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A mode-locked fiber laser with a chirped grating mirror
15 January 2000
Optics Communications 174 Ž2000. 205–214 www.elsevier.comrlocateroptcom
A mode-locked fiber laser with a chirped grating mirror J.W. Haus
a,)
, M. Hayduk b, W. Kaechele c , G. Shaulov d , J. Theimer e, K. Teegarden f , G. Wicks f
a
Electro-Optics Program, The UniÕersity of Dayton, Dayton, OH 45469, USA Air Force Research Laboratory, Sensors Directorate, 25 Electronics Parkway, Rome, NY 13441-4515, USA NaÕal Research Laboratory, Optical Sciences DiÕision - Code 5671, 4555 OÕerlook AÕenue, SW, Washington, DC 20375, USA d Tyco Submarine Systems Ltd., 250 Industrial Way West, Eatontown, NJ 07724, USA e Air Force Research Laboratory, Sensors Directorate, Wright-Patterson AFB, Dayton, OH 45430, USA f Institute of Optics, UniÕersity of Rochester, Rochester, NY 14627, USA b
c
Received 23 September 1999; accepted 5 November 1999
Abstract A novel fiber laser was built using a multiple-quantum well mode-locking element and a chirped fiber grating to balance dispersion and nonlinearity. Energetic pulses as short as 2 ps were generated in the cavity and propagated in a fiber to determine the pulse characteristics. Laser cavity modeling and pulse propagation simulations are in good agreement with experiments. q 2000 Published by Elsevier Science B.V. All rights reserved. PACS: 42.60.F; 42.55.A; 42.81.Q Keywords: Modelocked lasers; Optical fibers; Nonlinear waves
1. Introduction Mode-locked Erbium-doped fiber lasers have potential application where requirements demand inexpensive, compact sources of ultrashort Žf 1 ps. pulses in the 1.55 mm wavelength regime w1x. There are several potential applications for such ultrashort pulse sources; fiber lasers are of interest as high
) Corresponding author. Tel.: Ž937. 229-2394, fax: Ž937. 2292097, e-mail: [email protected].
repetition-rate sources for telecommunications, local area networks or RF digitization. Various fiber laser designs have been demonstrated using passive mode-locking techniques operating both in normal and anomalous dispersion regimes. The passive mode-locked laser has been successfully operated by methods employing either a saturable absorber in the cavity or when the nonlinear properties of a fiber play the role of an ‘artificial’ saturable absorber. The latter include additive-pulse mode-locking ŽAPM. using nonlinear polarization rotation with a polarizer for nonlinear transmission or a nonlinear loop mirror. Recently semiconductor saturable absorbers have been used to mode-lock linear fiber cavities w2–5x,
0030-4018r00r$ - see front matter q 2000 Published by Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 9 9 . 0 0 6 9 2 - 6
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J.W. Haus et al.r Optics Communications 174 (2000) 205–214
Žusually it is semiconductor material in the form of bulk, multiple quantum well ŽMQW. or single quantum well saturable Bragg reflector ŽSBR... Bandgap engineering and semiconductor growth technology was used to fabricate saturable absorbers with accurate control of the device parameters such as absorption wavelength, saturation energy, and carrier relaxation time. Analysis of mode-locked laser experiments has successfully applied the complex Ginzburg–Landau equations w6,7x to obtain further information about the lasers’ operation. Specifically, Kutz et al. w7x made a direct comparison between the simulations and experimental results on a fiber laser for cavities with different dispersion values. The fiber laser was designed as a low loss cavity with two quantum wells grown as part of a SBR. In this paper we present experimental and numerical studies of a fiber laser mode-locked by multiple quantum well ŽMQW. saturable absorber. In contrast to Kutz et al. w7x our cavity has a high loss and its nonlinear dynamics is dominated by the saturable absorber element. Previously, we reported modified versions of a compact, polarization insensitive, mode-locked fiber laser using a GaInAsrAlInAs MQW as the saturable absorber mirror element, but without a chirped Bragg grating w8–10x. The output pulses from the laser were 7–50 ps and the autocorrelation of the temporal shape was Gaussian. Here we modify the earlier fiber laser cavity by inserting a chirped fiber Bragg grating into the cavity. The compact Bragg grating element adds dispersion to the cavity, which is required for balancing the dispersive nonlinearity for the soliton shaping mechanism. The chirped Bragg grating adds about two orders of magnitude more dispersion in the cavity than contributed by the fiber alone. Stable pulses are generated whose pulse widths are on the order of 2 ps. Physical parameters for the laser are used in numerical simulations and results are compared with the experimental data. Pulse characteristics of the laser output pulses are measured after they propagate through a length of optical fiber. The laser operation is compared with results of simulations by applying the complex Ginzburg–Landau equation. The numerical solution of the nonlinear Schroedinger equation with stimulated Raman scattering is used to model the fiber propagation.
2. Experimental Passively mode-locked fiber lasers which rely on the nonlinear effects of the fiber to generate subpicosecond pulses via APM action have produced the shortest pulses from an erbium doped fiber attaining widths as narrow as 30 fs w11x. Lasers such as the figure-eight and fiber ring are two common examples of the types of lasers which use APM, but these lasers are rarely self-starting and require ultrasensitive control of the polarization state within the cavity. This sensitivity causes these lasers to suffer more from environmental fluctuations requiring delicate adjustment of the cavity birefringence to maintain optimal operation. These fluctuations have prevented passively mode-locked fiber lasers from finding widespread applications outside of the laboratory. Recent work has centered on the development of a compact, passively mode-locked fiber laser, which does not suffer from adverse birefringence effects. The integration of semiconductor devices as bulk saturable absorbers into fiber lasers has proven to be an effective method for creating self-starting mode-locked fiber lasers w12,13x. Loh et al. were the first to use a multiple quantum well saturable absorber as a nonlinear mirror in an integrated fiber Fabry–Perot fiber laser cavity w14x. This scheme had a distinct advantage over previously integrated fiber lasers in that it exhibited an immunity to changes in the cavity birefringence. This laser was capable of generating picosecond pulses with peak powers in excess of 30 W making this configuration an attractive choice as a compact picosecond pulse source. The construction of our compact, polarization insensitive, passively mode-locked erbium-doped fiber laser employing a MQW saturable absorber is shown in Fig. 1. One end of the cavity has a chirped fiber Bragg grating and the other end is capped by a
Fig. 1. Schematic diagram of the laser cavity. The output is removed as the pulse travels in either direction in the cavity.
