry the problem of solving the
two{dimensional carrier continuity equation can be reduced to solve four one{dimensional equations with the
consequent diminution of demands upon computational resources. Numerical solutions of the previous equations subject to the condition that N (r = 1; ; t) = 0 are obtained via a discretization of space and time. Time
and space integration steps of 0.1 ps and 0.075 m have been used throughout this work.
18
3. Multi{transverse mode Light-current characteristics
In this section the static behaviour of the VCSEL in the multi{transverse mode regime is studied by presenting
Light-Current characteristics. These are obtained by integrating Eqs. (3), (6) for short current ramps. Results
correspond to integration over a time of the order of 50 ns that is much smaller than typical thermal response
times (around s). Since the characteristic time in carrier-eld interaction (relaxation oscillations) is of the
order of ns, this ramp duration is slow enough to ensure that the results are obtained in a quasi-steady situation,
while being fast enough to avoid the eects of self-heating. In this way the active region temperature is kept
constant (and equal to the substrate temperature) during the current scan. The injected current density is taken
to be of the form j (r; ; t) = j R(r) where R(r) = 1 if rin < r < rout ; and 0 elsewhere. Here rin and rout are the
radii of the ring contact over which the current is injected. An uniform current injection over the active region
24
(disc contact) is considered and therefore, rin = 0 and rout = a. Light{current characteristics of the device are
shown in Fig. 3.
1.5
Power (mW)
LP01
(a)
LP02
1.0
c
s
LP11 and LP11
Total
0.5
0.0
(b)
−1
Modal Gain (cm )
1300
1250
1200
1150
2
4
6
2
j (kA/cm )
8
10
Figure 3. a) Steady{state power and b) modal gain of transverse modes for an uniform injection of current
in the active region.
For currents slightly above threshold fundamental transverse mode appears since its modal gain is the highest,
as it is shown in Fig. 3a. The intensity prole overlaps in an optimal way with the carrier density because this
is concentrated near the center of the laser due to the assumed disc contact geometry. At higher currents the
increasing stimulated recombination of carriers produces a hole in the carrier prole near the center of the device
and LP c and LP s modes reach threshold. These modes are excited with similar power since the symmetry of
the injected current does not privilege any in particular. Further increase of injected current also leads to the
appearance of LP in such a way that several transverse modes are able to coexist at the highest current value
of the considered range.
The previous results have been obtained by neglecting dierences of material gains between dierent transverse modes. The model also assumes a linear dependence of the material gain on the carrier density. This
simple model describes well the results obtained with a more realistic model of the quantum well material well.
This is due to the weak connement of the modes induced by the small waveguide index step. In this way variations of the carrier density over the mode proles are signicant, and then dierences in modal gain between
transverse modes dominate over dierences in material gain.
11
11
02
18
25
4. Relative intensity noise spectra
In this section the RIN characteristics of the VCSEL in the multi{transverse mode regime are studied. The RIN
spectra are calculated numerically by averaging over 200 trajectories, each of duration 10.24 ns, resulting in a
resolution of 97.6 MHz over a 25 GHz frequency range. A disc contact over the active region is considered. As it
is shown in Fig. 3 several transverse modes are excited when the injection current is increased. In Figs. 4{6 the
RIN spectra corresponding to the individual transverse modes and to the total intensity are plotted for dierent
values of the injection current.
−110
(a)
−130
RIN (dB/Hz)
−150
−110
(b)
−130
−150
−110
(c)
−130
−150
Figure 4.
0
1
2
3
4
5
6
Frequency (GHz)
7
8
9
10
RIN spectra for dierent injection currents corresponding to the individual transverse modes:
LP (solid line), LP c and LP s (dashed line), and LP (dotted line), and to the total intensity (bold solid
line). The values of the injection current in this gure are: a) j = 2 kA/cm , b) j = 3 kA/cm , j = 4 kA/cm .
01
11
11
02
2
2
2
For currents slightly above threshold the VCSEL operates in the fundamental transverse mode LP . The
RIN is similar to that of edge{emitting lasers and exhibits a resonance peak that corresponds to the relaxation
oscillation frequency (see Fig. 4a). A second harmonic peak can be also observed in the RIN. At higher currents
the LP c and LP s modes are excited due to SHB and another peak appears in the RIN at the relaxation
oscillation frequency of this mode (see Fig. 4b). Further increase of the injected current also leads to the
appearance of the LP mode and to another resonance peak in the RIN (see Fig. 5c). A frequency response
with several resonance peaks was demonstrated in RIN measurements of multitransverse mode VCSELs.
01
11
11
02
9
−110
(a)
−130
RIN (dB/Hz)
−150
−110
(b)
−130
−150
−110
(c)
−130
−150
Figure 5.
0
1
2
3
4
5
6
Frequency (GHz)
7
8
9
10
RIN spectra for dierent injection currents corresponding to the individual transverse modes:
LP (solid line), LP c and LP s (dashed line), and LP (dotted line), and to the total intensity (bold solid
line). The values of the injection current in this gure are: a) j = 5 kA/cm , b) j = 6 kA/cm , j = 7 kA/cm .
