Linewidth enhancement factor measurement based on FM-modulated optical injection: application to rare-earth doped active medium

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Linewidth enhancement factor measurement based on FM-modulated optical
injection: application to rare-earth doped active medium
Aur elien Thorette, Marco Romanelli, and Marc Vallet
Institut de Physique de Rennes, Universit e Rennes I - CNRS UMR 6251
Campus de Beaulieu, 35042 Rennes Cedex, France and
Corresponding author: [email protected]
A new method for measuring the linewidth enhancement factor of a laser is proposed. It is
based on frequency-modulated optical injection, combined with dual-frequency laser operation. The
linewidth enhancement factor is deduced from the experimental data using a theoretical analysis
based on a standard rate equation model. As the intracavity power is kept constant, the method
allows to free the process from the thermal eects that are usually present in AM/FM techniques.
Measurement of = 0:280:04 in a diode-pumped Nd:YAG laser demonstrates that the method
is well-suited for characterizing small values of .
The linewidth enhancement factor, also referred to
asHenry factor orfactor [1], quanties the phase-
amplitude coupling in a laser gain medium. The origin
ofcomes from an asymmetric gain prole or from a de-
tuning of the laser frequency with respect to the gain line
center. In semiconductors, for which can take rather
large values, this phase-amplitude coupling describes im-
portant characteristics of laser behavior, such as a large
broadening of the laser linewidth [2] or peculiar dynamics
under current modulation or optical injection [3]. While
it is fairly common to consider '0 for active media
with more symmetric gain proles such as diode-pumped
solid-state lasers, a small phase-amplitude coupling may
also have to be taken into account when targeting ap-
plications needing stabilized solid state lasers with very
low optical phase noise, such as gravitation wave de-
tection [4, 5] or optically carried radiofrequency gener-
ation [6].
Very extensive literature exist on -factor measure-
ments performed in all main types of semiconductor
lasers, i.e., quantum cascade lasers, quantum dots, VC-
SELs and so forth. The measurement methods include
direct estimation of the gain asymmetry [7], pump in-
duced phase modulation through AM/FM coupling [8]
and optical injection [9, 10]. Conversely, studies of the
phase-amplitude coupling in solid-state lasers have been
much less common. A value of = 0:250:13 has been
found in a Nd:YVO4 laser, using either injection [11] or
pump AM/FM modulation method [12], while a surpris-
ingly large 1 was reported in Nd:YAG microchip
lasers [13].
Any measurement of needs a way to either force,
as in injection methods [11], or measure, as in AM/FM
modulation [12], the optical phase. In both cases, one
needs to precisely control the optical frequency dier-
ence between the laser under study and an auxiliary op-
tical source. Here, we propose to use a laser operating in
a dual-frequency regime, thus providing simultaneously
the master and the slave oscillator. In this way, we can
take advantage of the intrinsic stability of the frequencydierence between the modes, and of their perfect mode-
matching (both due to the fact that they share the same
optical cavity) [14, 15]. However, while dual-frequency
operation facilitates the implementation of the method,
we stress that the latter does not require it, and could be
equally used in the standard optical injection congura-
tion.
The aim of this letter is thus to present an \FM/AM"
injection method based on the amplitude response of a
lasing mode to a frequency-modulated optical injection
of a second mode, and to show how its implementation
in a Nd:YAG dual-frequency laser leads to a rather
precise characterization of small factors.
