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when we are dealing with transverse electromagnetic TEM fields, where voltages andcurrents in the transmission lines are uniquely related to the transverse electric and magneticfields, r

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an idea borrowed from the phenomenon of injection locking of microwave oscillators by anexternal electronic signal [3] The close relationship between optical and microwaveprinciples suggests that it may be advantageous to apply microwave circuit techniques inmodelling of semiconductor lasers.

Engineers work best when using tools they are familiar with In particular, electrical andelectronic engineers are familiar with well-established electrical circuit models as tools toaid themselves in the understanding and prediction of behaviour of electrical machines orelectronic devices Since the early days of radio frequency (RF) and microwave engineering,microwave circuit theory has allowed us to explore fundamental properties ofelectromagnetic waves by giving us an intuitive understanding of them without the need

to invoke detailed and rigorous electromagnetic field theories [4–5] In the same spirit,microwave circuit formulation of the semiconductor laser diode enhances our understanding

of the device, which is otherwise obscured by hard-to-visualise mathematical formulations.Complex mathematical models are too sophisticated to be desirable for engineers, especiallythose who are not specialists in the field of laser physics but would like to have a quick-to-digest method of understanding and designing semiconductor laser devices It is far moreconvenient to work in terms of voltages, currents and impedances In fact, electromagneticfield theory and distributed-element circuits (transmission lines) give identical solutions

Distributed Feedback Laser Diodes and Optical Tunable Filters H Ghafouri–Shiraz

# 2003 John Wiley & Sons, Ltd ISBN: 0-470-85618-1

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when we are dealing with transverse electromagnetic (TEM) fields, where voltages andcurrents in the transmission lines are uniquely related to the transverse electric and magneticfields, respectively.

The attractiveness of using equivalent circuit models for semiconductor laser devicesstems from their ability to provide an analogy of laser theory in terms of microwave circuitprinciples In addition, microwave circuit models of laser diodes are compatible withexisting circuit models of microwave devices such as heterojunction bipolar transistors(HBTs) and field-effect transistors (FETs) – an attractive feature for optoelectronic integratedcircuit (OEIC) design [6] Equivalent circuit models have effectively helped many tounderstand, design and optimise integrated circuits (ICs) in the microelectronics industryand they have the potential to do the same for the optoelectronics industry The main theme

of this chapter is microwave circuit modelling techniques applied to semiconductor laserdevices

Two types of microwave circuit model for semiconductor lasers have been investigated:the simple lumped-element model based on low-frequency circuit concepts and the moreversatile distributed-element model based on transmission-line modelling The former(lumped-element circuit model) is based on the simplifying assumption that the phase ofcurrent or voltage across the dimension of the components has little variation This is truewhen considering only the modulated signal instead of the optical carrier signal In this case,Kirchoff’s law can be applied, which is nothing more than a special case of Maxwell’sequations [7–8] Strictly speaking, laser devices have dimensions in the order of theoperating wavelength, thus lumped-element models may not be suitable in ultrafastapplications where propagation plays an important role such as in active mode locking [9].However, the lumped-element circuit is reasonably accurate for microwave applications ifall the important processes and effects are modelled accordingly by the circuit on anequivalence basis

The latter of the two circuit modelling techniques (i.e transmission-line modelling) is amore powerful circuit model that includes distributed effects, which will be discussed indetail in this chapter It is worth pointing out that at microwave frequencies and above,voltmeters and ammeters for direct measurement of voltages and currents do not exist, sovoltage and current waves are only introduced conceptually in the microwave circuit tomake optimum use of the low-frequency circuit concepts

8.2 THE TRANSMISSION-LINE MATRIX (TLM) METHOD

The transmission-line matrix (TLM) was originally developed to model passive microwavecavities by using meshes of transmission lines [9–10] The numerical processes involved inTLM resemble the mechanism of wave propagation but they are discretised in both time andspace [10–11] Much work has been carried out using the TLM method for analysis ofpassive microwave waveguide structures (see [12] and references therein) Most of the workdone involved two-dimensional and three-dimensional TLMs, with the exception of theapplication to lumped networks [13–14], the heat diffusion problem [15] and semiconductorlaser modelling [16] Although the TLM is unconditionally stable when modelling passivedevices, the semiconductor laser is an active device and therefore requires more carefulconsideration The basics of the one-dimensional (1-D) TLM will be presented in thefollowing section, which forms the basis of the transmission-line laser model (TLLM) [17]

