Chapter 2 Theory
2.1 Optical Injection Locking
In 1980 Kobayashi and Kimura used GaAs lasers to demonstrate the injection locking experimental results. [1] Using two devices with close wavelength is necessary. In 2005 Lukas Chrostowski, Xiaoxue Zhao and Connnie J Chang-Hasnain used 1.55μm VCSEL to demonstrate the resonance frequency enhanced from 7GHz up to ~50GHz [2]. The technique is that one laser the master laser (ML) is external light source to inject photons in another laser the slave laser (SL). The resonant frequency enhancement might result from the external light injecting into the slave laser as a cavity increasing the photons in the slave laser. Using the technique has some advantages, such as side-mode suppression ratio(SMSR), enhancement of the relaxation oscillation, improvement of the nonlinear characteristic and chip frequency and so forth. It makes direct modulation more adequate for many applications.
The direct modulation of semiconductor lasers can be used for transmitting subcarrier-multiplexed signals at low cost of using injection locking to improve the nonlinear characteristic.[3] The resonance frequency enhancement by using the injection locking technique means that we will have much larger bandwidth can be used in optical communication. The technique can adequately improve the dynamic operation characteristic so that injection-locked lasers is a practicable way to use for future networks.
Figure 2.1.1 Optical Injection-Locking On Semiconductor Laser
2.2 Theoretical Model
The dynamics of injection-locked lasers are simulated by rate equations which couple the temporal variations of the amplitude, the phase and the number of carriers of the slave laser:
)
where A(t) is the field amplitude, defined as A2 (t)=S(t), where S(t) is the photon number.φ (t) is the phase difference between the temporal laser field of the slave laser and master laser. N(t) is the carrier number and J is the injection current. Nth is the threshold carrier number, g is the linear gain coefficient, γ P is the photon decay rate, κ (= 1/τ in ) is coupling coefficient, τ in is the cavity round-trip time of the slave laser, α is the linewidth enhancement factor of the slave laser, and γN is the carrier decay rate. N th also defines the carrier number at the onset of lasing, and contains both transparency carrier number and photon loss rate: gN =N tr+γ p/g .
By applying small signal linear approximation and stability analysis to the above rate equations [4, 5, 6], the injection locking range, ΔωL ,and locking stabilities can be derived and plotted as following:
,where A 0 is the stationary amplitude of the slave laser under optical injection.
The regions of injection locking stability are shown as a function of injection locking parameters in Figure 2.1.2. Equation 2.2.4 and Figure 2.1.2 illustrate that stronger optical injection broadens the stable injection locking range.
Figure 2.1.2 Injection locking stability as a function of injection ratio R and frequency detuning Δf.
2.3 Linewidth enhancement factor
In 1981, Fleming and Mooradian demonstrated the linewidth of a semiconductor laser and found the linewidth is different from Schawlow-Townes predicted. They were unable to explain the result [7]. In 1982, Charles H. Henry wrote a paper about the theory of the linewidth of semiconductor [8]. In general, the phase of the optical field fluctuation to influence the laser linewidth. The fluctuations are due to the spontaneous emission.Linewidth enhancement factor(α) is the deviation of the imaginary part and real part of the refraction index.
n
Linewidth enhancement factor is attributed to the change in refractive index with carrier density. Due to the Kramers-Kronig relations, we can find the change in the imaginary part of the susceptibility will change the real part of the susceptibility.
The refractive:
i
r
jn
n
n
The value of the linewidth enhancement factor is dependent on the dimension of the quantization. The complex refractive index changes by the carrier density. We can observer from the follower formulas [9].