J.W. Haus et al.r Optics Communications 174 (2000) 205–214
Fig. 2. Linear absorbance of the MQW element. The sample has 75 quantum wells.
semiconductor saturable absorber. The erbium-doped fiber is pumped through the Bragg grating and the output coupler extracts power either before or after the pulse interacts with the saturable absorber. This is a high loss cavity with an estimated 60% loss per round trip. The saturable absorber was coupled with several different cavity configurations in an attempt to determine the optimal laser cavity design. Previous experiments determined that the laser without a chirped Bragg grating operated well outside the soliton regime w9,10x with the MQW absorption contributions dominating over the cavity dispersion. The chirped grating increases the cavity dispersion required for stable pulse shaping and the self-phase modulation is provided by the MQW element; together the two provide the required pulse shaping. The saturable absorber used in these experiments was formed by thin semiconductor layers which when alternately stacked upon one another form one-dimensional quantum wells. These wells are capable of absorbing radiation in the near infra-red region around 1555 nm producing the desired optical modulation. The quantum well region which was grown by molecular beam epitaxy was comprised of 75 periods of alternating layers of 10 nm In 0.53 Ga 0.47 As quantum wells and 10 nm In 0.52 Al 0.48 As barriers grown lattice matched on a semi-insulating InP substrate. This region encompassed a total thickness of only 1.5 mm. This sample differed from other non-
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linear mirrors in that the etalon effects as a result of ` the total thickness were of concern. A distributed Bragg reflector grown under the MQW region would eliminate these effects and has been shown to work effectively in many fiber lasers employing a MQW nonlinear mirror w14x. However, lack of epitaxial growth resources limited the ability to grow a high reflectance distributed Bragg reflector between the substrate and the quantum well region. Another option was pursued in order to reduce the overall cavity loss. The InP substrate was thinned to approximately 50 microns and then coated with gold. This simple technique forms a high reflectivity nonlinear mirror in the laser cavity. The linear absorbance of the MQW region was characterized using a white light source and a 0.275 meter triple grating spectrometer. Fig. 2 shows the measured linear absorbance spectrum for the 75 period quantum well sample. Our erbium-doped fiber laser cavity consists of 2.5 m of heavily doped erbium fiber and is pumped at 980 nm with a MOPA laser diode. A short length of standard single mode fiber was spliced to the erbium-doped fiber which increased the cavity length to 6.7 m, corresponding to a fundamental repetition rate of 14.9 MHz. The erbium-doped fiber was characterized as having 1.0 dBrmW of absorption and a dispersion of y14.3 " 4 ps 2rkm. A 980r1550 nm WDM couples the pump laser into the laser cavity. A dispersion compensating chirped fiber Bragg grating centered at 1556.6 nm with a reflectivity of 72% and a bandwidth of 10.8 nm is used as the output coupler. The MQW saturable absorber serves as the reflector at the opposite end of the cavity. A free
Fig. 3. Autocorrelation spectrum and optical spectrum of output pulses from the laser cavity. The pulse width is 3.4 ps for this case and the spectral width is 0.92 nm.
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J.W. Haus et al.r Optics Communications 174 (2000) 205–214
space optical section consisting of two microscope objectives couples the fiber to the saturable absorber. This setup had an optimum coupling efficiency of approximately 30%. The pulse could be extracted from the cavity either before or after the pulse passes through he saturable absorber. Fig. 3 is a typical experimental autocorrelation and optical spectrum from the laser. Both the pulse and the spectrum have a symmetrical shape that can be approximated by a hyperbolic-secant-squared profile. The autocorrelation FWHM is 6.1 ps, which corresponds to a pulse width of about 3.4 ps and the spectral width is 0.92 nm. Pulses as short as 2 ps are produced with a corresponding spectral width of 1.2 nm. The weak spectral sidebands observed in some of our data are attributed to the continuum radiation shed by the pulse w1x. It was also determined that the pulse width is insensitive to the pump energy and the propagation direction within the cavity. The timing stability of the passively mode-locked laser was examined using the RF spectrum techniques previously discussed by von der Linde w15x. The rms timing jitter is estimated by measuring the power contained in the phase noise sidebands at higher harmonics of the cavity’s fundamental repetition rate. The phase noise power is proportional to the square of the cavity harmonic and is therefore best measured at the higher harmonics. Fig. 4 shows
the RF power spectrum at the 10th harmonic of the cavity’s fundamental repetition rate. Using this data, the timing jitter is found to be approximately 6 ps. The integration time is 50 ms. The jitter is typical of passively mode-locked lasers and this corresponds to a fraction of the spacing between pulses. No further elements were used to reduce the jitter, but we synchronized the output of our fiber laser by injecting a signal from an actively mode-locked fiber laser through the fiber grating w9,10x.
3. Modeling and simulation To understand the principal mechanisms of our mode-locked fiber laser’s operation we developed numerical simulations of the laser cavity. Chirped pulse propagation in an optical fiber was similarly compared to computations to determine its properties. The simulations incorporate the dominant physical processes that govern the formation and stability of pulses in a mode-locked laser cavity; i.e. linear dispersion, nonlinear absorption and dispersion, losses, gain, and gain bandwidth. Optical pulses propagating in a fiber are affected by the pulse chirp and energy, as well as, the fiber dispersion, nonlinearity and Raman processes. 3.1. Laser caÕity model
Fig. 4. RF spectrum of the tenth harmonic of the cavity fundamental repetition rate. The timing jitter is found to be about 6 ps. ns10.
The theory of mode-locking is based on the complex Ginzburg–Landau equation w16–20x derived under the condition that round-trip changes to the intra-cavity pulse are small. The saturable absorption was modeled by two-level system rate equation, which accounts for the free carrier generated refractive index changes in the semiconductor absorber. We numerically demonstrate that this contribution causes an additional frequency shift, which is analogous to Raman self-frequency shift in fibers. The cavity dispersion is determined from the chirped fiber grating and the cavity losses are estimated for the cavity based on the properties of the MQW sample. The dispersive nonlinearity is an additional parameter that is adjusted to get the output pulse width as small as 2 ps.
J.W. Haus et al.r Optics Communications 174 (2000) 205–214
The average equation belongs to the class of generalized complex Ginzburg–Landau ŽCGL. equations and has a form
EA Ez
E2
ž
si D
ž
E t2
q Dg
q d3 < A< 2 A
/
E2 E t2
y Ž l y g . q g 3 < A< 2 y g5 < A< 4 A .
