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#143182 - $15.00 USD Received 28 Feb 2011; revised 15 Apr 2011; accepted 16 Apr 2011; published 20 Apr 2011(C) 2011 OSA 25 April 2011 / Vol. 19, No. 9 / OPTICS EXPRESS 8802
1. Introduction
The development of frequency swept laser sources in the 1300 nm wavelength range with
tuning ranges of ~100 nm and sweep rates >15 kHz [1] has recently enabled a quantum leap in
system performance of optical coherence tomography (OCT) systems [2]. Yet, conventional
rapidly swept lasers are inherently limited in their achievable sweep rates due to the buildup
time of the lasing [3]. Recently, using the technique of Fourier domain mode-locking (FDML)
[4], greatly enhanced sweep rates of >5 MHz were achieved [5]. The typical instantaneous
linewidth of <0.1 nm corresponds to an instantaneous coherence length of up to several
millimeters. Currently, FDML lasers are the light sources of choice for OCT systems with
highest imaging speed [5,6].
In FDML operation, a narrowband optical bandpass filter is tuned synchronously to the
cavity roundtrip time of the laser. Therefore the laser field does not have to build up
repetitively as in the standard tunable laser and the sweep rate is only limited by the
mechanical response of the filter. Since FDML is a stationary laser operating regime [7] with
very long cavity photon lifetimes [8], substantial linewidth narrowing compared to the width
of the sweep filter can be observed [9]. Even though a connection between cavity dispersion
and linewidth has been revealed, the physics behind it is not understood and a quantitative
relation is currently not available. Further, a direct experimental, spectrally resolved
measurement of the instantaneous FDML linewidth is not possible, especially for very
narrowband emission, due to Fourier broadening [10]. Besides sweep range and rate, the
instantaneous linewidth or coherence length is the most important property of an FDML laser,
because it determines the maximum ranging depth in OCT imaging and it is sometimes
limiting system performance. Thus, theoretical access to this parameter is of high interest.
Using the previously presented theoretical model [7], the first goal of this paper is to
investigate the dependence of the FDML linewidth on cavity parameters and relevant physical
effects. The gained insight might be used to find ways to increase the instantaneous coherence
length from the mm range to the cm or m range in the future, enabling a whole new variety of
biological and non-biological imaging and sensing applications. The second goal is to
investigate if ASE and environmental instabilities ultimately limit the FDML linewidth
performance, as in typical semiconductor based lasers.
2. Experimental setup
In Fig. 1(a), the experimental setup of the FDML laser is shown. In order to study the
instantaneous linewidth at different points in the cavity, three output couplers are built into a
sigma ring geometry. Here, a semiconductor optical amplifier (SOA, Covega Corp., “BOA
1132”) is used as a gain medium, where the maximum of the gain lies at 1320 nm. In order to
ensure unidirectional lasing, two isolators (ISO) are built in before and after the SOA. The
sweeping action is performed by a tunable polarization maintaining (PM) Fabry-Perot
bandpass filter (FFP-TF, Lambda Quest, LLC.), with a bandwidth of 0.15 nm.
1
FFP-TF PBSFRM
50%
40%
50%
1.7km SMF
ISOISO
PM fiber
SOA
(a) (b)
Optical
spectrum
analyzerEOM
pulse
generator
function generator
laser input
2
3
1250
1300
1350
(t
) [n
m]
5 10 150
50
100(d)
t [µs]
P(t
) [m
W]
(c)
Fig. 1. (a) Setup of the polarization maintaining FDML laser with outcouplers numbered 1 to 3.
(b) Measurement setup for the instantaneous linewidth using a pulse generator, an electro-optical modulator and an optical spectrum analyzer. (c) Sweep filter center wavelength over
time. (d) Simulated output power over time.
#143182 - $15.00 USD Received 28 Feb 2011; revised 15 Apr 2011; accepted 16 Apr 2011; published 20 Apr 2011(C) 2011 OSA 25 April 2011 / Vol. 19, No. 9 / OPTICS EXPRESS 8803
The light coming from the SOA is coupled into a single mode fiber (SMF) with a length of
1.7 km by means of a polarization beam splitter (PBS). At the end of the SMF, a Faraday
rotating mirror rotates the light by 90° where it is transmitted back through the fiber to the
FFP-TF and the SOA. The FFP-TF filter is driven sinusoidally with a sweep frequency of 57.7
kHz in resonance to the cavity roundtrip time of T = 17.32 µs. In order to measure the
linewidth, a setup as shown in Fig. 1(b) is employed, similar to the one described in [10]. The
sweep range for this setup is 105 nm, see Fig. 1(c). The simulated time dependent output
power is shown in Fig. 1(d).
