This may be used to find the time-varying photon density for agiven drive current waveform or, alternatively, to find the frequency response of the devices 232 ANALYSIS OF DFB LASER DIOD
Trang 19.2 DFB LASER DIODES
As explained in Chapter 2, the feedback necessary for the lasing action in a DFB laser diode
is distributed throughout the cavity length This is achieved through the use of a gratingetched in such a way that the thickness of one layer varies periodically along the cavitylength The resulting periodic perturbation of the refractive index provides feedback bymeans of Bragg diffraction rather than the usual cleaved mirrors in Fabry–Perot laser diodes[1–3] Bragg diffraction is a phenomenon which couples the waves propagating in theforward and backward directions Mode selectivity of the DFB mechanism results fromthe Bragg condition When the period of the grating, , is equal to mB=2neff, where B isthe Bragg wavelength, neff is the effective refractive index of the waveguide and m is aninteger representing the order of Bragg diffraction induced by the grating, then only themode near the Bragg wavelength is reflected constructively Hence, this particular mode willlase whilst the other modes exhibiting higher losses are suppressed from oscillation Thecoupling between the forward and backward waves is strongest when m¼ 1 (i.e first-order
Distributed Feedback Laser Diodes and Optical Tunable Filters H Ghafouri–Shiraz
# 2003 John Wiley & Sons, Ltd ISBN: 0-470-85618-1
Trang 2Bragg diffraction) By choosing appropriately, a device can be made to provide distributedfeedback only at the selected wavelengths.
In recent years, DFB LDs have played an important role in the long-span and high-bit-rateoptical fibre transmission systems because of their stronger capability of single longitudinalmode operation To overcome the two modes’ degeneracy and achieve a pure single-modeoperation, quarterly-wavelength-shifted (QWS) DFB lasers have been proposed [4].However, in such QWS DFB lasers, spatial hole burning effects enhance the side modeswhen the coupling coefficient is large (i.e L > 3) In order to combat this effect, multiple-Phase shift DFB lasers have been proposed [5–8] It has been shown that side modes can
be effectively suppressed and a stable and pure single-mode operation results With thedevelopment of laser structures, efficient and relatively accurate simulation models arebecoming more and more important for laser designs and operation optimisation due tothe complication and expense involved in laser fabrication processes
Distributed feedback semiconductor lasers have a greater mode selectivity than Fabry–Perot devices and so are preferred as sources for long-haul high-capacity-fibre systems.However, dynamic single-mode (DSM) operation is still difficult Accurate multi-modedynamic computer models could help in designing DSM DFB devices Many DFB modelscalculate the individual mode threshold gains in an attempt to assess wavelength stability.However, these usually neglect the saturation and inhomogeneity of the gain which occurs atthe onset of lasing Dynamic models are available, but these assume a single oscillatingmode, making the study of mode stability impossible
The ideal semiconductor laser model would mimic the operation of the real devices inevery detail, simulating all characteristics of the laser while accounting for all variations indevice structure, processing, drive electronics and external optical components [9–10] Themodel could be connected to other device models to form an optical system model Such amodel would improve the design of photonic devices, circuits and systems It could also beused for detailed optimisation in particular applications
Limitations in computing resources require that simplifications and assumptions have to
be made before a model is developed Many optoelectronic device models use rate equations
to describe the interactions between the average electron and photon populations in thedevice [9–11] Numerous adaptations of this technique have been proposed For example,using a photon rate equation for each longitudinal laser mode gives the laser’s spectrumduring modulation [12] and dynamic frequency shifting (chirping) may be estimated fromthe transient responses of both populations [13] The laser rate equations may also describesaturation in laser amplifiers [14], the dynamic behaviour of model-locked lasers [15] andthe transient response of cleaved-coupled-cavity lasers [16]
The limitation of using photon density as a variable is that it does not contain opticalphase information Optical phase is important when there is a set of coupled opticalresonators such as in coupled-cavity lasers, external-cavity lasers, DFB lasers, or evenFabry–Perot lasers with unintentional feedback from external components In these cases,the output wavelength of the devices and its current to light characteristics are determined byoptical interference between the resonators Although rate equations can be used in simplecases, by calculating effective reflection coefficients at discrete wavelengths [16], findingthese wavelengths becomes difficult with multiple resonators exhibiting gain and variablerefractive indices, such as in the DFB laser [17]
A development of the rate equation approach is to use a SPICE-compatible equivalentcircuit of the laser diode This may be used to find the time-varying photon density for agiven drive current waveform or, alternatively, to find the frequency response of the devices
232 ANALYSIS OF DFB LASER DIODE CHARACTERISTICS BASED ON TLLM
Trang 3[18] This approach has an advantage in that it includes parasitic components in the laserchirp and mount and can be linked to models of the drive circuit for evaluation of thesystem’s response to modulation.