/
Ž 1. Here A is the optical field complex amplitude, scaled so that its square has units of optical power; t is the time variable and z is the distance divided by the cavity round-trip length L c . The first two terms on the right hand side are imaginary contributions; D is a cavity dispersion, d 3 is related to the effective Kerr nonlinearity of the cavity. The last four terms have real coefficients with the following physical meaning: Dg – gain and intracavity filter bandwidth limits; l y g is the net loss, where l and g are linear losses and gain per round-trip, respectively. Other physical parameters are defined as Dsy
Ž b 2 Lc q Dgr . 2
,
Dg s
g
V g2
1 q
V f2
,
d 3 s g Lc q ag 3 , where b 2 is a group velocity dispersion ŽGVD. for the fiber component, Lc is the optical cavity roundtrip length, Dgr is the dispersion from the chirped grating. Further parameters are: V g Žf . , gain Žfilter. bandwidth, the nonlinear absorption parameters are given by g 3 s q0rPs and g 5 s q0rPs2 , where q0 is the measured linear loss and Ps is the saturation power of the MQW element. The nonlinear dispersion has two contributions, one from the fiber g Lc , where g s 2.6 ŽW km.y1 is the fiber’s Kerr coefficient, and the other from the saturable absorber. The enhancement parameter, a , is adjusted to include the nonlinear index change of the saturable absorber. The physical parameters for our cavity are: L c s 13.4 m, Dgr s y13 ps 2 , b 2 s y15 ps 2rkm, Dg s 0.015 ps 2 , l s 0.6 and g s 0.59. The parameters q0 s 0.55 and Ps s 5 mW, give g 3 s 0.11 mWy1 , and g 5 s 0.022 ŽmW.y2 . The choice a s 5, yields d 3 s 0.5 mWy1 . Comparison between the dispersion from the chirped grating Dgr and from the fiber 2 b 2 L c reveals the that grating dispersion is domi-
209
nant by about two orders of magnitude. There are two open parameters in our model, the gain, g, and the enhancement parameter, a . Given the tight constraint from the absorber saturation parameters, the gain is restricted and must lie close to and below the value of the linear loss. This parameter is not critical in the analysis. The enhancement parameter is the main variable in adjusting the pulse width. Our parameter is of order unity. The simulations applied the beam propagation algorithm. The initial pulse envelope was chosen by using the hyperbolic secant profile solution with no chirp; this choice is motivated by the dominance of solitonic shaping in the laser cavity. The pulse is A s A 0 sech
t
ž/ t
;
Ž 2.
where the two parameters are related by the soliton area theorem A20t 2 s
2 < D<
d3
.
Ž 3.
A second relation is found from the energy balance of a stationary solution 1 Dg
1
2
y
g 3 A20 q
2
g A4 . Ž 4. 3 t 3 15 5 0 With the physical parameters given above the initial pulse full width at half maximum ŽFWHM. is t FW HM s 1.76t s 3.4 ps and P0 s A20 s 6.1 mW. This inigyls
Fig. 5. Pulse evolution using the CGLE in time and frequency regimes. The initial guess is close to a soliton. There is an additional chirp that develops on the pulse.
J.W. Haus et al.r Optics Communications 174 (2000) 205–214
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tial pulse evolves in the cavity using the full Ginzburg–Landau equations until a new steady-state is achieved. The initial pulse shape and spectrum are found on the left side of Fig. 5. The right side is the steady-state solution of Eq. Ž1., i.e. after 1000 round trips in the cavity. The initial and final solutions are quite close, which demonstrates that the pulse is well approximated by soliton parameters. The top right figure includes the pulse’s phase profile, which shows that the steady-state pulse is slightly chirped with a 2 value b s y dd tf2 t 2 s y0.25. Time-bandwidth product is 0.352, whereas experiment gives 0.387 before the saturable absorber and 0.429 after it Žsee Table 1., i.e. the output pulses have larger chirp. The final pulse width is t FW HM s 3.5 ps, which is close to our initial guess. Our simulations were also performed with the normal cavity dispersion, which would occur when the grating chirp is reversed in the cavity. For this case no stable pulses were formed using the parameters given above. This is also in accord with our experiments for the laser designed with this saturable absorber. The agreement between simulation and experiment is evidence that soliton shaping is important. Our initial experiments without the chirped grating in the cavity did provide mode-locked lasing, as mentioned earlier, but the pulse widths were longer and the autocorrelation shape was Gaussian and the optical spectrum was irregularly shaped. Our Table 1 Pulse characteristics after the propagation in fiber. Pump power is 72.4 mW Fiber length Žm.
A.C. pulsewidth Žps.
Spectral FWHA Žnm.
Power ŽmW.
DtDn
modeling parameters neglecting Dgr do not indicate mode-locking without soliton shaping w10,22,23x. 3.2. Pulse propagation The physical characteristics of the laser output pulses can be deduced by analyzing a pulse’s evolution through an optical fiber. We compare the experimental results against computations. The nonlinear Schroedinger equation, including stimulated Raman scattering, was simulated. Stimulated Raman scattering is relevant for the most energetic output pulses from the laser cavity; significant pulse shortening, as well as spectral changes were observed. Pulse propagation in the fiber is described by an equation with the form i
1 E 2u
Eu
q < u< 2 u s y
ia
uqtRu
E < u< 2
. Ž 5. Ez 2 ET2 2 ET Time is scaled to the pulse width t in this equation; the length has been scaled to the dispersion length L D s t 2r< b 2 <. b 2 is the group velocity dispersion ŽGVD. for the fiber. The field amplitude is scaled by u s A g L D , where g is the self-phase modulation coefficient for the fiber. a is the absorption coefficient scaled by the dispersion length; and t R s TR rt , where TR is related to the slope of the Raman gain w21x and is assumed to vary linearly with frequency in the vicinity of the carrier frequency. The field envelope was scaled to the amplitude of a fundamental soliton with the pulse width t . Parameters for the fiber were chosen appropriate for a standard singlemode fiber. The physical fiber parameters are: b 2 s y20 ps 2rkm, TR s 6 fs and the absorbance is 0.2 dBrkm. The initial pulse shape is q
(
u s N Ž sech T .
1q i b
;
After absorber 0 5.2 400 6.5 1800 9.0 2600 12.2 4400 8.7
1.02 0.92 0.56 0.42 0.36
0.1545 0.153 0.1374 0.126 0.1189
0.429 0.484 0.408 0.415 0.253
b is the chirp parameter and the amplitude is N s A 0 g L D . These parameters are used to fit the data for different pulse energies that are launched into the fiber.
Before absorber 0 5.2 400 4.8 1800 2.3 2600 2.5 4400 3.2
0.92 1.12 1.84 1.48 1.62
0.845 0.820 0.808 0.746 0.690
0.387 0.435 0.328 0.299 0.412
4. Propagation results
(
In the simulations the output pulses are chirped. This is already anticipated from the experiments by comparing the time-bandwidth product Žsee Table 1.