For the sake of clarity, we rst describe the method on
an ideal master-slave injection conguration. The rate
equation for the electric eld of an injected class-B laser
is generically [3] :
dE
dt= (1 +i)NE
2+i
E+ Einj (1)
Here,Eis the intracavity eld, Nthe active medium
gain, 
 is the detuning between the injected eld and the
free-running laser frequency,  is the injection eciency,
andEinjthe injected eld, whose phase is taken as refer-
ence. Separating phase and amplitude as E=Eexp(i')
leads to :
dE
dt=NE
2+ Einjcos' (2a)
d'
dt=N
2+ 
Einj
Esin' (2b)
We consider small perturbations of the injection-
locked, steady state regime. Thus, we write x=bx+x,
wherexstands forE;';N .bxdenotes the steady state
value ofxandxthe small perturbation. Linearization
of equation (2a) leads to :
dE
dt=bEN +bNE
2Einjsinb'' (3)arXiv:1703.03259v1  [physics.optics]  9 Mar 2017
2
This shows clearly that amplitude response to a phase
perturbation 'depends on the quantity sin b'. In partic-
ular, a zero response is expected when sin b'= 0. Using
the steady state equation (2b), this condition becomes
bN=2 =
, which we can transform using (2a) to the
more useful expression :

 =Einj
bE(4)
This detuning corresponds to a minimal amplitude re-
sponse to a phase perturbation, and this result shows
that it is directly related to . Consequently, it provides
a way to measure phase to amplitude coupling, and will
be at the root of our method. In the following, we will
denote this value as the minimal amplitude response de-
tuning 
 m. We point here that this method is only suited
to small values  < 1, because it relies on the measure-
ment of , which can only be derived from the span of
the injection locking region 
+
. This region is
roughlyj
j<p
1 +2Einj=bEfor low injection. Thus
we have=p
1 +2= 2
 m=(
+
), where the left-
hand term only has a dependance in 2for high values
>1, making any precise measurement impractical. On
the other hand, for  < 1, it scales as , which makes
this method well adapted to the low values expected for
solid-state lasers.
As for all injection methods, the measurement requires
a very stable and tunable master laser. In order to by-
pass this requirement, we now discuss the alternative
solution in which we will use the laser as its own ref-
erence, by making it dual-frequency, and using one of
the modes to inject on the other through a feedback.
The experimental setup is summarized in Fig. 1, which
is similar to the one described in [16]. It is centered
around a 5mm-long (111)-cut Nd3+:YAG crystal operat-
ing at 1064nm. The crystal is optically pumped with a
150mW single mode circularly polarized laser diode LD
at 808nm. The associated cavity is L= 6:5cm long.
One high-re
ection mirror ( M1) is directly coated on the
external face of the crystal and the cavity ends with a
10cm radius mirror ( M2) with 99% re
ectivity. In or-
der to achieve dual-frequency operation, a tunable phase
retardance is added in the cavity by using two quarter-
wave plates (QWP), of which one is rotated by an angle
. This induces an optical path dierence for the two po-
larization eigenmodes, which leads to a lasing frequency
dierence=yx=c=L . A 1mm-thick etalon
is also inserted to ensure single longitudinal mode opera-
tion for each polarization state. Dual polarization, dual-
frequency operation can thus be achieved, the frequency
oset between the two modes being tunable from 0 up
toc=4L. Here, we choose 200MHz. The two out-
put modes are combined by a polarizer P. The resulting
RF beatnote is the useful signal, which is monitored by a
photodiode PD followed by an electric spectrum analyser
ESA and on an oscilloscope.We then simply inject one mode in the other using
a feedback external cavity. An acousto-optic modulator
(AO) driven at a frequency fAOis used to shift the optical
frequencies so that x+ 2fAOy. Then a mirror M
and a QWP are used to inject back the x-polarization
in they-polarization in the laser cavity. The intensity
of this reinjection can be controlled by the diraction
eciency of the modulator, and the detuning between
the injected eld and the intracavity eld is then =
2fAO. When this detuning is kept small, stable phase
locking occurs between the output beatnote and the RF
synthetizer. The size of this locking range depends on
the feedback eciency, and can extend to 2MHz.
FIG. 1. Experimental setup. An intracavity birefringent el-
ement (here, two rotated quarter-wave plates QWPs) forces
the laser to produce two polarization modes. One of them is
frequency-shifted with an acousto-optic modulator (AO) close
to the other one, and injected back in the cavity. The out-
put signal is an optically-carried RF beat-note. See text for
details.
We then introduce a phase perturbation through a
modulation of the detuning . This can be conveniently
achieved using FM modulation of the RF synthetizer that
drives the acousto-optic modulator. This leads to a fre-
quency shift of 2 fAO=f0+f1cos(2fMt)). We choose
to modulate the detuning at a frequency fMclose of the
relaxation oscillations frequency fRof the laser that we
measure to be fR= 60kHz, so that the AM response is
maximized by the resonance. As the perturbation has to
be kept small, we take f1= 10kHz.