196 CIRCUIT AND TRANSMISSION-LINE LASER MODELLING (TLLM) TECHNIQUES

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8.2.1 TLM Link Lines

The TLM is a discrete-time model of wave propagation simulated by voltage pulsestravelling along transmission lines The medium of propagation is represented by thetransmission lines – a general or lossy transmission line consists of series resistance, shuntadmittance, series inductance and shunt capacitance per unit length, whereas an ideal orlossless transmission line has reactive elements only The transmission line may be described

by a set of telegraphist equations [7], which can be shown to be equivalent to Maxwell’sequations There are two types of TLM element that can be used as the building blocks of acomplete TLM network – they are the TLM stub lines and link lines [13] For a losslesstransmission line, the velocity of propagation is expressed by

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Similarly, the lumped shunt capacitor (C) is equivalent to a transmission line withcapacitance per unit length of Cd (Fig 8.1) where

Figure 8.2 TLM stub lines

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and has a parasitic capacitance expressed by

8.3 SCATTERING AND CONNECTING MATRICES

The most basic algorithm of TLM involves two main processes: scattering and connecting.When the incident voltage pulses, Vi, arrive at the scattering node, they are operated by ascattering matrix and reflected voltage pulses, Vr, are produced These reflected pulses thencontinue to propagate along the transmission lines and become incident pulses at adjacentscattering nodes – this process is described by the connecting matrix Formally, the TLMalgorithm may be expressed as

kVrT¼ SkViT ½Scattering

The terms ViT and VrT are the transpose matrices of the incident and reflected pulses,respectively The terms k and kþ 1 denote the kth and ðk þ 1Þth time iteration, respectively.The scattering and connecting matrices are denoted by S and C, respectively As the matricesinvolved in eqn (8.12) depend on the type of TLM sub-network, a worked example based onthe TLM sub-network of Fig 8.3 follows

The TLM sub-network consists of three ‘branches’ of lossy transmission lines as shown inFig 8.3, where scattering and connecting of the voltage pulses are clearly describedpictorially The normalised impedances are unity for the two lines connected to adjacent

SCATTERING AND CONNECTING MATRICES 199

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Figure 8.3 The TLM stub line: (a) incident pulses arriving at scattering node; (b) incident pulsesscattered into reflected pulses (scattering); (c) reflected pulses arriving at adjacent nodes (connecting).

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nodes and Zsfor the remaining branch (open–circuit stub line) The associated normalisedresistances are R and Rs, respectively The matrices ViTand VrTare given as

35

i

ð8:13Þ

where V1, V2 and V3 are the voltage pulses on ports 1, 2, and 3, respectively on eachscattering node (see Fig 8.3) The superscripts r and i denote reflected and incidentpulses, respectively It is convenient to break up the scattering matrix S and express it as[15–19]

where the matrices p, q, and r are defined in the following For the type of TLM sub-network

in Fig 8.3, we have

q¼ q½ 1 q2 q3where

qi¼

2ðRsþ ZsÞ

1þ R þ 2ðRsþ ZsÞ i¼ 1; 22ð1 þ RÞ

x[5], its generator or source

Figure 8.4 Thevenin equivalent circuit of the TLM sub-network

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26

3

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The connecting matrix depends on how the transmission lines are connected, that is, whichport(s) of one scattering node is/are connected to which port(s) of other adjacent scatteringnode(s) The matrix element in the connecting matrix is unity only if a connection allows apulse to travel from port i of node m to port j of node n In the example given in Fig 8.3, theconnecting matrix C may be expressed by 3 3 elements, where we have pulses travellingfrom

1 port 2 of node n 1 to port 1 of node n

2 port 3 of node n to port 3 of node n (open-circuit stub)

3 port 1 of node nþ 1 to port 2 of node n

Thus, the connecting matrices for node n and its adjacent nodes n 1 and n þ 1 are given as

3

A simple example of another TLM sub-network is shown in Fig 8.5(a), which consists of aresistor ‘sandwiched’ between two lossless transmission lines In the Thevenin equivalentcircuit shown in Fig 8.5(b), each transmission line is replaced by its characteristicimpedance in series with a voltage generator of twice the incident voltage pulse The

Figure 8.5 (a) A TLM sub-network and (b) its Thevenin equivalent circuit

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incident voltage pulses are denoted as Vi