J.W. Haus et al.r Optics Communications 174 (2000) 205–214
211
Fig. 6. Autocorrelation spectra pulses propagating in a standard optical fiber, up to 4.4 km in length. The laser output was taken before the saturable absorber; the pulse energy at the laser output coupler is 56.7 pJ.
with the result expected for hyperbolic-secant shapes or for Gaussian shapes. To quantify the laser pulse’s properties it is launched in a standard optical fiber and its evolution is followed. By varying the pulse energies launched into the fiber, and measuring their pulse shape and optical spectrum after several chosen lengths of fiber, the pulse chirp parameter can be determined. The laser’s output power was varied by changing the pump laser power and by choosing the output coupler direction to extract pulses; i.e. either before or after the pulse passed through the saturable absorber. Although there are small differences in the
shape of the pulses and their spectrum in the two cases, we assume that their properties are identical, except for the pulse energy. The fiber launched pulses propagate up to 4.4 km in the fiber. In Fig. 6 the pulse is extracted from the cavity after passing through the saturable absorber; its average energy is 10.4 pJ. The experimental autocorrelation spectra are shown on the left. During propagation the pulse spreads to about 8 ps, then shortens again. Pulse widths and spectral widths are extracted from the data and presented in Table 1. The spreading is due to the chirp and continuum radiation that is shed during propagation. The corresponding opti-
Fig. 7. Autocorrelation spectra pulses propagating in a standard optical fiber, up to 4.4 km in length. The laser output was taken after passing through the saturable absorber; the pulse energy is 10.4 pJ.
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J.W. Haus et al.r Optics Communications 174 (2000) 205–214
Fig. 8. Pulse widths from the experiment together with the simulation data. The launched pulse in the simulations has a chirp fixed at y0.45. Pulses are coupled out of the cavity before and after passing through the saturable absorber. The pulses are assumed to differ only in energy. Square symbols and solid line show experiment and simulations results, respectively, for pulses after saturable absorber, whereas triangle symbols and dashed line show experiment and simulations results, respectively, for pulses before saturable absorber.
cal spectra, shown on the right, remain symmetrical, but become sharper with a strong central peak and oscillatory side bands. Fig. 7 shows the evolution of the laser output when it is extracted before the pulse passes through the saturable absorber. This is a more energetic pulse with 56.7 pJ. In this case the pulse shape narrows during propagation, as shown by the autocorrelation on the left, and a small satellite pulse separates from the main pulse. The corresponding optical spectra Žright. become asymmetric with a tail at longer wavelengths. Pulse widths extracted from this data are found in Table 1 and plotted in Fig. 8. The comparison between simulation and experiment was used to establish the chirp parameter. Simulations for a hyperbolic-secant pulse with a chirp corresponding the y0.45 provided a reasonable fit to the experiment. Data was taken at different pump powers and the same parameters fit the data over a wide range. The optical spectra in the simulation also change as the pulse propagates. The experimental results after 4.4 km propagation distance are shown in Fig. 9. The corresponding simulation results are found in Fig. 10. The dashed lines are the input pulse shapes or input pulse spectra. The high energy output pulse
shown in Figs. 9 and 10 have red shifts. In the main pulse walks off the initial pulse local time by about 15 ps; a small satellite pulse also separates from the main pulse in our simulations. The intensity of the satellite is much smaller than observed in the experiments, but the magnitude of the time separation is close for both cases. The Raman scattering is suppressed during propagation of the low energy pulse; both the experiment and simulation are shown in Figs. 9 and 10, respectively. Notably, although no frequency shift is observed in the spectrum, the spectral peak increases as the pulse propagates for both experiment and simulation. The resemblance between experiment and simulation in the details of the spectra for different energies is also very close. The oscillations in the optical spectra are due to a four-wave mixing effect of the chirped pulse. The central peak becomes narrower and higher as the pulse propagates, increasing the nonlinear effects. The oscillation effect is absent in our simulations for pulses without chirp and for low intensity pulses with chirp. We have used a larger body of experimental data by varying the laser pump intensity; the simple model we provide here provides a good quantitative description of the experiment. The chirp parameter deduced from
Fig. 9. Autocorrelation spectra and optical spectra of the experimental pulses after propagating through 4.4 km optical fiber. The left side shows the autocorrelation spectra and the right side is the corresponding optical spectra. The top figures are the more energetic output pulses extracted from the cavity before the saturable absorber and the lower figures are the pulses extracted after passing through the saturable absorber.
J.W. Haus et al.r Optics Communications 174 (2000) 205–214
Fig. 10. Pulse temporal shapes Žleft. and corresponding optical spectra Žright. of the pulses from the simulation after propagating through 4.4 km of optical fiber. The pulse envelopes have a hyperbolic-secant shape with a chirp parameter of y0.45. The energetic Žbefore saturable absorber. pulse are plotted on the top half of the figure and the less energetic Žafter saturable absorber. ones are on the bottom half. Dashed lines show the shape and spectrum of initial pulse.
the experiments, y0.45, is larger than found in the laser cavity simulation, y0.25, which we attribute to the simplicity of our model, especially the introduction of the enhancement parameter and the neglect of the carrier lifetime on the cavity dynamics. 5. Conclusions A mode-locked fiber laser design incorporating a MQW saturable absorber and a chirped fiber Bragg grating is reliably mode-locked with pulse widths as short as 2 ps and no polarization control is required in the cavity. This is a simple, compact fiber laser design. The noise is low leading to timing jitter around 6 ps and the laser can be synchronized by injecting an external pulse at a different wavelength w9,10x. The increased dispersion and self-phase modulation in the cavity sustain energetic pulses whose energy in the cavity corresponds to about twenty solitons in a standard fiber. The complex Ginzburg–Landau equation is used to model pulse dynamics in the cavity. Soliton parameters dominate in our cavity; we find the steadystate solution is a pulse whose temporal width is nearly the same as found from soliton perturbation
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theory and it has a chirp. The value of the chirp parameter derived from the simulation is smaller than found from the propagation experiments. This is due in part to parameters that are not easily extracted from the experiment and in part due to the simplicity of the model. The pulse width evolution in the fiber were compared against numerical simulations of chirped pulse propagation in the fiber and we find quantitative agreement with all the data: autocorrelation and optical spectra. Also, two other refinements need consideration. When the cavity elements are lumped, then continuum radiation is shed in the spectrum and amplified in the cavity. Second, in addition to the fast relaxation time the saturable absorber also has a slow relaxation, which will lower the discrimination between the peak and tails of the pulse. Given the simplicity of the present model though, it provides a description of the laser cavity with predictive power.