An alternative variable to photon density is optical field, which contains phase informationand thus offers the possibility of dealing with multiple reflections The optical field within aresonator system may be solved in the frequency domain or in the time domain Frequency-domain models often use a transfer-matrix description of the laser that may be obtained
by multiplying together the transfer matrices describing each individual reflection [19–20].However, if the spectrum of a modulated laser is required, the multiplication has to beperformed for each wavelength at each time step [17] This is computationally inefficient.Time-domain models using optical fields are better suited to modulated devices withmultiple resonators than frequency-domain models because the former are simpler todevelop and require less computation Time-domain optical-field models are commonlybased on scattering matrix descriptions of the individual reflections and of the gain medium.The scattering matrices may be connected by delays (transmission lines) so that reflectedwaves out of one scattering matrix can be connected to each adjacent matrices after thedelay The delays represent the optical propagation time along a portion of the waveguide Asolution for the network is found by iteration, each iteration representing an increase in timeequal to the delay
At high-frequency modulation (16–17 GHz) [21], the dynamic characteristics of lasers areimportant and design methods that can help to predict the chirp and modulation efficiencyare needed The dynamic response of lasers is generally studied by solving a set of rateequations that govern the interaction between the carriers and photons inside the activeregion of the laser cavity In the earliest work, the equations are usually linearised to allowsolutions to be found for small-signal oscillations Although this gives insight to theimportant physical parameters, it has limited applicability Large-signal dynamics with non-linear effects such as gain saturation, spatial hole burning and changes of electron andphoton densities along the length of lasers are now essential in the study of DFB laserswhere these effects are more significant than in Fabry–Perot lasers [22–23] Thetransmission-line laser model based on the transmission-line modelling (TLM) method, isbeing developed to study many of the dynamic effects in lasers
Transmission-line laser modelling, which was developed by Lowery, employs domain numerical algorithms for laser simulation [24–33] This model splits the laser cavitylongitudinally into a number of sections In each section, TLLM uses a scattering matrix torepresent the optical process, such as stimulation emission, spontaneous emission andattenuation The matrices of these sections are then connected by transmission lines, whichaccount for the propagation delays of the waves From the iterations of scattering andconnecting processes, the output electric field in the time domain can be obtained Then, byapplying a Fourier transform, we can easily obtain the laser output spectra Large-signaldynamics with non-linear effects such as the changes of electron and photon densities alongthe length of the laser and spatial hole burning are key issues in the analysis of DFB laserdiodes These dynamic effects can be investigated easily by using transmission-line lasermodelling
time-TLLM models have been used to analyse QWS DFB LDs [32] With the insertion of
a zero-reflection interface (identity matrix) half way along the cavity, the effects of QWS
on laser operation have been simulated successfully However, using this method we canonly analyse DFB laser structures with one =2 phase shift at the centre of the cavity Wecannot use this technique to analyse other phase shift values or multi-phase-shift (MPS) lasers
Trang 49.3 TLLM FOR DFB LASER DIODES
In general, two operations, scattering and connecting, are involved in transmission-line lasermodelling The scattering operation takes voltage pulses incident on the nodes, kAi, andscatters them to give voltage pulses reflected from the nodes,kAr The reflected and incidentvoltage pulses are related together via the following scattering matrix, S which includesstimulation, emission, spontaneous emission and attenuation processes That is