3. Theory: propagation equation and extraction of instantaneous linewidth
The spectral sweep range for our FDML setup of 105 nm and the cavity length of 3.4 km lead
to a very large time-bandwidth product of 83 10 , inhibiting straightforward approaches to
FDML simulation. This problem is overcome in [7] by the introduction of a transformation
into the swept filter reference frame, reducing the number of required grid points by about two
orders of magnitude. The resulting evolution equation for the envelope function u(z,t)
22 3 2
0 0 0 2 0 3 2( ) 1 i ( ) i i -i i (i )z t s tu g a D D D u a u
(1)
contains all relevant effects, such as dispersion (second and third order dispersion, D2 and D3),
self-phase modulation γ and the linewidth enhancement α [11] of the SOA. Furthermore,
a(ω0) and g(ω0) represent the frequency-dependent loss and gain, also accounting for gain
saturation. Here, ω0(t) is the time dependent sweep filter center frequency. The term (i )s ta
represents the sweep filter. This model has successfully been applied to compute the time
dependent output power of an FDML laser [7] as shown in Fig. 1(d). Here we use it for the
first time to analyze the lasing linewidth.
Amplified spontaneous emission (ASE) is modeled as an equivalent noise source at the
input of the ideal SOA. It is implemented as additive white Gaussian noise [7,12] with a
constant spectral power density fP , computed from the noise power 3.2 mWnP measured
directly after the SOA for a blocked laser cavity together with the experimentally determined
small signal gain.
The values of D2 and D3 are 28 2 12.7603 10 s m and 41 3 11.2183 10 s m , respectively, and
γ is 0.00136 1 1W m . Furthermore, a typical value of α = 5 is assumed [13]. The sweep filter
function (i )s ta is implemented as a lumped element in form of a complex Lorentzian, as
described in [7]. The gain and loss have been carefully measured as shown in Figs. 2(a) and
2(b), respectively, and have been implemented accordingly in our simulation.
225 230 2350
100
200
300
(a)
Frequency [THz]
Gain
4.8 mW1.6 mW
0.5 mW
0.2 mW
0.06 mW
225 230 235
5.6
5.8
6
6.2
6.4
6.6
6.8(b)
Frequency [THz]
Lo
ss
Fig. 2. (a) Experimentally measured SOA power gain (linear scale) as a function of the optical
frequency for different values of the incident optical power. (b) Experimentally measured overall cavity power loss (linear scale) as a function of the optical frequency. The sweep filter
has been tuned to maximum transmission at each measured frequency.
#143182 - $15.00 USD Received 28 Feb 2011; revised 15 Apr 2011; accepted 16 Apr 2011; published 20 Apr 2011(C) 2011 OSA 25 April 2011 / Vol. 19, No. 9 / OPTICS EXPRESS 8804
The instantaneous linewidth can be obtained from the simulation data by Fourier
transforming the complex field envelope in the swept filter reference frame u(z,t), yielding the
instantaneous power spectrum |u(z,f)|2, where f denotes the frequency with respect to the
center frequency of the sweep filter. The duration T of one roundtrip can now be segregated
into a given number of subintervals, and the instantaneous power spectrum at different times
can be calculated by Fourier transforming u(z,t) for each interval. This way we can simulate
the temporal evolution of the instantaneous power spectrum. Because the simulation is
performed in the swept-filter reference frame, the simulation is NOT broadened by the
ongoing sweeping action during a time-gate as it is in the experiment. The instantaneous
power spectrum does not depend much on the position z within the laser cavity, and is here
computed after the SOA. The instantaneous linewidth corresponds to the full width at half-
maximum (FWHM) of the instantaneous power spectrum.
To experimentally measure the linewidth at different positions within the sweep, in the
setup the time gating is performed at various times t, corresponding to different sweep filter
center frequencies. The optical spectrum analyzer has a finite resolution of 20 pm,
corresponding to a spectral width of 3.5 GHz at 1310 nm. Furthermore, the 1.6 ns gating
window leads to a Fourier broadening of ~1 GHz according to the time-bandwidth product.