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Optics Communications 174 Ž2000. 205–214 www.elsevier.comrlocateroptcom
A mode-locked fiber laser with a chirped grating mirror J.W. Haus
a,)
, M. Hayduk b, W. Kaechele c , G. Shaulov d , J. Theimer e, K. Teegarden f , G. Wicks f
a
Electro-Optics Program, The UniÕersity of Dayton, Dayton, OH 45469, USA Air Force Research Laboratory, Sensors Directorate, 25 Electronics Parkway, Rome, NY 13441-4515, USA NaÕal Research Laboratory, Optical Sciences DiÕision - Code 5671, 4555 OÕerlook AÕenue, SW, Washington, DC 20375, USA d Tyco Submarine Systems Ltd., 250 Industrial Way West, Eatontown, NJ 07724, USA e Air Force Research Laboratory, Sensors Directorate, Wright-Patterson AFB, Dayton, OH 45430, USA f Institute of Optics, UniÕersity of Rochester, Rochester, NY 14627, USA b
c
Received 23 September 1999; accepted 5 November 1999
Abstract A novel fiber laser was built using a multiple-quantum well mode-locking element and a chirped fiber grating to balance dispersion and nonlinearity. Energetic pulses as short as 2 ps were generated in the cavity and propagated in a fiber to determine the pulse characteristics. Laser cavity modeling and pulse propagation simulations are in good agreement with experiments. q 2000 Published by Elsevier Science B.V. All rights reserved. PACS: 42.60.F; 42.55.A; 42.81.Q Keywords: Modelocked lasers; Optical fibers; Nonlinear waves
1. Introduction Mode-locked Erbium-doped fiber lasers have potential application where requirements demand inexpensive, compact sources of ultrashort Žf 1 ps. pulses in the 1.55 mm wavelength regime w1x. There are several potential applications for such ultrashort pulse sources; fiber lasers are of interest as high
) Corresponding author. Tel.: Ž937. 229-2394, fax: Ž937. 2292097, e-mail: [email protected].
repetition-rate sources for telecommunications, local area networks or RF digitization. Various fiber laser designs have been demonstrated using passive mode-locking techniques operating both in normal and anomalous dispersion regimes. The passive mode-locked laser has been successfully operated by methods employing either a saturable absorber in the cavity or when the nonlinear properties of a fiber play the role of an ‘artificial’ saturable absorber. The latter include additive-pulse mode-locking ŽAPM. using nonlinear polarization rotation with a polarizer for nonlinear transmission or a nonlinear loop mirror. Recently semiconductor saturable absorbers have been used to mode-lock linear fiber cavities w2–5x,
0030-4018r00r$ - see front matter q 2000 Published by Elsevier Science B.V. All rights reserved. PII: S 0 0 3 0 - 4 0 1 8 Ž 9 9 . 0 0 6 9 2 - 6
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J.W. Haus et al.r Optics Communications 174 (2000) 205–214
Žusually it is semiconductor material in the form of bulk, multiple quantum well ŽMQW. or single quantum well saturable Bragg reflector ŽSBR... Bandgap engineering and semiconductor growth technology was used to fabricate saturable absorbers with accurate control of the device parameters such as absorption wavelength, saturation energy, and carrier relaxation time. Analysis of mode-locked laser experiments has successfully applied the complex Ginzburg–Landau equations w6,7x to obtain further information about the lasers’ operation. Specifically, Kutz et al. w7x made a direct comparison between the simulations and experimental results on a fiber laser for cavities with different dispersion values. The fiber laser was designed as a low loss cavity with two quantum wells grown as part of a SBR. In this paper we present experimental and numerical studies of a fiber laser mode-locked by multiple quantum well ŽMQW. saturable absorber. In contrast to Kutz et al. w7x our cavity has a high loss and its nonlinear dynamics is dominated by the saturable absorber element. Previously, we reported modified versions of a compact, polarization insensitive, mode-locked fiber laser using a GaInAsrAlInAs MQW as the saturable absorber mirror element, but without a chirped Bragg grating w8–10x. The output pulses from the laser were 7–50 ps and the autocorrelation of the temporal shape was Gaussian. Here we modify the earlier fiber laser cavity by inserting a chirped fiber Bragg grating into the cavity. The compact Bragg grating element adds dispersion to the cavity, which is required for balancing the dispersive nonlinearity for the soliton shaping mechanism. The chirped Bragg grating adds about two orders of magnitude more dispersion in the cavity than contributed by the fiber alone. Stable pulses are generated whose pulse widths are on the order of 2 ps. Physical parameters for the laser are used in numerical simulations and results are compared with the experimental data. Pulse characteristics of the laser output pulses are measured after they propagate through a length of optical fiber. The laser operation is compared with results of simulations by applying the complex Ginzburg–Landau equation. The numerical solution of the nonlinear Schroedinger equation with stimulated Raman scattering is used to model the fiber propagation.
2. Experimental Passively mode-locked fiber lasers which rely on the nonlinear effects of the fiber to generate subpicosecond pulses via APM action have produced the shortest pulses from an erbium doped fiber attaining widths as narrow as 30 fs w11x. Lasers such as the figure-eight and fiber ring are two common examples of the types of lasers which use APM, but these lasers are rarely self-starting and require ultrasensitive control of the polarization state within the cavity. This sensitivity causes these lasers to suffer more from environmental fluctuations requiring delicate adjustment of the cavity birefringence to maintain optimal operation. These fluctuations have prevented passively mode-locked fiber lasers from finding widespread applications outside of the laboratory. Recent work has centered on the development of a compact, passively mode-locked fiber laser, which does not suffer from adverse birefringence effects. The integration of semiconductor devices as bulk saturable absorbers into fiber lasers has proven to be an effective method for creating self-starting mode-locked fiber lasers w12,13x. Loh et al. were the first to use a multiple quantum well saturable absorber as a nonlinear mirror in an integrated fiber Fabry–Perot fiber laser cavity w14x. This scheme had a distinct advantage over previously integrated fiber lasers in that it exhibited an immunity to changes in the cavity birefringence. This laser was capable of generating picosecond pulses with peak powers in excess of 30 W making this configuration an attractive choice as a compact picosecond pulse source. The construction of our compact, polarization insensitive, passively mode-locked erbium-doped fiber laser employing a MQW saturable absorber is shown in Fig. 1. One end of the cavity has a chirped fiber Bragg grating and the other end is capped by a
Fig. 1. Schematic diagram of the laser cavity. The output is removed as the pulse travels in either direction in the cavity.
