Chapter 1 Introduction
1.3 Organization
This thesis contains four chapters. Chapter1 gives the history of mode-locked laser in the development of experiments and theories, from solid state lasers to fiber lasers. Then we introduce the motivation of researching the long-cavity mode-locked fiber laser.
In chapter 2, we give a quick derivation for the master equation in the active mode-locking regime. Then we replace the active modulation term by the self-amplitude modulation term to describe the fast absorber scheme in passive mode-locking. In chapter 2.3, we introduce additive the original pulse mode-locking technique (APM), which requires the use of two resonators and the output pulse was from the interference of the output pulses of each resonator.
Then we explain the principle of Polarization APM (P-APM) mode-locking, which is a method without requiring an additional cavity, but by utilizing the nonlinear phase shifts of the two circular polarization states inside the fiber cavity.
In chapter 2.4, we introduce how the square pulse can be generated in the passive mode-locked fiber laser.
In chapter 2.5, we first show the characteristics of chirped Gaussian pulses in the linear chirp case only. Then we extend the formula to the second and third order chirp cases.
In chapter 3, we use the nonlinear polarization rotation to achieve an all-fiber mode-locked laser with a 400m-long single mode fiber as the main cavity. We show the experiment results of generating square pulses and also its stability by increasing the pump power, and short pulse. In chapter 3.2, we use the spectral filtering method to study the pulse characteristics in the square pulse and short pulse regimes. In chapter 3.3, the dual-wavelength phenomenon is observed.
We characterize it to be the additive of the square pulse and the short pulse. In chapter 3.4, experiment results of pulse compression on the filtered pulses will be demonstrated.
Finally, the conclusion of the thesis work and possible future work in the future is given in chapter 4.
Reference
[1.1] K. Gürs and R. Müller, “Breitband-modulation durch Steuerung der emission eines optischen masers (Auskopple-modulation),” Phys. Lett., vol. 5, pp. 179–181, 1963.
[1.2] K. Gürs, “Beats and modulation in optical ruby lasers,” in Quantum Electronics III, P.
Grivet and N. Bloembergen, Eds. New York: Columbia Univ. Press, pp. 1113–1119, 1964.
[1.3] H. Statz and C. L. Tang, “Zeeman effect and nonlinear interactions between oscillating laser modes,” in Quantum Electronics III, P. Grivet and N. Bloembergen, Eds. New York: Columbia Univ. Press, pp. 469–498, 1964.
[1.4] L. E. Hargrove, R. L. Fork, and M. A. Pollack, “Locking of He–Ne laser modes induced by synchronous intractivity modulation,” Appl. Phys. Lett., vol. 5, pp. 4–6, 1964
[1.5] H. W. Mocker and R. J. Collins, “Mode competition and self-locking effects in a Q-switched ruby laser,” Appl. Phys. Lett., vol. 7, pp. 270–272, 1965.
[1.6] D. I. Kuizenga and A. E. Siegman, “Modulator frequency detuning effects in the FM mode-locked laser,” IEEE J. Quantum Electron., vol. QE-6, pp. 803–808, 1970.
[1.7] E. P. Ippen, C. V. Shank, and A. Dienes, “Passive mode locking of the cw dye laser,”
Appl. Phys. Lett., vol. 21, pp. 348–350, 1972.
[1.8] H. A. Haus, “Theory of mode locking with a fast saturable absorber,” J. Appl. Phys., vol.
46, pp. 3049–3058, 1975.
[1.9] E. P. Ippen, H. A. Haus, and L. Y. Liu, “Additive pulse mode locking,” J. Opt. Soc.
Amer. B, Opt. Phys., vol. 6, pp. 1736–1745, 1989.
[1.10] R. J. Mears, L. Reekie, I. M. Jauncey, D. N. Payne “Low-noise erbium-doped fibre amplifier operating at 1.54μm,” Electron. Lett., vol. 23, pp. 1026-1028 , 1987.
[1.11] I. M. Jauncey, L. Reekie, R. J. Mears, D. N. Payne, C. J. Rowe, D. C. J. Reid, I.
Bennion, C. Edge. “Narrow-linewidth fibre laser with integral fibre grating,” Electron.
Lett., vol. 22, pp. 987, 1986.
[1.12] U. Sharma, C. S. Kim, and J. U. Kang, “Highly stable tunable dual-wavelength Q-switched fiber laser for DIAL applications,” IEEE Photon. Technol. Lett., vol. 16, pp.1277 , 2004.