01
11
02
11
2
2
2
The resonance peaks associated to transverse modes with azimuthal symmetry, LP and LP , appear in
the RIN of the individual modes and of the total intensity. However, unless at low frequencies in Fig. 1b,
the resonance peak associated to the LP c and LP s modes, , appears only in the noise spectra of these
modes. This peak is not found in the RIN that corresponds to the sum of intensities of these two modes with
azimuthal dependence. This corresponds to the anticorrelated power uctuations of LP c and LP s modes that
are observed at . Then the power redistribution occurs only in the azimuthal direction. As a consequence
power oscillations of transverse modes with dierent radial prole, such as LP and LP , are not excited at
. Only when is small a peak appears at this frequency in the RIN of the LP and of the total intensity
(Fig 1b). A possible explanation could be that carrier diusion can be eective in both the radial and azimuthal
directions at this small frequency. This point needs further investigation.
01
11
11
02
11
11
11
11
01
11
11
02
01
−100
(a)
−120
−140
−160
RIN (dB/Hz)
−100
(b)
−120
−140
−160
−100
(c)
−120
−140
−160
Figure 6.
0
1
2
3
4
5
6
Frequency (GHz)
7
8
9
10
RIN spectra for dierent injection currents corresponding to the individual transverse modes:
LP (solid line), LP c and LP s (dashed line), and LP (dotted line), and to the total intensity (bold solid
line). The values of the injection current in this gure are: a) j = 8 kA/cm , b) j = 9 kA/cm , j = 10 kA/cm .
01
11
02
11
2
2
2
It is also found that thepfrequency at which the highest resonance peak appears in the RIN of the total
intensity is proportional top (j ? jth ), where jth is the threshold current density (see Fig. 7). The standard
i being the power of LP i mode) seems to be valid also for the peaks
i (Pnm
linear relationship with Pnm
nm
associated to the individual transverse modes. However, when the power of the modes are of the same order the
resonance frequencies do not follow
q this behavior due to mode interaction. These resonance frequencies seem to
i is an eective mode volume given by
i , where Vn;m
be also inversely proportional to Vn;m
26
i =d
Vn;m
?R
2
j i j rdrd :
R n;m
i j rdrd
j n;m
2
4
(7)
Frequency of Maximum RIN (GHz)
10
8
6
4
2
0
0
1
2
3
1/2
(j − jth)
Figure 7.
level.
Resonance frequency corresponding to the highest peak in the RIN of the total power vs pumping
Finally we analyze the noise spectra at low frequencies. It has been experimentally observed that higher
transverse modes can lead to an increase in the RIN of the total power. Previous theoretical studies have
shown that the low frequency enhancement of RIN for individual modes is reduced when the modes have a small
spatial overlap. Our results show in Fig. 4b that when the LP c and LP c modes are excited, power uctuations
in the fundamental mode are larger than those of the total power by a factor of a few dBs. This increase is a
consequence of anti{correlated power uctuations of individual modes due to carrier competition (mode partition
noise). This eect is not signicant for the LP and LP modes due to their small spatial overlap (see Figs.
1 and 2). Then each mode interacts with its own carrier reservoir and the degree of anticorrelation of mode
power uctuations is reduced. As a consequence the RIN at low frequencies for the total power decreases when
the injection current is increased, even though the LP modes are excited (see Fig. 8). Therefore low RIN
operation can be maintained for low spatially overlapping modes.
At higher injected current the LP mode is excited (Fig. 5c). This mode has a signicant overlap with
the LP and LP modes (see Figs. 1 and 2). Then mode partition noise causes a 40{dB enhancement of the
RIN at low frequencies for these modes with respect to the RIN for the total power. It is also found that the
appearance of the LP mode leads to an increase of the total power uctuations at low frequencies, even though
the injected current increases (see Fig. 8). The dierent behavior for modes with dierent spatial distribution
have been also observed in small signal modulation response measurements. At higher currents the total power
uctuations seem to reach a constant level (Fig. 8).
3
15
11
01
11
11
11
02
01
11
02
13
−110
RIN (dB/Hz)
−120
−130
−140
−150
−160
2.0
Figure 8.
4.0
6.0
8.0
2
Current Density (kA/cm )
10.0
12.0
Relative intensity noise at 500 MHz for the total power vs the injected current density.
5. Conclusions
Relative intensity noise spectra of weakly index guided VCSELs working in the multi{transverse mode regime
have been studied. An ecient dynamical model taking into account all transverse modes supported by the
waveguide has been used. This model incorporates carrier diusion and spatial hole burning eects. Several
peaks are obtained in the noise spectra at the relaxation oscillation frequencies of the transverse modes. The
highest peak in the RIN of the total power appears at a frequency that increases with the pumping level in the
same way than for edge{emitting lasers.
Concerning the noise level at low frequencies, it is found that low RIN operation can be achieved in the multi{
transverse mode regime when the modes have a small spatial overlap. In this case the RIN at low frequencies
for the total power decreases when the injected current is increased, even though a higher order transverse mode
is excited. Mode partition noise is not signicant because each mode interacts with its own carrier reservoir.
However, when a transverse mode that overlaps with the lasing modes is excited a localised increase is found
in the RIN for the total power. Moreover the power uctuations in individual transverse modes are larger by
a factor of 40{dB than those of the total power. This enhancement is a consequence of anti{correlated power
uctuations of individual modes due to carrier competition.
ACKNOWLEDGEMENTS
Financial support has been provided by CICYT (Spain) Project TIC98-0418-C05-01.
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