J.W. Haus et al.r Optics Communications 174 (2000) 205–214
Fig. 2. Linear absorbance of the MQW element. The sample has 75 quantum wells.
semiconductor saturable absorber. The erbium-doped fiber is pumped through the Bragg grating and the output coupler extracts power either before or after the pulse interacts with the saturable absorber. This is a high loss cavity with an estimated 60% loss per round trip. The saturable absorber was coupled with several different cavity configurations in an attempt to determine the optimal laser cavity design. Previous experiments determined that the laser without a chirped Bragg grating operated well outside the soliton regime w9,10x with the MQW absorption contributions dominating over the cavity dispersion. The chirped grating increases the cavity dispersion required for stable pulse shaping and the self-phase modulation is provided by the MQW element; together the two provide the required pulse shaping. The saturable absorber used in these experiments was formed by thin semiconductor layers which when alternately stacked upon one another form one-dimensional quantum wells. These wells are capable of absorbing radiation in the near infra-red region around 1555 nm producing the desired optical modulation. The quantum well region which was grown by molecular beam epitaxy was comprised of 75 periods of alternating layers of 10 nm In 0.53 Ga 0.47 As quantum wells and 10 nm In 0.52 Al 0.48 As barriers grown lattice matched on a semi-insulating InP substrate. This region encompassed a total thickness of only 1.5 mm. This sample differed from other non-
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linear mirrors in that the etalon effects as a result of ` the total thickness were of concern. A distributed Bragg reflector grown under the MQW region would eliminate these effects and has been shown to work effectively in many fiber lasers employing a MQW nonlinear mirror w14x. However, lack of epitaxial growth resources limited the ability to grow a high reflectance distributed Bragg reflector between the substrate and the quantum well region. Another option was pursued in order to reduce the overall cavity loss. The InP substrate was thinned to approximately 50 microns and then coated with gold. This simple technique forms a high reflectivity nonlinear mirror in the laser cavity. The linear absorbance of the MQW region was characterized using a white light source and a 0.275 meter triple grating spectrometer. Fig. 2 shows the measured linear absorbance spectrum for the 75 period quantum well sample. Our erbium-doped fiber laser cavity consists of 2.5 m of heavily doped erbium fiber and is pumped at 980 nm with a MOPA laser diode. A short length of standard single mode fiber was spliced to the erbium-doped fiber which increased the cavity length to 6.7 m, corresponding to a fundamental repetition rate of 14.9 MHz. The erbium-doped fiber was characterized as having 1.0 dBrmW of absorption and a dispersion of y14.3 " 4 ps 2rkm. A 980r1550 nm WDM couples the pump laser into the laser cavity. A dispersion compensating chirped fiber Bragg grating centered at 1556.6 nm with a reflectivity of 72% and a bandwidth of 10.8 nm is used as the output coupler. The MQW saturable absorber serves as the reflector at the opposite end of the cavity. A free
Fig. 3. Autocorrelation spectrum and optical spectrum of output pulses from the laser cavity. The pulse width is 3.4 ps for this case and the spectral width is 0.92 nm.
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J.W. Haus et al.r Optics Communications 174 (2000) 205–214
space optical section consisting of two microscope objectives couples the fiber to the saturable absorber. This setup had an optimum coupling efficiency of approximately 30%. The pulse could be extracted from the cavity either before or after the pulse passes through he saturable absorber. Fig. 3 is a typical experimental autocorrelation and optical spectrum from the laser. Both the pulse and the spectrum have a symmetrical shape that can be approximated by a hyperbolic-secant-squared profile. The autocorrelation FWHM is 6.1 ps, which corresponds to a pulse width of about 3.4 ps and the spectral width is 0.92 nm. Pulses as short as 2 ps are produced with a corresponding spectral width of 1.2 nm. The weak spectral sidebands observed in some of our data are attributed to the continuum radiation shed by the pulse w1x. It was also determined that the pulse width is insensitive to the pump energy and the propagation direction within the cavity. The timing stability of the passively mode-locked laser was examined using the RF spectrum techniques previously discussed by von der Linde w15x. The rms timing jitter is estimated by measuring the power contained in the phase noise sidebands at higher harmonics of the cavity’s fundamental repetition rate. The phase noise power is proportional to the square of the cavity harmonic and is therefore best measured at the higher harmonics. Fig. 4 shows
the RF power spectrum at the 10th harmonic of the cavity’s fundamental repetition rate. Using this data, the timing jitter is found to be approximately 6 ps. The integration time is 50 ms. The jitter is typical of passively mode-locked lasers and this corresponds to a fraction of the spacing between pulses. No further elements were used to reduce the jitter, but we synchronized the output of our fiber laser by injecting a signal from an actively mode-locked fiber laser through the fiber grating w9,10x.
3. Modeling and simulation To understand the principal mechanisms of our mode-locked fiber laser’s operation we developed numerical simulations of the laser cavity. Chirped pulse propagation in an optical fiber was similarly compared to computations to determine its properties. The simulations incorporate the dominant physical processes that govern the formation and stability of pulses in a mode-locked laser cavity; i.e. linear dispersion, nonlinear absorption and dispersion, losses, gain, and gain bandwidth. Optical pulses propagating in a fiber are affected by the pulse chirp and energy, as well as, the fiber dispersion, nonlinearity and Raman processes. 3.1. Laser caÕity model
Fig. 4. RF spectrum of the tenth harmonic of the cavity fundamental repetition rate. The timing jitter is found to be about 6 ps. ns10.
The theory of mode-locking is based on the complex Ginzburg–Landau equation w16–20x derived under the condition that round-trip changes to the intra-cavity pulse are small. The saturable absorption was modeled by two-level system rate equation, which accounts for the free carrier generated refractive index changes in the semiconductor absorber. We numerically demonstrate that this contribution causes an additional frequency shift, which is analogous to Raman self-frequency shift in fibers. The cavity dispersion is determined from the chirped fiber grating and the cavity losses are estimated for the cavity based on the properties of the MQW sample. The dispersive nonlinearity is an additional parameter that is adjusted to get the output pulse width as small as 2 ps.
J.W. Haus et al.r Optics Communications 174 (2000) 205–214
The average equation belongs to the class of generalized complex Ginzburg–Landau ŽCGL. equations and has a form
EA Ez
E2
ž
si D
ž
E t2
q Dg
q d3 < A< 2 A
/
E2 E t2
y Ž l y g . q g 3 < A< 2 y g5 < A< 4 A .
