Tài liệu Laser điốt được phân phối thông tin phản hồi và các bộ lọc du dương quang P4 ppt

Similarly, other structures like the active planar Fabry–Perot FP section, thepassive corrugated distributed Bragg reflector DBR section and the passive planarwaveguide WG section can al

In dealing with such a complicated DFB laser structure, it is tedious to match all theboundary conditions A more flexible method which is capable of handling different types ofDFB laser structures is necessary In section 4.2, the transfer matrix method (TMM) [1– 4]will be introduced and explored comprehensively From the coupled wave equations, it isfound that the field propagation inside a corrugated waveguide (e.g the DFB laser cavity)can be represented by a transfer matrix Provided that the electric fields at the input plane areknown, the matrix acts as a transfer function so that electric fields at the output plane can

be determined Similarly, other structures like the active planar Fabry–Perot (FP) section, thepassive corrugated distributed Bragg reflector (DBR) section and the passive planarwaveguide (WG) section can also be expressed using the idea of a transfer matrix Byjoining these transfer matrices as a building block, a general N-sectioned laser cavity modelwill be presented Since the outputs from a transfer matrix automatically become the inputs

of the following matrix, all boundary conditions inside the composite cavity are matched.The unsolved boundary conditions are those at the left and right facets In section 4.3, thethreshold equation of the N-sectioned laser cavity model will be determined and the use ofTMM in other semiconductor laser devices will be discussed

An adequate treatment of the amplified spontaneous emission spectrum ðPNÞ is veryimportant in the analysis of semiconductor lasers [5], optical amplifiers [6 –8] and opticalfilters [9–10] In semiconductor lasers, PN is important for both the estimation of linewidth[11] and the estimation of single-mode stability in DFB laser diodes [12] In optical

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

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

amplifiers and filters, PN has also been used to simulate the bandwidth, tunability and thesignal gain characteristic In section 4.4, the TMM formulation will be extended so as toinclude the below-threshold spontaneous emission spectrum of the N-sectioned DFB laserstructure Numerical results based on 3PS DFB LDs will be presented.

By matching boundary conditions at the facets and the phase-shift position, the thresholdcondition of the single-phase-shifted DFB LD can be determined from the eigenvalueequation However, this approach lacks the flexibility required in the structural design ofDFB LDs Whenever a new structural design is involved, a new eigenvalue equation has to

be derived by matching all boundary conditions For a laser with the MPS DFB structure, theformation of eigenvalue equation becomes tedious since it may involve a large number ofboundary conditions

One possible approach to simplifying the analysis, whilst improving flexibility androbustness, is to employ matrix methods Matrices have been used extensively inengineering problems which are highly numerical in nature In microwave engineering[13], matrices are used to find the electric and magnetic fields inside various microwavewaveguides and devices One major advantage of matrix methods is their flexibility Instead

of repeatedly finding complicated analytical eigenvalue equations for each laser structure, ageneral matrix equation is derived Threshold analysis of various laser structures includingplanar section, corrugated section or a combination of them can be analysed in a systematicway Since they share the same matrix equation, the algorithm derived to solve the problemcan be re-used easily for different laser structures However, because of the numerical nature

of matrix methods, they cannot be used to verify the existence of analytical expressions in aparticular problem

In all matrix methods, the structures involved will first be divided into a number of smallersections In each section, all physical parameters like the injection current and material gainare assumed to be homogeneous As a result, the total number of smaller sections used variesand mostly depends on the type of problem For a problem like the analysis of transientresponses in LDs [14], a fairly large number of sections are needed since a highly non-uniform process is involved On the other hand, only a few sections are required for thethreshold analysis of DFB lasers since a fairly uniform process is concerned

For an arbitrary one-dimensional laser structure as shown in Fig 4.1, the wavepropagation is modelled by a 2
2 matrix A such that any electric field leaving (i.e ERðziþ1Þ

Figure 4.1 Wave propagation in a general 1-D laser diode structure

and ESðziÞ) and those entering (i.e ERðziÞ and ESðziþ1Þ that section are related to one anotherby

is performed in the time domain, whereas TMM works extremely well in the frequencydomain Table 4.1 summarises the characteristics of matrix methods

Using the time-domain-based TLM, transient responses like switching in semiconductorlaser devices can be analysed Steady-state values may then be determined from theasymptotic approximation However, it is difficult to use TLM to determine noisecharacteristics, and hence the spectral linewidth, of semiconductor lasers Due to the factthat most noise-related phenomena are time-averaged stochastic processes, a very longsampling time will be necessary if TLM is used In general, TLM is not suitable for theanalysis of noise characteristics in semiconductor laser devices

In 1987, Yamada and Suematsu first proposed using the TMM for analysing thetransmission and reflection gains of laser amplifiers with corrugated structures Thisfrequency-domain-based method works extremely well for both steady-state and noiseanalysis [6,9] In the present study, we are interested in the steady-state and noisecharacteristics of DFB lasers Hence, the use of TMM will be more appropriate