We have introduced a phase modulation and are now
interested in the resulting amplitude modulation, which
can be related to the expected phase-amplitude coupling
. We observe that the amplitude modulation response
depends strongly on the mean detuning f0, as shown
in Fig. 2. With no phase-amplitude coupling, the ampli-
tude response would be minimal for a null detuning, as
implied by equation (4). But experimentally we observe
that the amplitude modulation is minimal for a positive
value of the detuning, suggesting a measurable non-zero
value of. We notice that the minimal amplitude re-
sponse corresponds to an equal intensity for the two side
peaks atf0fM. This balance criterion between the
two sidebands is most convenient for an ecient mea-
surement, as shown by Fig. 2. Furthermore, Eq. 4 implies
3
that 
 mshould increase with the injection strength. We
have thus repeated the measurement for increasing val-
ues of the reinjection eciency, by varying the eciency
of the acousto-optic diraction. The results are plotted
in Fig. 3, and show indeed that 
 mdepends strongly on
the injection strength, again indicating a non-zero value
of the phase-amplitude coupling.
Traces (arb. units)
0 20 40
Time (ms)
−80−60−40−20∆0<∆mSpectrum (dB)
−80−60−40−20∆0= ∆m
180 120 60 0-60-120
Detuningδν−f(kHz)−80−60−40−20∆0>∆m
FIG. 2. Power spectrum for 
 0<
m, 
0= 
 mand 
 0>

m(with 
 0= (f0)=fR), and the associated time series
(black: modulation signal, red: output beatnote Ixy=jex+
eyj2). This shows that the balance of the two sidebands at
fRcorresponds to minimal amplitude response, and to 
phase shift between 
 0>
mand 
 0<
m
In order to have a quantitative comparison between
the experiments and the theory, and to extract a value of
, we have to rene the model of equations (1) to include
some features that are specic of dual-frequency opera-
tion (i.e. the coupling between the two eigenmodes in
the active medium). We can model our laser using the
following rate equations for the normalized elds ( ex;y)
and the normalized population inversions ( mx;y) [17], to
which we add the phase-amplitude coupling as an imag-
inary part for the gain:
0 1 2 3 4 5 6 7
Feedback strength Γ0.00.51.01.52.0Minimal response detuning ∆ mFIG. 3. Experimental values of 
 m(normalized to the relax-
ation oscillation frequency fR= 60 kHz). Blue curve: best
t from equation (7), corresponding to = 0:280:04. The
shaded region shows the computed values of 
 mwithin the
error range for .
dex
ds= (1 +i)mx+my
1 +ex
2(5a)
dey
ds= (1 +i)my+mx
1 +ey
2+i
ey+ ex(5b)
dmx;y
ds= 1(jex;yj2+jey;xj2)
"mx;y[1 + (1)(jex;yj2+jey;xj2)]
(5c)
The phase of the eld complex electric eld Exis
taken as a reference, and we use Ex=exexp(2ixt) and
Ey=eyexp(2i(x+ 2fAO)t). The time scale s= 2fRt
is related to the relaxation oscillations. Control param-
eters are the normalized feedback intensity  and the
normalized detuning 
 = ( 2fAO)=fR. Constant
parameters are pump rate , the coupling inside gain
medium, and"=
k=2fRthat accounts for the
populations lifetime 1 =
k. All these parameters can be
measured or experimentally controlled. The mode cou-
plingcan be infered from the frequency of antiphase
oscillations [18], which are observed by monitoring the
intensity noise of a single polarization mode. We measure
them to be at a frequency fAso that 
 = fA=fR= 0:66,
so we have = (1
)=(1 + 
) = 0 :200:05. Well
known values of 1 =
k= 230sleads to"0:01 [19].
is the ratio of the pump laser diode current to the
threshold current, and we have used a value of 1 :20:1.
The least known parameter here is , which is related
to the intensity injected from one mode to the other.