Longer gating times would suppress this effect, but lead to considerable smearing of the
linewidth due to the sweep filter dynamics. For a sweep filter driven by a cosine wave of the
form 0( ) cos(2 / )t t T , the broadening has its maximum value of 3.81 GHz at 5.3 and
3.3 µs, whereas at 1.3 and 7.3 µs, it is only 1.85 GHz due to the slower sweep speed at that
point. The combination of all these effects leads to a broadening of the experimentally
measured spectrum by about 4-8 GHz compared to the theoretically calculated values. In the
simulation, where the frequency axis moves along with the sweep filter, the gating window
can be chosen sufficiently long to avoid Fourier broadening. Here, we divide the axis into 16
intervals of 1.08 µs, summing up to the total roundtrip time of 17.32 µs.
4. Results
4.1 Agreement with experiment
In Fig. 3, the instantaneous power spectra are compared at different times t for the non-
detuned case, where the sweep filter frequency matches exactly the roundtrip time. In Fig.
3(a), t = 1.3 µs, where the sweep filter center frequency varies only slowly, and in Fig. 3(b) t
= 3.3 µs, where the cosine function is the steepest, thus the frequency changes fast. For this
reason, the experimental spectrum in Fig. 3(b) is considerably broadened as compared to the
simulated spectrum, with a full width at half-maximum (FWHM) of 12.07 GHz for the
experimental result vs. 5.81 GHz for the simulation.
-50 0 500
0.2
0.4
0.6
0.8
1(a)
Frequency [GHz]
Sp
ectr
ali
nte
nsity
[arb
. u
.]
-50 0 500
0.2
0.4
0.6
0.8
1(b)
Frequency [GHz]
Sp
ectr
ali
nte
nsity
[arb
. u
.]
Fig. 3. Experimental (red) and simulated (blue) instantaneous power spectra after the SOA at
(a) 1.3 µs and (b) 3.3 µs for no detuning.
#143182 - $15.00 USD Received 28 Feb 2011; revised 15 Apr 2011; accepted 16 Apr 2011; published 20 Apr 2011(C) 2011 OSA 25 April 2011 / Vol. 19, No. 9 / OPTICS EXPRESS 8805
For further validation, the detuned case is investigated, showing that good agreement
between theory and experiment is obtained under various conditions. In Figs. 4(a), 4(b) and
4(c), the instantaneous power spectrum is displayed at t = 7.3 µs for zero, 2 Hz and + 2 Hz
detuning, respectively. The detuning between the sweep period and the roundtrip time affects
not only the time dependent output power [7], but also the instantaneous power spectrum.
Specifically, we observe a pronounced high-frequency tail for both negative and positive
detuning, see Figs. 4(b) and 4(c). This asymmetry gets reduced for a smaller amount of
detuning, as can be seen by comparison with the non-detuned case shown in Fig. 4(a). The
main source of this asymmetry is found to be the third order dispersion term D3. In Fig. 4(b),
the asymmetry manifests itself as a small side peak, which is not resolved in the experiment
due to the limited resolution as discussed above.
-50 0 500
0.2
0.4
0.6
0.8
1
Frequency [GHz]
Spectr
al in
tensity [
arb
. u.]
(a)
-50 0 500
0.2
0.4
0.6
0.8
1
Frequency [GHz]
Spectr
al in
tensity [
arb
. u.]
(b)
-50 0 500
0.2
0.4
0.6
0.8
1(c)
Frequency [GHz]
Spectr
al in
tensity [
arb
. u.]
Fig. 4. Theoretical (blue) and experimental (red) power spectra at 7.3 µs for (a) no detuning,
for (b) a detuning of 2 Hz and for (c) a detuning of + 2 Hz.
4.2 Timing offset, linewidth enhancement factor and spectral shift
In our simulation, where the frequency axis moves along with the sweep, broadening of the
instantaneous power spectrum due to the sweep filter dynamics is eliminated. Thus, we can
analyze the instantaneous power spectrum averaged over the whole roundtrip time, which is
not possible in the experiment. In Fig. 5(a), the instantaneous power spectrum is plotted for
zero detuning and with the laser parameters as in the experiment, where the linewidth
enhancement factor is assumed to be α = 5 (red) [13] and α = 0 (blue). The sweep filter
transmission function is also shown for comparison. Figure 5(b) shows the same simulation,
but now with ASE only used at the start of the simulation to seed lasing. From Fig. 5(a) we
can extract that the frequency shift of the power spectrum is due to the linewidth enhancement
in the SOA, as also observed for conventional swept laser sources [13].