4.2.1 Formulation of Transfer Matrices

Based upon the coupled wave equations, one can derive the transfer matrix for a corrugatedDFB laser section From the solution of the coupled wave equations, one can express

EðzÞ ¼ ERðzÞ þ ESðzÞ ¼ RðzÞejb0zþ SðzÞejb0z ð4:2Þwhere ERðzÞ and ESðzÞ are the complex electric fields of the wave solutions, RðzÞ and SðzÞare two slow-varying complex amplitude terms and b0 is the Bragg propagation constant.From eqn (3.3), RðzÞ and SðzÞ have proposed solutions of the form

RðzÞ ¼ R1egzþ R2egz ð4:3aÞSðzÞ ¼ Segzþ S egz ð4:3bÞ

Table 4.1 Different types of matrix method

Scattering matrix ERðziþ1Þ and ESðziÞ ERðziÞ and ESðziþ1Þ frequency

TMM ERðziþ1Þ and ESðziþ1Þ ERðziÞ and ESðziÞ frequency

where R1, R2, S1 and S2 are complex coefficients which are found to be related to oneanother by [15]

Rðz1Þ ¼ R1egz1þ S2ejegz1 ð4:6aÞSðz1Þ ¼ R1ejegz1þ S2egz1 ð4:6bÞRðz2Þ ¼ R1egz2þ S2ejegz2 ð4:6cÞSðz2Þ ¼ R1ejegz2þ S2egz2 ð4:6dÞFrom eqns (4.6a) and (4.6b), one can express R1 and S2 such that

By substituting the above equations back into eqns (4.6c) and (4.6d), one obtains

E¼ eðz2 z 1 Þ; E1¼ eðz2 z 1 Þ ð4:8cÞFrom the above equations, it is clear that the electric fields at the output plane z2 can

be expressed in terms of the electric waves at the input plane By combining the aboveequations with eqn (4.2) we can relate the output and input electric fields through thefollowing matrix equation [6]

where matrix Uðz1j z2Þ represents any field propagation inside the section from z ¼ z2 to

z¼ z1 By comparing eqn (4.9) with eqn (4.11), it is obvious that

Uðz1j z2Þ ¼ Tðz½ 2 j z1Þ1 ð4:12Þ

where the superscript1 denotes the inverse of the matrix Due to conservation of energy,both matrices Tðz2j z1Þ and Uðz1j z2Þ must satisfy the reciprocity rule such that theirdeterminants always give unity value [4] In other words,

T

j j ¼ t11t22 t12t21¼ 1U

j j ¼ u11u22 u12u21¼ 1 ð4:13Þ

4.2.2 Introduction of Phase Shift (or Phase Discontinuity)

For a single PS DFB laser cavity as shown in Fig 4.3, the phase shift at z¼ z2 divides thelaser cavity into two sections

The field discontinuity is usually small along the plane of phase shift and any wavetravelling across the phase shift is assumed to be continuous As a result, the transfermatrices are linked up at the phase shift position as:

where Pð2Þis the phase-shift matrix at z¼ z2; zþ2 and z2 are the greater and lesser values of

z2, respectively, and 2 corresponds to the phase change experienced by the electric waves

ERðzÞ and ESðzÞ Alternatively, the physical phase shift of the corrugation may be used [9]

To avoid any confusion, we will use the phase shift of the electric wave hereafter

On combining eqn (4.14) with the transfer matrix shown earlier in eqn (4.9), the overalltransfer matrix chain of a single-phase-shifted DFB laser becomes

Without affecting the results of the above equations, one can multiply a unity matrix I aftermatrix T(1) This matrix I behaves as if an imaginary phase shift of zero or a multiple of 2
has been introduced As a result, the above matrix equation can be simplified such that

The use of the transfer matrix method is not restricted to the corrugated DFB laserstructure By modifying the values of  and  in the elements of the transfer matrix, otherstructures like the planar Fabry–Perot structure, the planar waveguide structure and thecorrugated Distributed Bragg Reflector structure can also be represented using the transfermatrix A DBR structure is different from the DFB structure because DBR structures have

no underlying active region The corrugated DBR structure simply acts as a partiallyreflecting mirror, the amount of reflection depending on the wavelength The maximumreflection occurs near the central Bragg wavelength Table 4.2 summarises all laserstructures that can be represented by transfer matrices The differences between them arealso listed

When the grating height g reduces to zero and the grating period approaches infinity, thefeedback caused by the presence of corrugations becomes less important At g¼ 0, becomes zero as does the variable  When becomes infinite, the detuning coefficient  isbecomes a planar structure Following eqns (4.9) and (4.10), the transfer matrix equation of

Table 4.2 Laser structures that can be represented using the TMM

Structure Active layer Corrugation Comments

the planar structure becomes

Similarly, the sign of  will decide whether a corrugated structure belongs to the DFB orDBR type By joining these matrices together as building blocks, one can extend the ideafurther to form a general N-sectioned composite laser cavity as shown in Fig 4.4 Laser

structures that comprise different combinations of the sections shown in Table 4.2 can bemodelled By joining these matrices together appropriately, one ends up with

where matrix YðzNþ1j z1Þ becomes the overall transfer matrix for the N-sectioned lasercavity Using the backward transfer matrix together with eqns(4.11) and (4.14), one obtains