/
Ž 1. Here A is the optical field complex amplitude, scaled so that its square has units of optical power; t is the time variable and z is the distance divided by the cavity round-trip length L c . The first two terms on the right hand side are imaginary contributions; D is a cavity dispersion, d 3 is related to the effective Kerr nonlinearity of the cavity. The last four terms have real coefficients with the following physical meaning: Dg – gain and intracavity filter bandwidth limits; l y g is the net loss, where l and g are linear losses and gain per round-trip, respectively. Other physical parameters are defined as Dsy
Ž b 2 Lc q Dgr . 2
,
Dg s
g
V g2
1 q
V f2
,
d 3 s g Lc q ag 3 , where b 2 is a group velocity dispersion ŽGVD. for the fiber component, Lc is the optical cavity roundtrip length, Dgr is the dispersion from the chirped grating. Further parameters are: V g Žf . , gain Žfilter. bandwidth, the nonlinear absorption parameters are given by g 3 s q0rPs and g 5 s q0rPs2 , where q0 is the measured linear loss and Ps is the saturation power of the MQW element. The nonlinear dispersion has two contributions, one from the fiber g Lc , where g s 2.6 ŽW km.y1 is the fiber’s Kerr coefficient, and the other from the saturable absorber. The enhancement parameter, a , is adjusted to include the nonlinear index change of the saturable absorber. The physical parameters for our cavity are: L c s 13.4 m, Dgr s y13 ps 2 , b 2 s y15 ps 2rkm, Dg s 0.015 ps 2 , l s 0.6 and g s 0.59. The parameters q0 s 0.55 and Ps s 5 mW, give g 3 s 0.11 mWy1 , and g 5 s 0.022 ŽmW.y2 . The choice a s 5, yields d 3 s 0.5 mWy1 . Comparison between the dispersion from the chirped grating Dgr and from the fiber 2 b 2 L c reveals the that grating dispersion is domi-
209
nant by about two orders of magnitude. There are two open parameters in our model, the gain, g, and the enhancement parameter, a . Given the tight constraint from the absorber saturation parameters, the gain is restricted and must lie close to and below the value of the linear loss. This parameter is not critical in the analysis. The enhancement parameter is the main variable in adjusting the pulse width. Our parameter is of order unity. The simulations applied the beam propagation algorithm. The initial pulse envelope was chosen by using the hyperbolic secant profile solution with no chirp; this choice is motivated by the dominance of solitonic shaping in the laser cavity. The pulse is A s A 0 sech
t
ž/ t
;
Ž 2.
where the two parameters are related by the soliton area theorem A20t 2 s
2 < D<
d3
.
Ž 3.
A second relation is found from the energy balance of a stationary solution 1 Dg
1
2
y
g 3 A20 q
2
g A4 . Ž 4. 3 t 3 15 5 0 With the physical parameters given above the initial pulse full width at half maximum ŽFWHM. is t FW HM s 1.76t s 3.4 ps and P0 s A20 s 6.1 mW. This inigyls
Fig. 5. Pulse evolution using the CGLE in time and frequency regimes. The initial guess is close to a soliton. There is an additional chirp that develops on the pulse.
J.W. Haus et al.r Optics Communications 174 (2000) 205–214
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tial pulse evolves in the cavity using the full Ginzburg–Landau equations until a new steady-state is achieved. The initial pulse shape and spectrum are found on the left side of Fig. 5. The right side is the steady-state solution of Eq. Ž1., i.e. after 1000 round trips in the cavity. The initial and final solutions are quite close, which demonstrates that the pulse is well approximated by soliton parameters. The top right figure includes the pulse’s phase profile, which shows that the steady-state pulse is slightly chirped with a 2 value b s y dd tf2 t 2 s y0.25. Time-bandwidth product is 0.352, whereas experiment gives 0.387 before the saturable absorber and 0.429 after it Žsee Table 1., i.e. the output pulses have larger chirp. The final pulse width is t FW HM s 3.5 ps, which is close to our initial guess. Our simulations were also performed with the normal cavity dispersion, which would occur when the grating chirp is reversed in the cavity. For this case no stable pulses were formed using the parameters given above. This is also in accord with our experiments for the laser designed with this saturable absorber. The agreement between simulation and experiment is evidence that soliton shaping is important. Our initial experiments without the chirped grating in the cavity did provide mode-locked lasing, as mentioned earlier, but the pulse widths were longer and the autocorrelation shape was Gaussian and the optical spectrum was irregularly shaped. Our Table 1 Pulse characteristics after the propagation in fiber. Pump power is 72.4 mW Fiber length Žm.
A.C. pulsewidth Žps.
Spectral FWHA Žnm.
Power ŽmW.
DtDn
modeling parameters neglecting Dgr do not indicate mode-locking without soliton shaping w10,22,23x. 3.2. Pulse propagation The physical characteristics of the laser output pulses can be deduced by analyzing a pulse’s evolution through an optical fiber. We compare the experimental results against computations. The nonlinear Schroedinger equation, including stimulated Raman scattering, was simulated. Stimulated Raman scattering is relevant for the most energetic output pulses from the laser cavity; significant pulse shortening, as well as spectral changes were observed. Pulse propagation in the fiber is described by an equation with the form i
1 E 2u
Eu
q < u< 2 u s y
ia
uqtRu
E < u< 2
. Ž 5. Ez 2 ET2 2 ET Time is scaled to the pulse width t in this equation; the length has been scaled to the dispersion length L D s t 2r< b 2 <. b 2 is the group velocity dispersion ŽGVD. for the fiber. The field amplitude is scaled by u s A g L D , where g is the self-phase modulation coefficient for the fiber. a is the absorption coefficient scaled by the dispersion length; and t R s TR rt , where TR is related to the slope of the Raman gain w21x and is assumed to vary linearly with frequency in the vicinity of the carrier frequency. The field envelope was scaled to the amplitude of a fundamental soliton with the pulse width t . Parameters for the fiber were chosen appropriate for a standard singlemode fiber. The physical fiber parameters are: b 2 s y20 ps 2rkm, TR s 6 fs and the absorbance is 0.2 dBrkm. The initial pulse shape is q
(
u s N Ž sech T .
1q i b
;
After absorber 0 5.2 400 6.5 1800 9.0 2600 12.2 4400 8.7
1.02 0.92 0.56 0.42 0.36
0.1545 0.153 0.1374 0.126 0.1189
0.429 0.484 0.408 0.415 0.253
b is the chirp parameter and the amplitude is N s A 0 g L D . These parameters are used to fit the data for different pulse energies that are launched into the fiber.
Before absorber 0 5.2 400 4.8 1800 2.3 2600 2.5 4400 3.2
0.92 1.12 1.84 1.48 1.62
0.845 0.820 0.808 0.746 0.690
0.387 0.435 0.328 0.299 0.412
4. Propagation results
(
In the simulations the output pulses are chirped. This is already anticipated from the experiments by comparing the time-bandwidth product Žsee Table 1.
