This thesis is consisted of four chapters. Chapter 1 is an introduction of history of fiber lasers and our motivation for demonstrating this research. In chapter 2, it will describe the theory of mode-locked fiber lasers including the active mode-locking and some methods about how to stabilize ML-EDFLs. In chapter 3, it presents experimental setup and results. Analyses of the results are also included. In chapter 4, by using VPI, the simulation results of our ML-EDFL structure are shown. It will be also compared with the results of the experiment in chapter3. Finally, in chapter 5, we make a brief conclusion and discuss results that we have successfully achieved. Some possible improvements on the laser configuration are also proposed.
Mirror Mirror
CHAPTER 2
The Theory of Mode-Locked Lasers
2.1 Theory of mode-locked lasers
Two techniques used for generating short optical pulses from lasers are known as Q-switching and mode locking. Mode locking means that the phases of longitudinal modes in the laser cavity have fixed relation. In order to explain it, at first, we can assume that every optical field in the laser cavity has irregular intensity and different resonance frequencies as shown in figure 2.1. It means that the laser is not a single mode laser. [18]
Figure 2.1 initial state of optical field in laser cavity
However, when the laser starts to become stable, there are only some modes can exist in the cavity. The output of the laser consists of these modes. These modes which can exist in the cavity are dependent on the cavity length. We could express it as following equation,
L c
q T
q
= ⋅
=
−
=
∆ω ω +1 ω 2π π (2.1.1) where L is cavity length. Analyzing in time domain, we assume that there are three
frequencies ωq,ωq+1,ωq+2 can exist in the laser cavity and they have the same amplitude. The interval between any two of these frequencies is∆ω. The electron field can be expressed as
] Re[
)
( 1 (ω φ1) 2 (ω 1 φ2) 3 (ω 2 φ3)
ε t = E ej qt+ +E ej q+t+ +E ej q+ t+ (2.1.2) The output of electric field is the function of time. The intensity can be written as following these phases are the function of time and their interval are not fixed, the ratio of beat frequency of these three frequencies will change in time.
Figure 2.2 (a) the electric field of individual frequency ωq,ωq+1,ωq+2 and they have fixed relation of the phase. (b) total electric field (c) total optical field.
(a)
(b)
(c)
0 T 2T 3T 2π/T ωq-1 ωq ωq+1
0 T ωq
ωc
2π/T ωq-1 ωq+1
0 T 2T ωq
ωc
2π/T ωq-1 ωq+1
However, if the electric field of these three frequencies all have their maximum values at T=0 and fixed phases; we can get a periodic peak power at the output as shown in figure 2.2. The mediate figure is expressed as total electric field. The peak of the total electric field is triple times value of the single electric field. Therefore, we can extrapolate that the maximum intensity of total optical field is nine times value of the single optical field. Then, the pulse trains occur.
Analyzing in frequency domain, the distribution of electric field of one round trip time is shown as figure 2.3 (a). The range of the distribution of electric field is only in 0 < t < T, where T is round trip time and ε(t) is zero at other times. Using the Fourier transformation, it can be transformed to frequency distribution, E(ω) as shown in figure 2.3 (a) at the right hand side. Where ε(t) is the pulse signal which consists of sine waves in the cavity and its pulsewidth τp <<T .
Figure 2.3 the distribution of electric field (left hand side) of once (a), two (b), three round trip times (c) of the cavity and its optical spectrum (right hand side) (a) one period: signal ε(t)
(b) two period: signal ε(2)(t)
(c) three period: signal ε(3)(t)
Optical spectrum E(ω)
Optical spectrum E(2) (ω)
Optical spectrum E(3) (ω)
In this example, we can take a notice thatωc, carrier frequency, is unnecessary to equal to the longitudinal modes ωq inside the cavity. The ε(2)(t) of two round trip time is twice reiteration of the same signal ε(t) of one round trip time. We could write it asε(2)(t)≡ε(t)+ε(τ −T), where T is a delay time in time domain. After
The distribution of intensity in frequency domain could be written as )
Equation (2.1.5) can be illustrated as figure 2.3 (b) at the right hand side. After more than twice of the round trip times, only frequency which is also one of the longitudinal modes of the cavity can exist in this cavity as shown in figure 2.3 (a)-(c) and the others which cannot be allowed to exist will become zero.
Fiber lasers typically operate simultaneously in a large number of longitudinal modes falling within the gain bandwidth. Therefore, the total optical field can be written as among 2M+1 modes. If all modes operate independently of each other with no definite phase relationship among them, the interference terms in the total intensity
|E(t)|2 averages out to zero. This is the situation of multimode CW fiber laser [19].