ERð Þz1

ESð Þz1

¼YN m¼1

In the above equation, P ðmÞ1

is the inverse of the phase shift matrix PðmÞand Zðz1j zNþ1Þ

is the overall backward transfer matrix Comparing eqns (4.20) with (4.21), it is clear thatmatrices YðzNþ1j z1Þ and Zðz1j zNþ1Þ are inverse to one another such that

Zðz1j zNþ1Þ ¼ Yðz½ Nþ1j z1Þ1 ð4:23Þwhere the superscript1 indicates the inverse of the matrix From the property of the inverse ofmatrix products, individual transfer matrices GðmÞand FðmÞare related to one another That is

GðmÞ¼

FðmÞ1 for m¼ 1 to N ð4:24ÞThe above equation shows the equivalence between the forward and the backward transfermatrices in the general N-sectioned laser cavity Unless stated otherwise, the forwardtransfer matrix is assumed hereafter

4.2.3 Effects of Finite Facet Reflectivities

It was discussed in Chapter 3 that the lasing characteristic of the DFB laser depends on thefacet reflectivity In this section, the facet reflectivity will be implemented using the TMM

Figure 4.5 Schematic diagram showing reflections at the laser facets of a DFB LD

In Fig 4.5, a simplified schematic diagram for the reflections at the facets of the N-sectionedlaser cavity is shown.

In Fig 4.5, ^r1and ^r2 are the amplitude reflections at the left and right facets, respectivelyand medium 1 is the active region of the LD In most practical cases, medium 2 is air Due tothe finite thickness of coating on the laser facets, any electric field passing through maysuffer a phase change of iði ¼ 1; 2Þ Depending on the direction of propagation, all theoutgoing electric fields at the left facet (i.e ERðzþ

ERðzþ1Þ ¼ e

j 1ffiffiffiffiffiffiffiffiffiffiffiffiffi

1 ^r2 1

p ERðz1Þ þ ^r1e

j 1ffiffiffiffiffiffiffiffiffiffiffiffiffi

1 ^r2 1

ESðz Nþ1Þ

4.3 THRESHOLD CONDITION FOR THE N-SECTIONED

At¼ ESðz



1Þ

ESðzþ Nþ1Þ¼

1

When the matrix element h22approaches zero, the transmission gain becomes infinite and aresonant cavity is formed Physically, a laser that operates in this condition is said to havethe threshold condition satisfied After substantial manipulation of eqn (4.28), the thresholdcondition becomes

y22ðzNþ1j z1Þ þ ^r1y21ðzNþ1j z1Þ  ^r2y12ðzNþ1j z1Þ  ^r1^r2y11ðzNþ1j z1Þ ¼ 0 ð4:33ÞFor DFB semiconductor lasers having finite facet reflections, one needs to find all theelements of the propagation matrix YðzNþ1j z1Þ For a mirrorless DFB laser cavity where

^r1 ¼ ^r2¼ 0, the above threshold equation is simplified such that

y22ðzNþ1j z1Þ ¼ 0 ð4:34Þ

In fact, eqn (4.33) is a general expression that can be used to determine the lasing thresholdcharacteristics of semiconductor laser devices These include FP lasers, conventional DFBlasers (both mirrorless and those having finite facet reflections), single-phase-shifted DFBlaser structures, multiple-phase-shifted DFB laser structures [17–19] as well as multipleelectrode DFB laser structures [16,20–23] By increasing the number of sections, TMM can

be used to represent a tapered or chirped DFB laser structure [3– 4,24] Similarly, the transfermatrix method has also been used in surface emitting devices [25–26] which have receivedworldwide attention in recent years Table 4.3 summarises the minimum number of transfermatrices and phase shifts required in the threshold analysis of some popular semiconductorlaser structures [27]

EMISSION SPECTRUM USING THE TMM

In the previous section, the threshold equation of the N-sectioned laser cavity was definedusing the transfer matrix In fact, the TMM can also be applied to the below-threshold

analysis In semiconductor-based optical amplifiers and filters, the spontaneous emissionspectrum has been used to determine the bandwidth, tunability and signal gaincharacteristics Based on the use of Green’s function [28] for the noise calculation of theopen resonator, the transfer matrix formulation will now be extended to include the outputspontaneous emission spectrum taken from the right laser facet.

4.4.1 Green’s Function Method Based on the Transfer Matrix Formulation

In this section, we refer once again to the general N-sectioned laser cavity as shown earlier inFig 4.4 The amplitude reflections at the left and right laser facets are ^r1and ^r2, respectively,and perfect index coupling is assumed Following Henry [28], the one-dimensionalinhomogeneous wave equation of a transversely and laterally confined laser mode in thecomposite longitudinal structure can be expressed as

section only

section–Different corrugation period

in each section

corrugation period

corrugation depth g

laser