E ch
The change of the linewidth enhancement factor with different quantization dimension alters the term of g
ch. g
ch is the density-of-states of the electron-hole pair. It is expresses by the step and delta functions, for a quantum well and quantum box. The change in the imaginary part of the susceptibility (gain or loss) will be influence by a corresponding change in its real part (refractive index) through the Kramers-Kronig relations. A symmetrical gain curve will lead to the dispersion curve of the refractive index has a zero at the frequency corresponding to the gain peak[10]. Large value will result in chirp under direct modulation in optical fiber communication.There are several methods to measure the linewidth enhancement factor, such as RF-modulation measurement, the interferometric measurement, the amplified spontaneous emission (ASE) method and using the locking region measurement [11]
2.4 Simulation theory
We start with a wave-guiding structure and define the z-direction when the direction of propagation. The basic equation is the Maxwell equations. However, we would like to go further and obtain a set of equations directly usable in solving the system. Under the assumption of the scalar wave, the wave equation can be solved with the variable separation technique.We assume that the solution to the wave equation can be written as the product
)
instance the effective index method or the beam propagation method.The simulation software solves the problem including all the governing semiconductor equations. The-dependent part of the electric field satisfies the equation.
) simple physical explanation is that the spontaneous emission noise generates or excites the photons, amplified by the the optical gain. The optical power is determined both by the spontaneous emission and the optical gain at any bias condition. The complex propagation constant k(z) contains information about the solution of the transverse and lateral dimensions, or in other words, it is calculated from the effective
Eq.2-1.1
Eq.2-1.2
index at a specific cross section in the xy-plane. The effective index and the k(z) are dependent on the frequency, material properties and the photon density ,as a result of the non-linear gain suppression.
Using Green's function in the analysis of DFB lasers was first proposed by C.H.
Henry [12] and later extensively used by Tromborg [13] in deriving analytical formulas for DFB lasers. We has used the Green's function method because of its accuracy in treating spontaneous emission,and its conceptual simplicity.It is also suitable for numerical implementation.The Green's function method starts with the wave equation, 2-1.2. The objective is to obtain a compact expression for the solution to the noise driven wave equation. In the Green's functions method, the solution to Eq.
2-1.3 can be written as the Wronskian of the wave equation. The integration is over the diode cavity length . The Wronskian W is a functional of the distribution of the wavenumber [k(w,N(z))]
in general. The interpretation of Eq.2-1.3 is very simple according to the basic principle of the Green's function method. The Green's function method says that for any linear differential equation with a driving source term, the solution can be found by decomposing the source into many smaller pieces in space. The Green's function can be written as homogeneous wave equation, which satisfies the boundary condition at the left laser
Eq.2-1.3
Eq.2-1.4
facet and internal interfaces for multi-section lasers but may not satisfy the boundary condition at the right laser facet. ZR(z) is the corresponding solution which satisfies the boundary condition at the right facet and the internal interfaces. The Wronskian can be written as
) ( )
( )
( )
(
Z z
dz z d Z z dz Z z d Z
W
L R
R LSince ZL(z)and ZR(z)are solutions to the homogeneous wave equation, it follows
from simple algebra that 0 dz
dw in each waveguide section. This means that W is
position independent.Therefore, under a particular bias condition, the Wronskian is only a function of the frequency or wavelength.
For the detailed 2D E-field distribution, we need to use a more elaborated way to simulate this problem.The basic equations used to describe the semiconductor device behavior are Poisson’s equation and the current continuity equations from Maxwell’s equations for electrons and holes:
, where V is electrical potential, n and p are electron concentration and hole concentration, ND and NA are doping of shallow donors(D) and shallow acceptors(A),fD and fA are occupancy of donor (D) and acceptor(A) levels, Ntj is density of j th deep trap,ftj is occupancy of the j th deep trap level , Jn and Jp are current flux densities,Rntj and Rptj are electron and hole recombination rate for
Eq.2-1.7 Eq.2-1.6
Eq.2-1.8 Eq.2-1.5
quantum well, Rsp is spontaneous recombination rate, Rst is stimulated recombination rate, Rau is auger recombination rate. By solving the above equation sets, we could calculate more precisely the field distribution within the cavity.