[1.13] L. Chen, M. Zhang, C. Zhou, Y. Cai, L. Ren and Z. Zhang, “Ultra-low repetition rate linear-cavity erbium-doped fibre laser modelocked with semiconductor saturable absorber mirror,” Electron. Lett., vol. 23, pp. 1026-1028 , 2009.
[1.14] A. Killi, J. Dörring, U. Morgner, M. J. Lederer, J. Frei,and D. Kopf, “High speed electro-optical cavity dumping of mode-locked laser oscillators,” Opt. Express 13, pp.
1916-1922, 2005.
[1.15] S. H. Cho, B. E. Bouma, E. P. Ippen, and J. G.Fujimoto, “Low-repetition-rate high-peak-power Kerr-lens mode-locked TiAl2O3 laser with a multiple-pass cavity,”
Opt. Lett., vol. 24, pp. 417-419 , 1999.
[1.16] K. H. Fong, S. Y. Kim, K. Kazuro, H. Yaguchi, and S. Y. Set, “Generation of low-repetition rate high energy picosecond pulses from a single-wall carbon nanotube mode-locked fiber laser,” Opt. Amp. and Appl. Conf., paper OMD4, 2006.
[1.17] W. H. Renninger, A. Chong, and F. W. Wise, “Highly-chirped dissipative solitons in anomalous-dispersion fiber lasers,” CLEO., paper CTuFF6, May 2008,
[1.18] W. H. Renninger, A. Chong, and Frank W. Wise, “Giant-chirp oscillators for short-pulse fiber amplifiers,” Opt. Lett., vol. 33, pp. 3025–3027, 2008
[1.19] S. Kobtsev, S. Kukarin, and Y. Fedotov, “Ultra-low repetition rate mode-locked fiber laser with high-energy pulses,” Opt. Express, vol. 16, pp. 21936–21941, 2008.
Chapter 2
Principle of passive mode-locked fiber laser
2.1 Master equation in the active mode-locked regime
We follow the treatment in the paper by H. A. Haus, (“Mode-Locking of Lasers”, [2.1]) to derive the mater equation for explaining the mode-locking mechanism. The following equation shows the optical field amplitude change of the n-th mode in each roundtrip pass. The first term in the right hand side means the change of the n-th mode after passing through the gain medium and the linear loss of the whole cavity. The second term is caused by the amplitude modulator in each roundtrip. is the amplitude of the n-th mode,
The following two terms show how the modulator makes the (n-1)-th and (n+1)-th modes contribute to the n-th mode after modulation.
Fig. 2.1 shows the gain profile and linear loss. The modes which experience positive net gain ( -loss) can exist in the cavity.
Fig. 2.1 The gain distribution of different modes with linear loss [2.1]
The following three assumptions will be used:
(1) The frequency dependent gain can be expanded to second order of . (2) Discrete frequency spectrum is replaced by a continuum spectrum, as a
continuous function of .
(3) becomes in the continuum limit,
supposing that the spectrum is dense.
We can get a differential equation describing mode-locking for the amplitude change per round trip.
In steady state, , we can get a Gaussian pulse solution, which demonstrates how a mode-locked pulse can be formed theoretically.
2.2 Fast Saturable Absorber Mode-Locking
Fig. 2.2 Schematic of passively mode-locked laser with a fast saturable absorber and the time dependence of the pulse and the net gain
In passive mode-locked lasers, the modulator was replaced by a saturable absorber. The theoretical model was first developed by H. A. Haus in the reference [2.2], who had done fantastic work in the fast saturable absorber
mode-locking theory. When the relaxation time of the absorber was much longer than the pulse generated in the laser, the slow saturable absorber mode-locking theory needs to be used [2.3].
Here we first introduce the simple model to explain how the fast saturable absorber works. First, the transmission function is given by the follow
expression:
(2.2-1) where
If the saturation is relatively weak, we can expand this formula by Taylor expansion.
, is the effective area of the mode. Thus the transmission function becomes
, e is the self-amplitude modulation (SAM) coefficient. We replace the active modulator term by the fast saturable absorber term in the master equation (2.2-1), and the term merge into the loss l term. We finally have
.
This second order differential equation simply described the mechanism of fast saturable absorber mode-locking. The solution is a hyperbolic secant function of time:
, where
and . is the pulse-width and we can see that it is inverse proportional to the gain bandwidth and proportional to peak gain .