1 and Vi

2, while the reflected pulses are Vr

1 and Vr

2.The impedances of the lossless transmission lines are Z1and Z2 From Fig 8.5(b), the nodalvoltages (v1 and v2) may be expressed as

i1¼2V

i 1

Z1

i 2

Rþ Z2

2Vi 1

Rþ Z1

2Vi 2

Z2

ð8:22Þand

v2¼ Vi

2þ Vr 2

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Another simple but useful TLM sub-network is shown in Fig 8.6(a) and its Theveninequivalent is given in Fig 8.6(b) This is similar to one of the sub-networks in Fig 8.3 but it

is lossless in this case and each branch has a different value of line impedance Now, thecommon nodal current is defined by

i¼2V

i 1

Z1

þ2V

i 2

Z2

þ2V

i 3

Z3

ð8:28ÞFrom Fig 8.6(b), the total admittance at the scattering node is expressed as

Figure 8.6 (a) A TLM sub-network and (b) its Thevenin equivalent circuit

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From Millman’s thereom [19] the common nodal voltage can be found from

The TLM sub-network in Fig 8.6(a) forms the basis of the matching network There aremany other TLM sub-networks that can be formed by using series and shunt resistorstogether with the TLM link lines and stub lines For example, periodically unmatchedboundaries of TLM link lines can be used to mimic corrugated gratings [20] and shuntconductances can be included to model gain-coupled DFB lasers [21] A TLM sub-networkwhich consists of a series resistor and two reactive stub lines will now be used to model thewavelength dependence of semiconductor laser gain in the transmission-line laser model

8.4 TRANSMISSION-LINE LASER MODELLING (TLLM)

The transmission-line laser model is a wide-bandwidth dynamic laser model that takes intoaccount important considerations such as inhomogeneous effects, multiple longitudinalmodes, spectral dependence of gain, carrier-induced refractive index change, andspontaneous emission noise The TLLM is very flexible and has successfully been used tomodel a wide range of laser devices including Fabry–Perot lasers [16], DFB (index-coupledand gain-coupled) and DBR lasers [21–23], quantum well (QW) lasers [24], cleaved-coupled-cavity (CCC) lasers [25] external cavity (EC) mode-locked lasers [26], fibre gratinglasers [27] and laser amplifiers [28–29] Electrical parasitics and matching circuits can also

be included as part of the model [29]

The TLLM can be thought of as a pedagogical model to aid in the physical understanding

of the laser in terms of more familiar circuit techniques The topology of the model closely

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mimics the physical structure of the laser It was first developed by Lowery [16] and wasbased on the 1-D TLM concepts discussed in the preceding section Since the TLLM is atime-domain model its main application is transient analysis, where laser non-linearity isimportant.

8.5 BASIC CONSTRUCTION OF THE MODEL

In the TLLM, the laser cavity is divided into many smaller sections as shown in Fig 8.7,hence the longitudinal distributions of carrier and photon density can be modelled Theindividual sections are connected to one another by dispersionless transmission lines (i.e.group velocity equal phase velocity) with characteristic impedance of Z0 Voltage pulsestravel along these transmission lines in both forward and backward directions, analogous to

the propagation of optical waves inside the laser cavity The model time step, tð Þ, describesthe time required for the voltage pulse to propagate from one section to the adjacent section

At the centre of each section is the scattering node, where the TLM scattering matrix isplaced When incident voltage pulses arrive at the scattering nodes, the TLM scatteringmatrix operates on them to produce reflected voltage pulses (see section 8.3)

The fundamental processes of the laser such as gain (stimulated emission), loss(absorption), and noise (spontaneous emission) are contained in the scattering matrix

Figure 8.7 The transmission-line laser model (TLLM) and its components

BASIC CONSTRUCTION OF THE MODEL 207

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Spectral dependence of the material gain, which the optical waves experience, is alsoincluded Coupling between counter-propagating waves that occurs in DFB lasers can also

be modelled At the laser facets, the reflections that provide feedback to achieve lasingaction are simulated by unmatched terminal loads