where k is the iteration number and Is is the spontaneous wave As discussed in Chapter 8,the scattering operation can be derived from a knowledge of the impedances of thetransmission lines and associated components, such as resistors at the nodes Equation (9.1)can be modified to include the source voltage pulses,kAs, so
kAr ¼ S kAiþkAsþ Is ð9:2ÞThe reflected pulses that propagate to the next scattering nodes become new incident pulsesfor the next scattering operation This process can be expressed as
In eqn (9.3) C is the connection matrix that can be derived from the topology of the network
It should be noted that for all pulses to arrive at the nodes synchronously, the transmissionlines must have equal delay times Each delay time should also be equal to the iteration timestep t In the numerical calculation, we need to initialise the value of voltage Aiand thenrepeat eqns (9.1) and (9.2) to find the time evolution of the voltage Aior Ar In transmission-line laser models, the voltage pulses represent the optical fields along the cavity A chain oftransmission lines connects these fields from the laser rear facet via optical cavity to the laserfront facet The scattering matrices represent the optical processes of stimulated emission,spontaneous emission and attenuation The local carrier density will be updated according tothe rate equation model at each time step and the magnitudes of these processes at aparticular matrix will also be re-calculated with the new information of the carrier density Itshould be noted that the local carrier density should be updated at each time step ðtÞaccoding to the rate equation model The updated carrier density will then be used to set themagnitude of the optical processes in the scattering matrix
9.4 A DFB LASER DIODE MODEL WITH PHASE SHIFT
In a DFB laser diode, the forward and backward waves are coupled along the entire cavitylength because of the refractive index modulation along the cavity This coupling can be
Figure 9.1 The TLLM model for uniform DFB laser diodes
234 ANALYSIS OF DFB LASER DIODE CHARACTERISTICS BASED ON TLLM
Trang 5represented by impedance discontinuities placed between the model sections as shown inFig 9.1 However, a model for the phase shift is needed to model such DFB laser diodes.
In doing so, phase stubs are employed and connected to the main transmission line In thismodel circulators are used (see Fig 9.2) to send the waves out of the stubs in the correctdirection For example, a forward wave will enter the first left-hand circulator (port 1) and isdirected to the stub port (port 2) Since the stub presents an impedance mismatch, part of thewave will be reflected back into port 2 The circulator then directs this reflected wave to port
3, where it continues on as a forward wave The remainder of the wave enters the stub to bedelayed before returning to port 2 to be directed to port 3 Backward waves simply pass fromport 3 to port 1 of this first circulator A second set of three-port circulators is used to delaythe backward waves
The phase delay caused by a stub can be varied by altering its impedance For example aninfinite stub impedance gives a reflection with zero phase shift; a matched capacitive stubgives a phase shift ofð2ptf Þ radians; a zero impedance stub gives p radians; a matchedinductive (shorted) stub gives ð2p f tÞ radians where f is the optical frequency Otherphase shifts are available over a limited bandwidth by using other reflection coefficients
A complete DFB laser diode model with phase shift is shown in Fig 9.3 Here, scatteringmatrices have been inserted between the circulators of each section Also, alternate con-necting transmission lines have different impedances This creates impedance mismatches atthe section boundaries, which couple the forward and backward waves [28] Each section has anassociated carrier rate equation model to enable the local gain, refractive index and spontaneousnoise to be calculated from the injection current and the carrier recombination rates [24]