2.5 Couple-Wave Equations
As was discussed in the previous section, the problem has been reduced to solving for the Green's function and the corresponding Wronskian. These in turn require the knowledge of the solution of the homogeneous wave equation
0 composition and indirectly through the carrier density and photon density variations along the waveguide.
In DFB and DBR lasers, corrugations are made along the wave guides which introduce coupling between forward and backward waves. The purpose is to perturb the propagation constant k (z) to achieve desirable scattering effects to the propagating waves. In a laser with a grating of period Lg, its effective refractive index can be written as
) )
where we consider the general case that the grating period may vary as a function of position. denotes the slow varying part of the complex index and 2 n( ) is the magnitude of the index variation which is again a complex quantity. We define a reference wave number which is usually set to the Bragg wave number in simple grating structures at threshold condition such that
L
g
0
where
used to denote the average grating period. 0is a constant independent of the frequency and injection conditions. In general, we can assume that the change in the grating period is smooth and write the following expansion,)
where
ch(z)is caused by some form of chirp grating variation of grating period.Based on Eq. 2-2.3, we re-write the wave number as
]
Since only the optical frequencies close to the Bragg condition are considered, the
+higher order terms.
which is then substituted into Eq. 2-2.11. To simplify the notation, we introduce a slow varying function
neglect the second derivatives involving ( ))
( 2
0
or in matrix notation:
2.6 Theory of relative intensity noise(RIN)
RIN peak is a good show of the relaxation frequency of the device. The driving force not input current is the Langevin force (F
s, F
n and Fφ) of the field due to the spontaneous emission. The Langevin force is assumed to be irrelative white Gaussian noise [14]. The relative intensity noise (RIN) spectrum is frequency dependence. The former can be derived from the rate equations.
The intrinsic relative intensity noise(RIN) of a device is defined as
P
small-signal analysis of the rate equations for a single-mode laser, we can derive the noise spectrum of the device. The relative intensity noise spectrum of external light injected locked device can be derived using the follow rate equations [15].Eq 2-3.1
S , and N are the photon number, the phase and the carrier number inside the slave laser cavity. G0 is the gain coefficient, N0 is the transparency carrier number, τp
is the photon lifetime, τn is the carrier lifetime, I is the slave laser bias current, ε is the gain compression factor, and α is the linewidth enhancement factor. Fs , Fφ and Fn are the noise terms. Δω is the detuning between the master and slave laser. S is the inj
photons injection into the slave laser. kc is the coupling coefficient, which determined by the photon injected into the cavity-round trip time.
We based model of injection-locked rate equation is usually used to describe the interaction between photons and carriers inside a laser cavity. When an additional light source is injected into the cavity, the system preserves the general form of the original equations, but with extra terms describing the effects of the injection.
t
Substituting into the injection-locked rate equation
For the phase part:
Finally, for the carrier part:
( )Equations (2-3.3), (2-3.4), (2-3.5) are written in matrix form:
considering I as the small signal modulation current.
The modulation response transfer function will be
) (
) ) (
( 1
I H SThe light injected into the cavity of the slave laser and depletes the carrier density. It makes the spontaneous emission rate reduced and more photons are coupled in phase into the amplified injection field. The more photons in phase and the relaxation frequency should enhance. The RIN spectrum shows that the relaxation frequency peak becomes higher with injections. At a lower injection level directly adds photons into the slave laser cavity by using more carriers, compensating the gain saturation and enhances the relaxation peaks of the slave laser. Under the stronger injections condition, the injected photons deplete the most of the available carriers, saturate the signal and decrease the relaxation peaks finally. It prevents the further improvement of the relaxation frequency.