2.3 Addittive Pulse Mode-Locking (APM)
The additive pulse mode-locking is a technique for passive mode-locked lasers [2.4] and is usually used to generate pulses with the width from picosecond to femtosecond [2.5], [2.6]. The early APM techniques contained two resonators as shown in Fig. 2.3. It was later found that the APM effect can be also achieved in a single resonator cavity by utilizing the two light polarizations. This is illustrated in Fig. 2.4.
Fig. 2.3 The scheme of APM
From Fig. 2.4, the linearly polarized light is transformed into the elliptically polarized light by a wave plate. It then passes through an isotropic Kerr–medium which will rotate the elliptically polarized light by an intensity dependent angle.
If the output light is again linearly polarized by an analyzer, the output light of the system will be intensity dependent. In this way this effect realizes an equivalent fast saturable absorber. It is a good method for the fiber laser systems since all the components in this scheme can be made as the fiber-optic type.
Consider the two circularly polarized eigen-modes in the nonlinear fiber [2.7] as follows:
(2.3-1) Here are the fields of two circularly polarized waves and is the Kerr nonlinearity coefficient. Equation (2.3-1) shows how these two polarization states affect each other in the Kerr medium. It also shows that both circular polarization modes acquire different phase shifts that are proportional to intensities. The angle of the elliptic polarization will rotate an angle as shown in Fig. 2.4 [2.7]. When the elliptic polarization light passes through the analyzer, the intensity dependent transmission effect is basically equivalent to a saturable absorber. This mechanism is called the polarization additive pulse mode-locking (P-APM), which is a common technique used in passive mode-locked fiber lasers.
Fig. 2.4 Rotation of elliptical polarization in an isotropic Kerr-medium
2.4 Square pulse generation
Nanosecond square pulses can be generated in the Er-doped mode-locked fiber laser with a long cavity. This is due to the peak power clamping effect [2.8].
Equation (2.4-1) shows that the round trip transmission of the light depends on , the beat length of the birefringent element, the cavity length L and the azimuth angle of the polarization-dependent isolator with respect to the fast axis. Here is the rotation angle of the polarization controllers. The net transmission is a nonlinear function of the power as shown in Fig 2.5. It is caused by the nonlinear polarization rotation (NPR) effect.
( 2 . 4 - 1 ) The effective beat length will change with the input power, as indicated by equation (2.4-2). is the power dependent beat length
(2.4-2) This will be contained in the equation (2.4-3) and thus affect the power dependent switching power .
(2.4-3) Equation (2.4-3) shows how L and are related to the switching power
. We can find that a longer cavity length results a lower switching power.
This is also the reason why square pulses are much easier generated in a long
fiber cavity.
Fig. 2.5 shows the simulation normalized transmission T in the cases of
and . When the transmission profiles
do not have a flat top shape. But when , the power peaks near p=2.5 and p=8.5 clearly exhibit a flat top. So it means by adjusting the polarization controllers one can change the transmission profile, letting the system operated in the flat top power range to generate square pulses.
Fig. 2.5 Transmission T in the cases of and [2.9]
2.5 High Order Chirp
Considering a Gaussian pulse with a linear chirp , the electric field in the spectral domain with normalize amplitude can be written as
If is the inverse Fourier transform of , i.e., the time domain electric field, then
The variable is determined by the spectral width. We can use the following equation (2.5-1) to calculate by substituting into the full-width-half-maximum spectral width, :
(2.5-1) Since
the Gaussian shape pulse has the pulse-width given by
It shows how the linear spectral chirp term contributes to the time domain width.
This is the main effect why the optical pulses get wider in fibers. If , the optical pulse is called transform limited. The time bandwidth product for a Gaussian pulse becomes
=
Now we want to estimate the second order chirp of the pulse. The idea comes from the linear chirp case. We consider a Gaussian pulse with a linear chirp and a second order chirp (caused by third order dispersion). The electric field in the spectral domain with normalize amplitude is given by equation (2.5-2), assuming in the moving frame of the pulse.
(2.5-2) Supposing that is small enough, let us multiply the electric field by a
narrowband filter function with a central frequency . Denoting
and expanding the phase term to third order around the filter central frequency by Taylor expansion, one has:
We will neglect the third order term in the right hand side by assuming is small enough. We will also assume the filter bandwidth is much narrower than the pulse spectral width.