Figure 9.2 The TLLM model representing a phase shift
Figure 9.3 A complete DFB laser diode model with phase shift p is a phase-shift stub, l and c aregain-filter stubs and i is the injection current
Trang 6The single scattering matrix S shown in Fig 9.3 represents a section of laser with length
L This matrix operates on the forward- and backward-travelling incident waves toproduce a set of reflected waves These are then passed along the transmission lines ready tobecome new incident waves upon adjacent scattering nodes after one iteration time step Iftwo sections of the model were to be used to represent each period of the DFB grating on thereal device, the number of sections and hence the computational task would be excessive.However, it is possible to represent an odd number of grating periods with a single pair ofmodel sections without compromising the model’s accuracy [28] This technique relies onthe model having a square grating modulation This can be decomposed into a number ofsinusoidal gratings at harmonics of the grating period by Fourier techniques One of theseharmonics models the real device’s grating period
Note that the amplitude of each harmonic decreases with the harmonic number, that is, thefifth harmonic produces a coupling of one-fifth of the amplitude of the fundamentalharmonic For this example, the coupling of each period of the square grating has to beincreased by a factor of five over the coupling of the real laser’s grating to compensate Asimpler and much neater rule is that the coupling per unit length must be equal for modeland real devices [28] If a small number of sections is used, the optical field will be sampledless than once per wave period This under-sampling is essential for realistic computer times.Under-sampling has been used in all TLLMs and does not compromise accuracy if thesampling rate (section length/group velocity) is more than twice the bandwidth of the opticalwave [24] The use of two sections per grating period ensures that the DFB’s spectrumalways lies near the centre of the modelled spectrum
9.5 ANALYSIS OF TLLM FOR DFB LASER DIODES
Once the transmission-line representation of the device has been derived, an algorithm can
be produced One of the advantages of TLLM is that the algorithm is always an exactrepresentation of the transmission-line model; no inaccuracies are introduced once thetransmission-line representation has been formulated This means that all approximationshave physical meaning because they are associated with the parameters of the transmissionlines The terms in eqns (9.1) to (9.3) will now be derived for the DFB laser model Note thatthe travelling optical electric fields are represented by voltage pulses A (forwards) and B(backwards) in the model Thus, a unity constant m, with dimension of metres, is used toconvert between electric field and voltage to maintain dimensional correctness
The scattering matrix can be split into two scattering matrices, one for each wave direction.This is possible as there is no cross coupling between the wave directions in the scatteringoperation In a uniform DFB LD, the scattering process for the forward wave, with incidentpulses from the previous section AiðnÞ, the gain filter’s capacitive stub Ai
CðnÞ and the gainfilter’s inductive stub AiLðnÞ, may be expressed as [27]
AðnÞ
ACðnÞ
ALðnÞ
24
35
35
i
þ
IsZp=200
24
35
S
ð9:4Þ
236 ANALYSIS OF DFB LASER DIODE CHARACTERISTICS BASED ON TLLM
Trang 7Su¼1y
ðg þ yÞ 2YC 2YL
g 2YC y 2YL
g 2YC 2YL y
26
3
IS¼ ffiffiffiffiffiffiffiffi
i2 S
It should be noted that, as mentioned in section 2.3.4, due to the dispersive properties ofthe semiconductor, the actual material gain g given in eqn (9.12) is also affected by theoptical frequency f, and hence the wavelength l So far, the gain has been assumed to be atthe resonant frequency However, if the optical frequency is tuned away from the resonantpeak, the exact value of the gain becomes different from the peak value On the basis ofexperimental observation, Westbrook [33] extended the linear peak gain model further so
Trang 8a2¼ 4 105cm1eV2