Active mode locking occurs when phases of various longitudinal modes are synchronized such that the phase difference between any two neighboring modes is locked to a constant value φ such that φm-φm-1=φ, as we explain previously. Such
a phase relation implies that φm=mφ+φ0 which means that they have fixed phase relation. The mode frequency ωm can be written as ωm=ω0+2mπΔν. If we use these relations in equation (2.1.6) and assume for simplicity that all modes in the cavity have the same amplitude E0, we can express it analytically. The total intensity is given as
2
2 2
2 0
sin [(2 1) / 2]
| ( ) |
sin ( / 2)
M t
E t E
t
π ν φ π ν φ
+ ∆ +
= ∆ + (2.1.7) It is a periodic function of time with period τr =1/Δν, which is just the round trip time inside the cavity. φ is the phase difference between any two neighborhood modes. The simulation of the total intensity |E(t)|2 is shown in figure 2.4 for nine coupled modes (M=4).
Figure 2.4 Illustration of the mode-locked pulse train as M=4
Typically, mode-locked laser could be distinguished into active mode-locked laser and passive mode-locked laser. The generation of femtosecond short pulses could be easily made by passive mode–locked laser using nonlinear effect in the cavity. There are many kinds of passive mode-locking which can be used, such as mode-locking with slow or fast saturation absorber [20], additive pulse mode-locking (APM) [21], nonlinear polarization rotation mode-locking (P-APM) [22], and Kerr lens mode-locking (KLM) [23] etc la.
Active mode locking could be achieved through directly modulation of the intensity or phase of light by using the active components, such as Electro-Optical modulator and acoustic-optical modulator. It will be discussed more details in following section 2.2. Beside, to overcome the influence of thermal and environmental perturbation becomes more and more important for a short pulse and high repetition rate fiber laser used for optical communication system. In order to increase the bit error rate (BER), it is necessary to decrease the timing jitter and amplitude jitter of fiber laser, especially operating at ultra-high repetition rate.
Therefore, we will discuss how to stabilize the active mode-locked fiber laser in section 2.3.
2.2 Active mode-locked lasers
2.2.1Amplitude modulation mode-locking
Amplitude modulation mode-locking is a method to produce a short pulse train and high repetition rate by directly modulating the optical amplitude of the light. It can be analyzed both in the time and frequency domains [24].In the time domain, the amplitude modulation provides a time dependent loss so that only the pulses which pass through the modulator at the lowest loss will exist. As the pulses pass through the modulator continually, the pulsewidth will get shorter and shorter. However, shorter pulses will experience larger dispersion and finally the two forces balance each other to form the steady state pulse shape. In this way, the modulation time period must be equal to a multiple of the roundtrip time for producing stable pulses. Figure 2.5 shows the active mode-locking process in the time domain.
Figure 2.5 Principle of active mode-locking explained in the time domain [24]
In frequency domain, we can assume that the center frequency of signal gain profile is ν0, and the amplitude of the central mode without amplitude modulation is expressed as ε( )t =E0cos(ω0t). The transform function of the active amplitude modulator, which controls the loss of light in the cavity, can be written as
exp[ (1 cos )]
am m m
t% = −∆ − ω t ≅exp[− ∆12 mωm2 2t ], where ωm =2πfm =2πN∆f and f m is modulation frequency, such as the signal after modulation can be expressed as following,
where ∆ is modulation index. It is clear from this equation that the center m frequency ν0 induces two side modes with fixed phase relationship (ν0± fm) while it experiences modulation of active modulator. Similarly, after these two side modes which are made by center frequency ν0 go through the active amplitude modulator, there will also increase other new side modes (ν0±2fm) with fixed phase relationship. These sidebands can injection-lock the neighboring modes sequentially and finally the mode-locking is achieved. (See figure 2.6)
The modes that are separated every f will be phase-locked, and short pulses m can be formed in the time domain. When N equals to 1, the laser is mode locked at the fundamental repetition rate. When N is an integer greater than 1, the laser is harmonic mode locked. We will discuss it detailed in the following section. (See section 2.2.3)
Figure 2.6 Principle of actively mode-locking explained in the frequency domain
Sideband coupling
ω
Mode-locked frequency behavior
1 2 3 4 5 6
-1 -0.5 0.5 1
2.2.2 Phase modulation mode-locking
Phase modulation mode-locking is a method to produce a short pulse train by modulating the optical phase. It can also be analyzed both in the time and the frequency domains [25].