The model assumes single transverse mode operation of the laser diode, and thatvariations of the carrier and photon densities in the lateral and transverse dimensions are notsignificant, except for broad-area lasers such as the tapered waveguide structure [30] Today,advanced fabrication techniques allow single transverse and longitudinal mode laser devices

to be achieved, typical of these are strongly index-guided laser structures such as buriedheterostructure (BH) lasers By spatially averaging the lateral x- and transverse y-dependence ofthe optical field amplitude, the model is simplified into a 1-D model [16] The 1-D TLLM is

a reasonable approach, which has been supported by many other 1-D dynamic laser modelssuch as the transfer matrix model (TMM) [31], time-domain model (TDM) [32–33], andpower matrix model (PMM) [33–34]

The reservoir of carriers (Nn where n is the section number) in the carrier density modelinteracts with the photon density Sð Þ through the laser rate equations This dynamic carrier–nphoton interaction is independently modelled in each and every section The carriers aresupplied by the current sources placed in every section of the model In this way,electrically-isolated multi-electrode lasers can easily be modelled by injecting differentcurrent amplitudes into separate parts of the model Non-uniform current pumping is onemethod of ensuring a more evenly distributed carrier concentration in the laser cavity,especially when there is severe spatial hole burning such as in quarterly-wavelength-shiftedDFB lasers In the presence of significant spatial hole burning, the longitudinal carrierdiffusion effect can also be included in the model [35]

Therefore, the TLLM is a powerful laser model and yet it is relatively easy to visualise,being a distributed-element circuit model The outputs of the model are collected as opticalfield samples, which can be coupled to other TLLM-compatible models such as optical am-plifiers, optical filters and photodiodes [36] The samples of optical field and optical powerare readily fast Fourier transformed (FFT) to obtain the optical and RF spectra, respectively.The accuracy of the model can be enhanced by using a smaller model time step tð Þ but atthe expense of longer computation time

8.6 CARRIER DENSITY MODEL

The carrier–photon resonance that leads to the transient phenomenon of relaxationoscillations is governed by the laser rate equations The carrier rate equation is given as

an equivalent circuit (see Fig 8.8) at each and every section of the TLLM, where the carrierdensity is represented by the voltage, V In the equivalent circuit of Fig 8.8, the current, I ,

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represents the current injected into the active region, the storage capacitor, C, representscarrier build-up/depletion, the resistor, Rsp, represents the spontaneous emission rate, and thecurrent, Istim, represents the stimulated emission rate The corresponding circuit equation isgiven as

Figure 8.8 Equivalent circuit model of the carrier density rate equation

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where neff is the effective refractive index of the semiconductor material Since the powerdensity may be defined in terms of the transverse fields alone, the wave impedance is used

to relate the transverse components of the electric and magnetic fields [5,7] This enables thepower transmitted to be expressed in terms of only one of the transverse fields Hence,the optical power, P, escaping from the output facet may be defined in terms of the trans-verse electric field as

dI

ðnL z¼ðn1ÞL

ðGg scÞdz

InL ¼ Iðn1ÞLexp½ðGg scÞL

ð8:39Þ

where Iðn1ÞLis the initial optical intensity, InLis the intensity after propagating a distance

of L; G is the optical confinement factor, g is the gain coefficient that is frequencydependent, and sc is the loss factor As the optical field amplitude is proportional to thesquare root of the intensity, we can write

is shown in Fig 8.9 The frequency (wavelength) dependence of the small amplified

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signal term, GgL=2ð ÞE0, is modelled by the RLC bandpass filter, which is then added back

to the incoming signal Eð 0Þ and attenuated by the factor exp ð scL=2Þ to produce theoutput amplified signal ðELÞ The lumped circuit representation of the second-orderbandpass filter in Fig 8.9 (rectangular dashed lines) may be converted into its TLM

counterpart, as shown in Fig 8.10 The bandpass filter has a Lorentzian response, assumingthat the laser is a homogeneously broadened two-level system [6,37] The Thevenin-equivalent circuit of Fig 8.10 is shown in Fig 8.11 The lumped inductance and capacitance

of the bandpass filter are modelled as short-circuit and open-circuit TLM stubs, respectively

Figure 8.9 Block diagram of the gain filter

Figure 8.10 The TLM stub filter

Figure 8.11 Thevenin equivalent circuit of the TLM stub filter

Tiêu đề	Circuit and Transmission-Line Laser Modelling (TLLM) Techniques
Trường học	John Wiley & Sons, Ltd
Chuyên ngành	Optoelectronics
Thể loại	Chương
Năm xuất bản	2003
Thành phố	Shiraz

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