References
[1] S. Kobayashi and T. Kimura,"Coherence on injection phase-locked AlGaAs semiconductor laser," Electronics Letters, vol. 16, pp. 668-670, 1980
[2] Lukas Chrostowski, Xiaoxue Zhao and Connie J. Chang-Hasnain, “50 GHz Directly-Modulated Injection-Locked 1.55 μm VCSELs,” Optical Society of America, 2005
[3] Erwin K Lau,"High-Speed Modulation of Optical Injection-Locked Semiconductor Lasers," Electrical Engineering and Computer Sciences University of California at Berkeley, 2006
[4] S. Mohrdiek, H.Burkhard, and H. Walter, "Chirp reduction of directly modulated semiconductor lasers at 10 Gb/s by strong CW light injection," J. Lightw.
Technol., vol. 12, no. 3, pp. 418-424, Mar. 1994.
[5] R. P. Braun, G. Grosskopf, R. Meschenmoser, D. Rohde, F. Schmidt, and G.
Villino,"Microwave generation for bidirectional broadband mobile communications using optical sideband injection locking," Electron. Lett., vol. 33, no.
16, pp. 1395-1396, Jul. 1997.
[6] X. Lixin, W. H. Chung, L. Y. Chan, L. F. K. Lui, P. K. A. Wai, and H. Y. Tam,
"Simultaneous all-optical waveform reshaping of two 10-Gb/s signals using a single injection-locked Fabry-Perot laser diode," IEEE Photon. Technol. Lett., vol. 16, no. 6, pp. 1537-1539, Jun. 2004.
[7] M. W. Fleming and A. Mooradian, “Fundamental line broadening of single-mode(GaA1)As diode lasers,’’ Appl. Phys. Lett., vol. 38, p. 511, 1981.
[8] CHARLES H. HENRY, “Theory of the Linewidth of Semiconductor Lasers,’’
IEEE journal of quantum electronics, vol. QE-18, no. 2, February 1982
[9] Yasunari Miyake and Masahiro Asada, “Spectral Characteristics of Linewidth
Enhancement Factor α of Multidimensional Quantum Wells, “ Japanese journal of applied physics, vol 28 pp1280-1281, 1989
[10] MAREK OSINSKI and JENS BUUS, “Linewidth Broadening Factor in semiconductor Lasers-An Overview” Quantum Electronics, IEEE Journal of, 1987 [11] G. Liu, X. Jin, and S. L. Chuang,“Measurement of Linewidth Enhancement Factor of Semiconductor Lasers Using an Injection-Locking Technique” IEEE photonics technology letters, VOL. 13, NO. 5, MAY 2001
[12] C. H. Henry, ``Theory of spontaneous emission noise in open resonators and its application to lasers and optical amplifiers,'' J. Lightwave Technol., LT-4, 288-297 (1986).
[13] B. Tromborg, H. Olesen, and X. Pan, ``Theory of linewidth for multi-electrode laser diodes with spatially distributed noise sources,'' IEEE J. Quantum Electron., QE-27, 178-192 (1991).
[14] X. Jin and S. L. Chuang , “Relative intensity noise characteristics of injection-locked semiconductor lasers, “APPLIED PHYSICS LETTERS, vol 77, NUMBER 9 , 28 AUGUST (2000)
[15] Lukas Chrostowski, “Optical Injection Locking of Vertical Cavity Surface Emitting Lasers,” Fall (2003)
Chapter 3.
Simulation Result of the Self-pulsation Laser
3.1 Background on designIn this section a single gapped FP laser diode will be introduced which forms the basis for our platform.The single gap laser is fabricated by etching into the waveguide of the FP laser diode. The gaps act as reflection centers and produce a modulation of the reflection and transmission spectra dependent on the characteristics of the slot such as gap position, gap depth to which it is etched and slot width. Even if the gap is not etched into the active regions it will still interact with the mode of the electric field of the waveguide as the mode profile is not fully confined to the active region and will expand into the surrounding cladding regions. The 1D first order electric field mode profile modeled using the finite difference time domain technique for a simple laser structure with active region depth of 1 µm, upper cladding region of 1 µm and lower cladding of 1 µm with active region refractive index of 3.55 and cladding region refractive index 3.41, which are normal values for an InGaAsP active region sandwiched between InP cladding regions, are shown below in Fig. 3.1.1.