Thus
By applying the inverse Fourier transform, the term leads to constant
phase shift, and the term leads to a shift in time domain. Thus we can simplify the expression for IFT as follows:
where . The pulse-width in the time domain will be
(2.5-3) where (in the unit of second square) and
(in the unit of inverse of second square). Here
is given by
and is the filter band width in the unit of nm. is the difference of the central wavelength of the pulse and the central wavelength of the filtered optical band in angular frequency. We should note that the central frequency of original spectral field is zero (2.5-2), therefore and .Suppose is much smaller than . By neglecting the term in equation (2.5-3), one has
(2.5-4) We get a simple formula to describe the higher order chirp effect on the filtered pulses. The relation in equation (2.5-4) predicts that the filtered pulse-width will have a linear dependence with the filter central wavelength if there is a second order chirp.
Now we extend the formula to more general case which including third order spectral chirp. Similarly, we first consider the following electric field equation
(2.5-5) Let the electric field multiply a filter function with a central wavelength which is , then we expand the phase term to second order term by Taylor expansion around the filter central frequency and do the Inverse Fourier Transform translating the spectral field to electric field in time domain.
The resulted pulse-width in time domain is given by,
(2.5-6) Equation (2.5-6) give the pulse-width of filter band which have a filter band central frequency difference to central frequency to original spectrum, i.e. . By expanding (2.5-6) in the power series of and neglecting third order term and higher order terms of it, we can get the following equation that is parabolic function to .
(2.5-7)
Reference
[2.1] H. A. Haus, “Mode-Locking of Lasers,” IEEE J. Selected Topics Quantum Electron. vol.
6, pp. 1173-1185, 2000.
[2.2] H. A. Haus, “Theory of mode locking with a fast saturable absorber,” J. Appl. Phys., vol.
46, pp. 3049–3058, 1975.
[2.3] E. G. Arthurs, D. J. Bradley and A. G. Roddie, “Buildup of picosecond pulse generation in passively mode-locked rhodamine dye lasers,” Appl. Phys. Lett., vol. 23, pp. 88–90, 1973.
[2.4] E. P. Ippen, H. A. Haus, and L. Y. Liu, “Additive pulse mode locking,” J. Opt. Soc. Am., vol. 6 , pp. 1736 , 1989.
[2.5] J. Mark, L. Y. Liu, K. L. Hall, H. A. Haus, and E. P. Ippen, “Femtosecond pulse generation in a laser with a nonlinear external resonator”, Opt. Lett., vol. 14 , Issue 1, pp.
48-50 , 1989.
[2.6] J. Chen, J. W. Sickler, E. P. Ippen and F. X. Kärtner, ” High repetition rate, low jitter, low intensity noise, fundamentally mode-locked 167 fs soliton Er-fiber laser,” Opt. Lett., vol. 32, pp. 1566-1568, 2007.
[2.7] H. A. Haus and E. P. Ippen, “Mode-locked Fiber Ring Lasers,” OSA TOPS on Ultrafast Electronics and Optoelectronics, vol.13, pp. 6-12, 1997
[2.8] Y. Li, X. Gu, M. Yan, E. Wu and H. Zeng, “Square nanosecond mode-locked Er-fiber laser synchronized to a picosecond Yb-fiber laser,” Opt. Express, vol. 17, pp.
4526-4532, 2009.
[2.9] V. J. Matsas, T. P. Newson, D. J. Richardson, D. N. Payne, “Self-starting passively mode-locked fiber ring soliton laser exploiting nonlinear polarization rotation,” Electron.
Lett., vol. 28, pp. 1391-1393, 1992.
Chapter 3
Experimental results of long-cavity length mode-locked fiber lasers
3.1 Pulse measurement
3.1-1 Experimental setup
Fig. 3.1 Experimental setup of the long-cavity length fiber laser and the output pulse measurement
Our experiment setup is illustrated in Fig. 3.1. The fiber laser system is 30% output
pumped by 980nm diode lasers. The pump lights are coupled into the fiber cavity by using WDM couplers. We use a 1.65 meter long, Er-doped gain fiber with the pump absorption coefficient of 80 dB/m. The 400 m single mode fiber (SMF)contributes to the main cavity length. The cavity also contains two polarization controllers and one polarization dependent isolator to implement the nonlinear polarization rotation technique for achieving mode-locking.
The generated square pulse is shown in Fig. 3.2 (a), which has the pulse-width of 3.57 ns. The optical spectrum is shown in Fig. 3.2 (b) with the spectral width of 10.5 nm.