In the time domain, the phase modulator provides a periodic phase change for the optical pulse. If the pulsewidth is much smaller than the modulation period, the change of the optical phase produced by the phase modulator can be expressed as:
... neglected . The first order term
dt dφ
will influence the central frequency of the pulses and shifting magnitude of influence depends on its value. Therefore, if t ≠0
dt dφ
, the central frequency of optical pulse will be changed. In another word, the pulse will experience smaller gain and center frequency will still be changed, if t
dt dφ
is still not equal to zero. This is unstable and will not lase. Only the pulses which pass through the PM modulator and experience maximum gain is at t=0
dt dφ
, and its every round trip is able to be stable. Then, it will lase. (See figure 2.7)
Figure 2.7 Time domain of phase modulation
As regarding the second order term 2 2 2
Also, it will affect the optical bandwidth of the pulses. The effect can be expressed in mathematics by:
Figure 2.8 Development of pulse train in time domain by superposition of modes
In the frequency domain, we can assume that the central frequency isν0. When it passes through the phase modulator, the electric field of the pulse can be written as:
))E(t)=E0cos(2πν0t+∆mcos(2πfmt where f is the modulating frequency of phase modulator, m ∆ is the modulation m index, and J is the n-th order Bessel function. If n ν0 is one of the harmonic modes in the laser cavity and f is N times magnitude of the fundamental harmonic m
frequency of the cavity, these harmonic modes (ν0 +Kfm) will have the fixed phase relation with the ν0, where K=±1, ±2, ±3, , ,etc la. Therefore, all these harmonic modes will have fixed phase relation. In time domain, these harmonic modes will create constructive interference at periodic time and destructive interference at other times by injection-locking. (See figure 2.8)
2.2.3 Harmonic mode-locking
A continuous wave erbium ring laser can be actively mode-locked by using an amplitude or phase modulator to generate pulses at the modulation frequency f m
c where f is the cavity mode-spacing frequency, c is the speed of light, L is the cavity c length and n is the refractive index of the cavity. These pulses have a round trip time of tr, which is related to f and the pulse width τ as following, c
c tr 2nL
= (2.2.7) This is known as fundamental mode-locking, and it produces pulses at repetition rate equal to f . c
The cavity mode-spacing frequency of a typical laser cavity is of the order of 0.5~6MHz. To increase the pulse repetition rate, pulses could be produced at integer harmonics of the cavity mode-spacing by modulating at a frequency f , given by m
p
m Pf
f = (2.2.8) where P is an integer representing the number of longitudinal modes locked, and ranges from a few hundred to tens of thousand. This is known as harmonic mode
locking, these longitudinal modes with equal intervalPf is called as supermodes, p and its new round trip time shows as following,
P c tr 2nL 1
⋅
= (2.2.9) In 1970s, the KS (Kuizenga and Siegman [26]) theory predicted that with amplitude mode-locking the time bandwidth product is 0.441 for a chirp-free Gaussian pulse and 0.315 for a Sech2 pulse. Furthermore, it states that the pulsewidth τ is inversely proportional to
( )
δ 1/4 and(
fm⋅∆f3dB)
1/4, so that gain band width of the laser cavity and K is a pulse shape-dependent constant. It is clear from this equation that with increasing modulation frequency and increasing modulation amplitude, the optical pulsewidth will be narrowed.However, though we can use this way to promote higher repetition rates, the drawback of harmonic mode-locking is not stable for a long time. We will discuss it later.
Actually, the smallest pulsewidth and chirp of the pulses can be estimated by using the time-bandwidth product of transform limited. For chirp free Gaussian sharp, time bandwidth product is 0.441. For Sech2 sharp, time bandwidth product is 0.315.
We can use this transform-limited to appraise our laser. However, the exact estimate is not possible since the cavity dispersion is not considered in equation (2.2.10).
2.2.4 Rational harmonic mode-locking
Using mode-locked fiber ring laser to generate pulses at very high repetition rates requires a small cavity path length and high-speed modulators and signal generators, because of long cavity length. An alternative solution is using a rational mode-locking technique to generate pulses at rational harmonics of the fundamental locking frequency, thus enabling the use of longer cavities and slower components to generate pulses at higher bit rate [27].
Ahmed and Onodera [28] introduced the idea of mode-locking a laser cavity at rational harmonics of the modulator frequency in order to generate pulses at a repetition rate higher than the modulator frequency. This is achieved by slightly detuning the cavity frequency f so that it is now related to the modulator frequency c
f expressed as following equation (2.2.11), m
c where k is the rational number which can have values range from 1 to no more than 20. This leads to a pulse repetition rate of f , given by p
c
p kp f
f =( ±1)⋅ (2.2.12) Therefore, rationally mode-lock ring lasers generate optical pulses at a repetition rate which is k times value of the modulation frequency.