, E0is the gain peak energy at the transparency and a3is dE0=dN,the gain peak position carrier dependence a 3¼ 1:4 1020eVcm3
9.5.2 Scattering Matrix for the DFB Laser Diode with Phase Shift
For a DFB LD with phase shift, the scattering process for the forward wave, with incidentpulses from the previous section AiðnÞ, the gain filter’s capacitive stub Ai
CðnÞ, the gain filter’sinductive stub Ai
LðnÞ and the phase shifting stub Ai
377
377
i
þ
k
ISZC=2000
266
377
2
6
6
377
ð9:16Þwhere
tan pneffl
238 ANALYSIS OF DFB LASER DIODE CHARACTERISTICS BASED ON TLLM
Trang 9ð9:20Þfor a low–high impedance boundary, and
kþ1
Aðn þ 2ÞBðn þ 1Þ
ð9:21Þ
for a high–low impedance boundary
k is the iteration number and i denotes incident pulses to the main scattering matrices S,
is the coupling coefficient and r denotes reflected pulses from the main scattering matrices.Note that there is a one-iteration time step delay as the optical field samples move along thelines For a standard DFB device, eqns (9.20) and (9.21) are applied alternately along thedevice length, i.e n¼ ð1; 3; 5; 7; Þ For QWS grating devices [31] a zero-reflectioninterface (identity matrix) is inserted half way along the cavity Figure 9.4 shows the
schematic diagram of the TLLM for the QWS DFB laser diode When a facet is placed at alow–high impedance boundary a simple resistive termination can be used giving:ffiffiffi kþ1BiðnÞ ¼R
9.6.1 Connection Matrix C for the Stubs Within a Section
There are also equations governing the reflections at the ends of the transmission line stubs.These are half a time step long to ensure that pulses arrive back at the originating scattering
Figure 9.4 The TLLM model for a QWS DFB laser diode
Trang 10matrix after a delay of one time step For the inductive stubs in each section, the reflectioncoefficient is negative, giving
9.7 CARRIER DENSITY RATE EQUATION
Neglecting the effect of diffusion along the laser cavity we may apply the carrier density rateequation for each section of the model However, this assumption may not be accurate forlaser diodes with low facet reflectivities, including laser diode amplifiers Electrons may beinjected into the conduction band in the active region of the laser by sandwiching it betweentwo higher-bandgap semiconductor layers and applying a forward bias across the structure.The electrons may leave the region into which they have been injected by diffusion to otherregions The electrons may also recombine to the valence band by stimulated or spontaneousrecombination Electrons involved in these processes may be accounted for using a carrierdensity rate equation given by
Trang 11The power exiting the front facet, P, is related to the wave incident on the facet from thesection ArðnÞ as [24]
F¼1L
ðL 0
where PðzÞ is the normalised longitudinal field intensity distribution along the laser cavitywith length L and P is the average of PðzÞ To minimise the longitudinal spatial hole burningeffect and thus get single longitudinal mode oscillation, it has been proposed that the flatness
F should be below 0.05 in DFB laser diodes [24]
9.8 RESULTS AND DISCUSSIONS
Based on the TLLM MPS DFB model introduced in section 9.5, the operation of QWS DFBLDs as well as 3PS DFB LDs have been analysed Parameters used in our analysis are given
in Table 9.1 The reflectivity at both facets is assumed to be 0 To make it easy to understand,
a schematic diagram of the 3PS DFB LD structure is shown in Fig 9.5
Table 9.1 Parameter values used in the TLLM model
Physical parameter Symbol and value
Operating wavelength 0¼1550 mm
Cavity length L¼ 200 mm
Active layer thickness d¼ 0:15 mm
Active layer width w¼ 2 mm
Spatial gain constant a¼ 5 1017cm2
Optical confinement factor ¼ 0.3
Transparency carrier density N0¼ 9 1017cm3
Carrier lifetime ¼ 4.0 ns
Spontaneous emission coupling factor ¼ 104
Internal attenuation t¼ 0
Laser facet reflectivities R¼ 0
Monomolecular recombination coefficient A¼ 108s1
Bimolecular recombination coefficient B¼ 8:6 1011cm3s1
Auger recombination coefficient C¼ 4:0 1029cm6s1
Number of model sections S¼100
Grating coupling per unit length ¼ 160 cm1
Initial carrier density Ni¼ 1:5 1018cm3