0 5 10 15 20
0.0000 0.0005 0.0010 0.0015 0.0020 0.0025
Intensity (Watt)
Time (nanosecond)
pulsewidth~3.576ns
Fig. 3.2 (a) Square pulse measured by oscilloscope
1480 1500 1520 1540 1560 1580 1600 1620 1640 1660
Fig. 3.2 (b) Optical spectrum in the square pulse regime
0 100 200 300 400
Fig. 3.3 (a) Output power versus pump power
0 5 10 15 20 -0.00005
0.00000 0.00005 0.00010 0.00015 0.00020 0.00025 0.00030
Intensity (Watt)
Time (nanosecond)
Pulse width versus pump power 1.631 ns
3.163 ns 4.31 ns
Fig. 3.3 (b) Square pulses in different pump power
Fig. 3.3 (c) Pulse-width of square pulses and output power in different pump power
Fig. 3.3(a) shows that the output power is increasing linearly with the pump power. Fig. 3.3 (b) and Fig. 3.3 (c) shows that the width of output square pulse is increasing linearly with the pump power while peak power of the square pulses remain constant. The mode-locked region starts at 133 mW pump power and remains mode-locked at least till the pump power of 450 mW (limited by our available pump power). The obtained maximum pulse energy of 142 nJ breaks the record in [3.1.1], which only has the highest pulse energy of 120 nJ without pulse breaking.
At the pump power of 133 mw and the output power of 21 mw, this laser can also produce short pulses with the pulse-width of 647 ps and with a much wider spectral width of 25 nm as shown in Fig. 3.4 (a) (b).
0 2 4 6 8 10
-0.00004 0.00000 0.00004 0.00008
Intensity (Watt)
Time (nanosecond)
pulsewidth~647.7(ps)
Fig. 3.4 (a) Short pulse measured by oscilloscope
1500 1530 1560 1590 1620 1650 -70
-60 -50 -40 -30
dBm
Wavelength (nm)
spectral width~24nm
Fig. 3.4 (b) Optical spectrum of the short pulse
3.2 Spectral filtering measurement
The previous section shows that the laser can be operated in the square pulse and short pulse regimes. In this section, we want to estimate the spectral linear chirp and second order chirp characteristics of the laser output pulses by sliding-filtering the optical spectrum.
3.2-1 In the square pulse regime
Fig. 3.5 Experimental setup of the spectral filtering method
5 10 15 20
-0.0001 0.0000 0.0001 0.0002 0.0003 0.0004
Intensity (Watt)
Time (nanosecond)
Pulse width~3.9ns
Fig. 3.6 The square pulse measured by oscilloscope
Fig. 3.5 shows the experimental setup for measuring the filtered pulses. Fig.
3.6 showed the original square pulse with the pulse-width of 3.9 nanosecond.
We put an external optical filter outside the cavity to filter the pulse shown in Fig. 3.6. By selecting the different optical bands as shown in Fig. 3.7 and 3.8, the resulted pulses are recorded by the oscilloscope as shown in Fig. 3.9
The original spectrum has the spectral width of 11.2 nm in Fig. 3.7. The central wavelength is at 1564.3nm. By using a filter bandwidth (FBW) of 1.6nm and by varying the filter central wavelength (FCW) from 1558.7 nm to 1569.9 nm, the resulted pulses are shown in Fig. 3.9. We can observe that these nanosecond filtered pulses still maintain square pulse-like shapes.
In Table 3.1, the numerical values of the pulse-width have a concave down parabolic-like trace versus the central wavelength of the band-pass filter as illustrated in Fig. 3.10. It is caused by linear chirp and second order chirp and third order chirp. We should note that the FCWs have been shifted in another reference frame as shown in Fig. 3.10. The 1564.3 nm is shifted to 0 nm, so does other filter bands. The blue dash line is the fitting parabolic curve of the black solid line and the fitting coefficients are put in Table 3.1 and we will use them to calculate these spectral chirp parameters by result of Chap 2.5. These calculated chirp parameters is shown in Table 3.2. Linear chirp is in the unit of square of seconds (s). We can observe that the largest pulse-width occurs in the bands near the central wavelength.
1520 1540 1560 1580 1600 1620 1640
The original pulse has central wavelength 1564.3nm spectral width~11.2nm,FBW=1.6nm
dBm
Wavelength (nm)
Central wavelength of each filter band
1558.7nm
Fig. 3.7 Optical spectrum with FBW=1.6nm in square pulse regime
Fig. 3.7 Optical spectrum with FBW=1.6nm in square pulse regime