Rational mode-locking of erbium-doped fiber lasers at repetition rate of up to 200GHz have been reported [29], using values of k as high as 15. Actually, when k ≧ 3, problems arise because different pulses experience different losses in the modulator. This may lead to large amplitude fluctuations between consecutive pulses in the pulse train. These fluctuations greatly increase the error rate and are not tolerable in optical communication systems.
2.3 Stabilization of mode-locked laser
A stable optical short pulse source with a high repetition rate is very important to ultrahigh speed optical communication. The active mode-locking of erbium-doped fiber lasers is one of the most attractively potential ways of achieving this, because it can produce a transform-limited, picosecond pulse train with ultrahigh repetition.
However, problems arise with fiber lasers due to their susceptibility to mechanical vibrations and temperature variations affecting the length of the fiber cavities. The resulting instability and the difficulty of tuning the repetition frequency have been the major barriers to the application in communications where minor perturbations can cause intolerable bit errors, especially operating at ultra-high speed repetition rate.
Therefore, the additional scheme of stability of mode-locked fiber laser becomes necessary.
2.3.1 Regenerative mode-locking
The cavity length in an actively mode-locked fiber laser is typically much longer than that of other mode-locked lasers. When a small perturbation is applied to the cavity, the absolute frequency compared with that in ordinary lasers will change. This means that it is difficult to maintain the optimum operational conditions over a long period, although it is possible to generate clean short pulses in a short period.
Also, this problem can be overcome with a regenerative mode-locking technique [30]. The regeneratively mode-locked fiber laser has long term stability since the modulation frequency is extracted from the mode-locked pulse itself. The structure is shown as figure 2.9. Regenerative mode-locking is accomplished by feeding back the longitudinal self-beat signal which is detected by a high speed photodetector and a high Q filter. Because the phase between the pulse and the modulation signal is adjusted, the pulses will always experience maximum transmission when they pass
1480 nm LD
through the modulator. Thus, complete mode-locking is achieved automatically because the ideal feedback signal, which reflects the instantaneous frequency change between the longitudinal beats, is used as the modulation frequency even when some perturbation are applied.
Figure 2.9 Experimental setup for the harmonically and regeneratively mode-locked erbium-doped fiber laser [30].
Actually, regenerative mode-locking has several similarities to passive mode-locking. Lasing is initiated by the noise through the use of artificial loss modulation, and the technique automatically adjusts the modulation frequency as the cavity length changes. However, the technique has one drawback in that the repetition rate fluctuates with time in a free running condition when the optical path length in the cavity is not stable. Therefore, it is necessary to have another mechanism to stabilize the repetition rate of the regeneratively mode-locked fiber laser at the fixed frequency, such as phase-locked loop.
2.3.2 Phase-locked loop (PLL)
In 1992, for active mode-locked fiber laser, the stabilization mechanism used relied on locking the electrical phase of output optical pulse to that of the drive source have been demonstrated [7]. To stabilize the laser, they developed the phase-locking circuit shown within the dashed line. (See figure 2.10)
A length of erbium-doped fiber wound on a piezoelectric crystal (PZT). It works as voltage control oscillator (VCO). A fraction of the laser output is detected by a photodiode, amplified and filtered with a narrow bandwidth filter to generate a sinusoidal signal. The frequency mixer work as a phase detector which compares the phase of the pulse train and that of the drive source (synthesizer). The error signals will occur from the mixer while there are influences affecting the fiber cavity with time. Simultaneously, the displacement of the PZT which was controlled by error signals feedback adjusts the length of the fiber cavity to compensate the perturbation of the fiber cavity caused by mechanical vibrations or temperature variations.
Figure 2.10 Mode-locked Erbium-doped fiber ring laser and stabilization scheme (Dashed line), where “PC” is polarization controller and “SID” is step index fiber [7]
PZT
In this paper, once the output pulse is locked in this way, it is possible to tune the synthesizer by ±5~6 KHz with a very little change in either pulsewidth or bandwidth. Actually, this tuning range corresponds to the change of the fiber length wound on PZT, which is also limited by the amplifier output voltage range. Besides, the speed of the phase-locking circuit is limited by the response of the high voltage amplifier. As a result, only thermal drifts and vibrations of a few hundreds of Hertz can be effectively suppressed. However, it is enough to overcome the influence of generally environmental perturbation to the fiber laser.
Recently, in order to stabilize mode-locked fiber laser with ultra-high speed repetition rate used for communications, phase-locking technology is publicly used. It is also utilized in regeneratively mode-locked fiber laser to stabilize its repetition rate
Recently, in order to stabilize mode-locked fiber laser with ultra-high speed repetition rate used for communications, phase-locking technology is publicly used. It is also utilized in regeneratively mode-locked fiber laser to stabilize its repetition rate