J.W. Haus et al.r Optics Communications 174 (2000) 205–214
211
Fig. 6. Autocorrelation spectra pulses propagating in a standard optical fiber, up to 4.4 km in length. The laser output was taken before the saturable absorber; the pulse energy at the laser output coupler is 56.7 pJ.
with the result expected for hyperbolic-secant shapes or for Gaussian shapes. To quantify the laser pulse’s properties it is launched in a standard optical fiber and its evolution is followed. By varying the pulse energies launched into the fiber, and measuring their pulse shape and optical spectrum after several chosen lengths of fiber, the pulse chirp parameter can be determined. The laser’s output power was varied by changing the pump laser power and by choosing the output coupler direction to extract pulses; i.e. either before or after the pulse passed through the saturable absorber. Although there are small differences in the
shape of the pulses and their spectrum in the two cases, we assume that their properties are identical, except for the pulse energy. The fiber launched pulses propagate up to 4.4 km in the fiber. In Fig. 6 the pulse is extracted from the cavity after passing through the saturable absorber; its average energy is 10.4 pJ. The experimental autocorrelation spectra are shown on the left. During propagation the pulse spreads to about 8 ps, then shortens again. Pulse widths and spectral widths are extracted from the data and presented in Table 1. The spreading is due to the chirp and continuum radiation that is shed during propagation. The corresponding opti-
Fig. 7. Autocorrelation spectra pulses propagating in a standard optical fiber, up to 4.4 km in length. The laser output was taken after passing through the saturable absorber; the pulse energy is 10.4 pJ.
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J.W. Haus et al.r Optics Communications 174 (2000) 205–214
Fig. 8. Pulse widths from the experiment together with the simulation data. The launched pulse in the simulations has a chirp fixed at y0.45. Pulses are coupled out of the cavity before and after passing through the saturable absorber. The pulses are assumed to differ only in energy. Square symbols and solid line show experiment and simulations results, respectively, for pulses after saturable absorber, whereas triangle symbols and dashed line show experiment and simulations results, respectively, for pulses before saturable absorber.
cal spectra, shown on the right, remain symmetrical, but become sharper with a strong central peak and oscillatory side bands. Fig. 7 shows the evolution of the laser output when it is extracted before the pulse passes through the saturable absorber. This is a more energetic pulse with 56.7 pJ. In this case the pulse shape narrows during propagation, as shown by the autocorrelation on the left, and a small satellite pulse separates from the main pulse. The corresponding optical spectra Žright. become asymmetric with a tail at longer wavelengths. Pulse widths extracted from this data are found in Table 1 and plotted in Fig. 8. The comparison between simulation and experiment was used to establish the chirp parameter. Simulations for a hyperbolic-secant pulse with a chirp corresponding the y0.45 provided a reasonable fit to the experiment. Data was taken at different pump powers and the same parameters fit the data over a wide range. The optical spectra in the simulation also change as the pulse propagates. The experimental results after 4.4 km propagation distance are shown in Fig. 9. The corresponding simulation results are found in Fig. 10. The dashed lines are the input pulse shapes or input pulse spectra. The high energy output pulse
shown in Figs. 9 and 10 have red shifts. In the main pulse walks off the initial pulse local time by about 15 ps; a small satellite pulse also separates from the main pulse in our simulations. The intensity of the satellite is much smaller than observed in the experiments, but the magnitude of the time separation is close for both cases. The Raman scattering is suppressed during propagation of the low energy pulse; both the experiment and simulation are shown in Figs. 9 and 10, respectively. Notably, although no frequency shift is observed in the spectrum, the spectral peak increases as the pulse propagates for both experiment and simulation. The resemblance between experiment and simulation in the details of the spectra for different energies is also very close. The oscillations in the optical spectra are due to a four-wave mixing effect of the chirped pulse. The central peak becomes narrower and higher as the pulse propagates, increasing the nonlinear effects. The oscillation effect is absent in our simulations for pulses without chirp and for low intensity pulses with chirp. We have used a larger body of experimental data by varying the laser pump intensity; the simple model we provide here provides a good quantitative description of the experiment. The chirp parameter deduced from
Fig. 9. Autocorrelation spectra and optical spectra of the experimental pulses after propagating through 4.4 km optical fiber. The left side shows the autocorrelation spectra and the right side is the corresponding optical spectra. The top figures are the more energetic output pulses extracted from the cavity before the saturable absorber and the lower figures are the pulses extracted after passing through the saturable absorber.
J.W. Haus et al.r Optics Communications 174 (2000) 205–214
Fig. 10. Pulse temporal shapes Žleft. and corresponding optical spectra Žright. of the pulses from the simulation after propagating through 4.4 km of optical fiber. The pulse envelopes have a hyperbolic-secant shape with a chirp parameter of y0.45. The energetic Žbefore saturable absorber. pulse are plotted on the top half of the figure and the less energetic Žafter saturable absorber. ones are on the bottom half. Dashed lines show the shape and spectrum of initial pulse.
the experiments, y0.45, is larger than found in the laser cavity simulation, y0.25, which we attribute to the simplicity of our model, especially the introduction of the enhancement parameter and the neglect of the carrier lifetime on the cavity dynamics. 5. Conclusions A mode-locked fiber laser design incorporating a MQW saturable absorber and a chirped fiber Bragg grating is reliably mode-locked with pulse widths as short as 2 ps and no polarization control is required in the cavity. This is a simple, compact fiber laser design. The noise is low leading to timing jitter around 6 ps and the laser can be synchronized by injecting an external pulse at a different wavelength w9,10x. The increased dispersion and self-phase modulation in the cavity sustain energetic pulses whose energy in the cavity corresponds to about twenty solitons in a standard fiber. The complex Ginzburg–Landau equation is used to model pulse dynamics in the cavity. Soliton parameters dominate in our cavity; we find the steadystate solution is a pulse whose temporal width is nearly the same as found from soliton perturbation
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theory and it has a chirp. The value of the chirp parameter derived from the simulation is smaller than found from the propagation experiments. This is due in part to parameters that are not easily extracted from the experiment and in part due to the simplicity of the model. The pulse width evolution in the fiber were compared against numerical simulations of chirped pulse propagation in the fiber and we find quantitative agreement with all the data: autocorrelation and optical spectra. Also, two other refinements need consideration. When the cavity elements are lumped, then continuum radiation is shed in the spectrum and amplified in the cavity. Second, in addition to the fast relaxation time the saturable absorber also has a slow relaxation, which will lower the discrimination between the peak and tails of the pulse. Given the simplicity of the present model though, it provides a description of the laser cavity with predictive power.
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