The cause of this frequency shift can be understood if we separately investigate the gain
term containing α in Eq. (1), 0( ) 1 i ,zu g u in the frequency domain. We found that
for α>0 and an asymmetric gain function g(ω0) which falls off more rapidly on its high-
frequency side (compare Fig. 2(a)), the power spectral peak of a pulse u gets shifted to lower
frequencies after propagation through the gain medium.
In our case, we additionally observe a significant broadening of the linewidth from 7.25
and 7.41 GHz to 10.08 and 9.99 GHz, respectively, indicating that SOAs with low or
optimized values for α might be preferred. This finding might also indicate that for light
sources with very narrow instantaneous linewidth, post amplification as presented in [3] might
lead to decreased coherence properties, depending on α.
The qualitative and quantitative agreement of experimental and simulation results
presented above show the validity of our model. Furthermore, these results clearly indicate
that the linewidth is NOT dominated by external noise sources, such as fluctuations of the
pump current, frequency or amplitude instabilities of the filter drive waveform or acoustic
vibrations, since such effects are not contained in the simulation. Our model can now be used
to identify the physical effects governing the instantaneous linewidth by successively
switching on and off the effects in the simulation, which is not possible in experiment. The
#143182 - $15.00 USD Received 28 Feb 2011; revised 15 Apr 2011; accepted 16 Apr 2011; published 20 Apr 2011(C) 2011 OSA 25 April 2011 / Vol. 19, No. 9 / OPTICS EXPRESS 8806
central question is if the linewidth is dominated by ASE, as is usually the case for
semiconductor and other lasers in the absence of external noise sources [11]. In Fig. 5(b), the
linewidth is displayed as obtained with ASE used only for initial seeding. Comparison with
Fig. 5(a) shows that the power spectrum is virtually unchanged without ASE. Rather, the
instantaneous lineshape is governed directly by the FDML dynamics due to the sweep filter
and gain action, dispersion and self-phase modulation.
-50 0 500
0.2
0.4
0.6
0.8
1(a)
Frequency [GHz]
Spectr
alin
ten
sity
[arb
. u
.]
-50 0 500
0.2
0.4
0.6
0.8
1(b)
Frequency [GHz]
Spe
ctr
alin
ten
sity
[arb
. u
.]
Fig. 5. (a) Simulated instantaneous power spectrum for α = 5 (red) and α = 0 (blue), the sweep filter transmission function is drawn in black. (b) The instantaneous power spectrum for α = 5
(red) and α = 0 (blue) but without ASE.
5. Conclusions
In conclusion, the instantaneous power spectrum of an FDML laser is theoretically and
experimentally investigated. The linewidth enhancement factor results in a frequency shift
relative to the sweep filter center frequency as well as a broadening, and third order dispersion
leads to an asymmetry of the instantaneous power spectrum. Good agreement between
simulation and measurement is obtained for both the non-detuned and the detuned case,
confirming the validity of our theoretical model. The simulations reveal that the instantaneous
linewidth is not governed by external noise sources or ASE, but results directly from the
FDML dynamics due to the sweep filter and gain action, dispersion and self-phase
modulation. Such a theoretical understanding of the effects governing the instantaneous power
spectrum is important for a further optimization of the linewidth and thus the coherence
properties of FDML lasers.
Acknowledgments
S. Todor and C. Jirauschek acknowledge support from Prof. P. Lugli at the TUM, and B.
Biedermann and R. Huber would like to acknowledge support from Prof. W. Zinth at the
LMU Munich. This work was supported by the German Research Foundation (DFG) within
the Emmy Noether program (JI 115/1-1 and HU 1006/2-1) and under DFG Grant No. JI
115/2-1, as well as by the European Union project FUN OCT (FP7 HEALTH, Contract No.
201880). S. Todor additionally acknowledges support from the TUM Graduate School.
#143182 - $15.00 USD Received 28 Feb 2011; revised 15 Apr 2011; accepted 16 Apr 2011; published 20 Apr 2011(C) 2011 OSA 25 April 2011 / Vol. 19, No. 9 / OPTICS EXPRESS 8807