Since it has been shown that phase locking between
the laser beatnote and the RF synthetizer occurs when
j
j< and that it is characterized by a single peak at
4
2fAOon the RF power spectrum,  can be obtained by
measuring the phase-locking range. The experimental
beat-note signal can be compared with the computed
quantityIxy=jex+eyj2.
−50−40−30−20−1010 log|A(fR)|
0.0 0.5 1.0 1.5
Mean detuning ∆ 0−π/20−π/2πarg [A(fR)]α= 0 0.1 0.2 0.3
FIG. 4. Maximum value of the transfer function A(fR) for
amplitude response to a phase perturbation for dierent val-
ues of the mean detuning 
 0. One gets closer to zero response
when approaching 
 0= 
 m, and a phase shift is observed
when 
 0crosses 
 m.
We introduce the phase modulation through the detun-
ing, thus taking in account the FM modulation applied
to the acousto-optic modulator. Therefore we consider

 = 
 0+acos(2fMt).
The response of the system to this modulation accord-
ing to the model is obtained by linearizing the equations
around the equilibrium. We can then introduce the mod-
ulation as a small perturbation, and obtain the transfer
functionA(fM) for the output intensity. As expected, it
features a resonant maximum at fR, which suggests we
should modulate at this particular frequency. The am-
plitude and phase of the maximum value A(fR) of this
transfer function is plotted in Fig. 4. While for = 0, the
minimal (zero) response is obtained at 
 0= 0, it is not
anymore the case when 6= 0. The minimal response
is shifted to a particular value 
 m>0, that depends
strongly on . This value also corresponds to a phase
jump for the transfer function. This conrms that the
balance of the modulation sidebands in the spectrum is
a good way to measure 
 m.
The same reasoning that led to equation (4) also ap-
plies in this more complex case, resulting in the relation

m=bex
bey(6)similar to equation (4). For low injection level, one can
consider that bex;ydo not dier appreciably from their
equilibrium values in the free-running regime, so that
equation (6) further simplies to 
 m=. In the gen-
eral casebex;yhave to be computed from the system equa-
tions (5a-5c), and inserted into equation (6). This pro-
cedure leads to a 4th degree polynomial for 
 m, which
can be solved for a single real value

m=(1 +f(;;)) (7)
wherefis a correction function that cancels for  = 0,
and has the rather cumbersome expression to the rst
order in :
f(;
;) =
(
222
2)
2
+
222(2
4=22
2+2+ 3
4+
2
3
4+ 
2) + 2
2(
2++ 2
2) + 
41
2

3(
2++ 2
2) + 
2(8)
This shows that for higher injection level, the minimal
response detuning is not simply , but it depends on
other parameters of the model.
We can now use equation (7) to extract the value of
from the measured results of Fig 3. From each point
of Fig 3, a value of can be estimated, so that the nal
result can be obtained simply by averaging :
=
m
(1 +f(;;))
(9)
To take in account that the the uncertainty on a single
measurement is higher for low values of , we use a
weighted average with weights 1 =, and we nd a value
of= 0:280:04. The uncertainty is computed from
the precision of the frequency measurements, which we
estimate to be around 8kHz, and also includes the uncer-
tainties on and. This leads to a satisfying reduced
chi-squared value of 1.15.
In conclusion, a new method for characterizing has
been demonstrated through the measurement of the in-
herent small phase-amplitude coupling in a Nd3+:YAG
solid-state laser. It is based on frequency modulated
injection, so that it completely frees the measure from
pump modulation and associated strong thermally in-
duced AM/FM coupling. The use of laser operating in a
dual-frequency regime makes it easier to solve the prob-
lem of the mode-matching of master and slave lasers in
injection setups, and circumvents the need of a stable
reference laser for heterodyning.
5
The physical origin of the phase-amplitude coupling
in rare-earth active medium was left out of the scope
of this letter, but we can already note that the value
= 0:280:04 is quite close to the 0 :250:13 reported
for Nd3+:YVO 4in Ref. [12], suggesting very little in
u-
ence of the host crystal matrix on . Our method can be
applied to Er3+-doped bulk or bered medium. For in-
stance, this may give clues to the potential contribution
ofto the AM/FM noise conversion process during low
phase noise microwave or THz generation [20, 21].