Figure. 3.1.1 Mode profile of the fundamental mode and refractive index profile through the laser structure.[1]
From Fig. 3.1.1 the fundamental mode is seen to penetrate into the cladding region so any perturbation in this area will influence the mode profile. The scattering matrix method is a easy and accurate technique which can be used to determine the reflection and transmission from gaps etched into the laser cavity.Numerous texts deal with the SMM of which is a good introduction.Of particular importance in a laser structure is the ability to determine loss using the method. This is an important advantage of the SMM over that transmission matrix method (TMM). A FP laser with one etched slot can be described as three cavities with different interface reflections and transmissions as described below in Fig 3.1.2.
Figure 3.1.2. Schematic description of single slot laser diode.[1]
In fig. 3.1.2, ni refers to the effective refractive index in these section of the laser structure, while ri refers to the reflection from the interfaces as shown above. Each section can be described as a separated cavity and the total reflection and transmission is then found.The back section amplitude reflection from the left side and right side is described as
the back section cavity length. The back section amplitude transmission from the left side is described asgiving a power reflection and transmission is Rbl = rbl2
and Tbr = tbr2
respectively. The reflection and transmission of the back section and gap region is found by including the back section reflection and transmission in the SMM calculation as follows
Figure 3.1.3 Calculated reflection spectrum of a single gap laser (1550 nm).
3.2 Wave intensity distribution analysis
To solve this problem, initially we took an finite difference technique [4] for the start. The variation in the third dimension is assumed to be uniform for now. When we simulate the device structure, the pumped region was indexed a little higher to mimic the optical source field. Fig. 3.2.1 shows the two dimensional field distribution.
When calculating the axial field intensity, we can find out a sharp increase of confinement when the Dgap increases more than 2um as shown in Fig. 3.2.2.
For the detailed 2D E-field distribution, we need to use a more elaborated way to simulate this problem.The basic equations used to describe the semiconductor device behavior are Poisson’s equation and the current continuity equations from Maxwell’s equations for electrons and holes:
, where V is electrical potential, n and p are electron concentration and hole concentration, ND and NA are doping of shallow donors(D) and shallow acceptors(A),fD and fA are occupancy of donor (D) and acceptor(A) levels, Ntj is density of j th deep trap,ftj is occupancy of the j th deep trap level , Jn and Jp are current flux densities,Rntj and Rptj are electron and hole recombination rate for quantum well, Rsp is spontaneous recombination rate, Rst is stimulated recombination rate, Rau is auger recombination rate.
Figure 3.2.1. 2-dimensional simulation of pumped dual-section cavity
Figure 3.2.2. Calculation of confinement of E-field in the quantum well axial direction.
99.0%
99.2%
99.4%
99.6%
99.8%
100.0%
100.2%
0 0.5 1 1.5 2 2.5 3 3.5
Dgap (m)
Confinement of E-field (%)
Figure 3.2.3. (a) Field distribution: Wgap=5μm, but with no air gap; (b) Field distribution on Wgap=5μm,Dgap=5 μm
Wgap Dgap =1μm Dgap =3μm Dgap =5μm
2μm 50.3% 90.1% 90.4%
3μm 49.7% 90.8% 93%
5μm 55.4% 93.4% 94%
Table 2. Field intensity ratio on the pumped cavity
By solving the above equation sets, we could calculate more precisely the field distribution within the cavity. First of all, we started the simulation under R1=R2=0.32 and I2 off .We focus now on the wave intensity distribution with an air gap of different depths and widths.Figure 3.2.3(a) shows the E-field distribution without any gap. Figure 3.2.3(b) shows that the wave intensity is re-distributed when the depth of the air gap is increased to 5μm, the field is hardly penetrated into the right section.
We calculated the wave intensity distribution ratio of the pumped cavity versus the
We calculated the wave intensity distribution ratio of the pumped cavity versus the