The authors thank Marc Brunel for fruitful discussion,
and Steve Bouhier and Ludovic Frein for technical sup-
port in electronics.
[1] C. Henry, IEEE J. Quantum Electron. 18, 259 (1982).
[2] K. Petermann, Laser Diode Modulation and Noise , Vol. 3
(Springer Science & Business Media, 1988).
[3] S. Wieczorek, B. Krauskopf, T. Simpson, and D. Lenstra,
Phys. Rep. 416, 1 (2005).
[4] F. Acernese, M. Alshourbagy, F. Antonucci, S. Aoudia,
K. G. Arun, P. Astone, G. Ballardin, F. Barone, L. Bar-
sotti, M. Barsuglia, and others, Phys. Rev. A 79(2009),
10.1103/PhysRevA.79.053824.
[5] B. P. Abbott, R. Abbott, T. D. Abbott, M. R. Aber-
nathy, F. Acernese, K. Ackley, C. Adams, T. Adams,
P. Addesso, R. X. Adhikari, and others, Phys. Rev. Lett.
116, 241103 (2016).
[6] G. Pillet, L. Morvan, M. Brunel, F. Bretenaker, D. Dol,
M. Vallet, J.-P. Huignard, and A. Le Floch, J. Light.
Technol. 26, 2764 (2008).
[7] B. W. Hakki and T. L. Paoli, J. Appl. Phys. 46, 1299(1975).
[8] J.-G. Provost and F. Grillot, IEEE Photonics J. 3, 476
(2011).
[9] R. Hui, A. Mecozzi, A. D'ottavi, and P. Spano, Electron.
Lett. 26, 997 (1990).
[10] K. Iiyama, K.-i. Hayashi, and Y. Ida, Opt. Lett. 17, 1128
(1992).
[11] S. Valling, T. Fordell, and A. M. Lindberg, Phys. Rev.
A72(2005), 10.1103/PhysRevA.72.033810.
[12] T. Fordell, S. Valling, and b. M. Lindberg, Opt. Lett.
30, 3036 (2005).
[13] C. Szwaj, E. Lacot, and O. Hugon, Phys. Rev. A 70
(2004), 10.1103/PhysRevA.70.033809.
[14] G. W. Baxter, J. M. Dawes, P. Dekker, and D. S.
Knowles, IEEE Photonics Technol. Lett. 8, 1015 (1996).
[15] M. Brunel, N. D. Lai, M. Vallet, A. Le Floch, F. Brete-
naker, L. Morvan, D. Dol, J.-P. Huignard, S. Blanc, and
T. Merlet, in Proc. SPIE 5466, Microwave and Terahertz
Photonics (2004) p. 131.
[16] M. Romanelli, L. Wang, M. Brunel, and M. Vallet, Opt.
Express 22, 7364 (2014).
[17] J. Th evenin, M. Romanelli, M. Vallet, M. Brunel, and
T. Erneux, Phys. Rev. A 86(2012), 10.1103/Phys-
RevA.86.033815.
[18] E. Lacot and F. Stoeckel, JOSA B 13, 2034 (1996).
[19] M. Brunel, O. Emile, M. Vallet, F. Bretenaker,
A. Le Floch, L. Fulbert, J. Marty, B. Ferrand, and
E. Molva, Phys. Rev. A 60, 4052 (1999).
[20] F. Quinlan, T. M. Fortier, M. S. Kirchner, J. A. Taylor,
M. J. Thorpe, N. Lemke, A. D. Ludlow, Y. Jiang, and
S. A. Diddams, Opt. Lett. 36, 3260 (2011).
[21] A. Rolland, G. Ducournau, G. Danion, G. 'hen Loas,
M. Brunel, A. Beck, F. Pavanello, E. Peytavit, T. Akalin,
M. Zaknoune, J.-F. Lampin, F. Bondu, M. Vallet,
P. Szriftgiser, D. Bacquet, and M. Alouini, IEEE Trans.
Terahertz Sci. Technol. 4, 260 (2014).