Design Optimization of Indium-Gallium-Arsenide-Phosphide Multi-Quantum Well Electroabsorbtion Modulators

A theoretical analysis of the performance of InGaAsP multi-quantum well electroabsorption modulators is presented. A comprehensive model of the quantum confined Stark effect is described to determine the absorption and index change spectra versus applied field. This model is based on previously developed models for GaAs/ AlGaAs structures, but includes improvements in the handling of exciton line broadening and the variation of exciton oscillator strength with field. The analysis of line broadening due to composition fluctuations is presented, revealing a previously neglected factor. Two numerical methods for calculating the line broadening, based on the resonant tunneling method, are presented and compared. A theoretical analysis of barrier composition fluctuation broadening is presented, which separately calculates the contribution from the electron and hole and their different penetration into opposite barriers when field is applied. The total linewidth model is compared with published linewidi:h measurements. Theoretical results of absorption spectra versus applied field were compared with two sets of experimental measurements. With appropriate choice of several unknown factors related to the quantum well fabrication quality, the theory and experimental data were \vell matched in shape of the absorption edge, shift of the edge with field, and decrease in the exciton oscillator strength with field. The theoretical model was used to optimize modulator device design, through the calculation of thousands of design combinations of device length, well number, well width, barrier width, well composition, and applied voltage. For each design, bandwidth, contrast ratio, loss, detuning, and several chirp parameters were calculated. It is shown that long distance transmission performance may be optimized with negative values of a specific chirp parameter called the 3dB Henry factor. Modulator design can be optimized for such values by operating close to the exciton and accepting high optical loss. Loss may be reduced by optimum choice of device length, well number, and barrier width, while it can be compensated by an optical amplifier. The optimum design changes considerably when the requirement for negative chirp is eliminated. Such designs use more quantum wells and tune the device further from the exciton. Finally, the model provides a means to choose the optimum well width.

structures, but includes improvements in the handling of exciton line broadening and the variation of exciton oscillator strength with field.
The analysis of line broadening due to composition fluctuations is presented, revealing a previously neglected factor. Two numerical methods for calculating the line broadening, based on the resonant tunneling method, are presented and compared. A theoretical analysis of barrier composition fluctuation broadening is presented, which separately calculates the contribution from the electron and hole and their different penetration into opposite barriers when field is applied. The total linewidth model is compared with published linewidi:h measurements.
Theoretical results of absorption spectra versus applied field were compared with two sets of experimental measurements. With appropriate choice of several unknown factors related to the quantum well fabrication quality, the theory and experimental data were \vell matched in shape of the absorption edge, shift of the edge with field, and decrease in the exciton oscillator strength with field.
The theoretical model was used to optimize modulator device design, through the calculation of thousands of design combinations of device length, well number, well width, barrier width, well composition, and applied voltage. For each design, bandwidth, contrast ratio, loss, detuning, and several chirp parameters were calculated.
It is shown that long distance transmission performance may be optimized with negative values of a specific chirp parameter called the 3dB Henry factor. Modulator design can be optimized for such values by operating close to the exciton and accepting high optical loss. Loss may be reduced by optimum choice of device length, well number, and barrier width, while it can be compensated by an optical amplifier.
The optimum design changes considerably when the requirement for negative chirp is eliminated. Such designs use more quantum wells and tune the device further from the exciton. Finally, the model provides a means to choose the optimum well width.

PREFACE
There has been considerable interest in the development of external modulator devices for optical communications. The conventional methods based on direct modulation of a diode laser's drive current tend to be limited by the laser's frequency chirp associated with the modulation. This chirp results in system bandwidth limitations through pulse dispersion in long fiber optic links. With external modulation the laser is operated in a continuous output mode and modulation is provided by an external electrooptic modulator device. Laser chirp is eliminated and the system is limited instead by the lesser chirp introduced by the modulator. Elimination of the modulator function also allows optimization of the laser design for other performance aspects.
A key contender for the role of external modulator is the multiple quantum well (MQW) electro-absorption modulators based on the quantum confined Stark effect (QCSE). These devices have demonstrated high data rate capability and sufficient contrast ratio and loss performance.1-3 Electro-absorption modulators operate at a wavelength which is in the transmissive wavelength region above the device's absorption edge in the "on" state. In the "off' state a voltage is applied which shifts the absorption edge and results in high absorption at the operating wavelength. With the QCSE the absorption edge of the semiconductor is considerably sharper or more abrupt with wavelength due to excitonic absorption. In a bulk semiconductor, Wannier excitons consist of an electron in the conduction band and a hole in the valence band which are bound together through their Coulomb interaction to form a state similar to a hydrogen atom. In a bulk semiconductor the excitons are seen only at low temperature and quickly disassociate with applied field. In a quantum well, the electrons and holes are confined in close proximity in one direction in the quantum well. The excitons persist at room v tennperature and with high fields applied perpendicular to the quantum well layers. As field is applied, the shape of the potential wells for electron and hole changes and the subband levels within the well change. The result is that the energy of the exciton absorption shifts with applied field. This effect is known as the quantum confined Stark effect and it results in large movements of a sharp absorption edge, both aspects being conducive to good modulator performance.
Historically, these modulators were first developed in the GaAs/AlGaAs material system for use at wavelengths near 0.8 µm.4-5 Given the importance of the low loss and low dispersion 1.3 µm and 1.55 µm communications windows in optical fiber, MQW modulator development for those wavelengths is necessary. The wavelength of absorption in such devices is a function of the semiconductor's bandgap and thus different material systems are required for operation at 1.3 and 1.55 µm. The quaternary In1-xGaxAsyP1-y system is a major contender.
Quaternary semiconductor systems give additional degrees of design freedom, allowing separate optimization of both the well width and the well/barrier compositions.
Initial work by Nojima and Wakita6 indicated that large well width leads to large exciton shift with field but decreased oscillator strength. It is very important to optimize this tradeoff as well as other tradeoffs of the various design parameters and performance factors. The potential appears to exist to greatly improve performance of InGaAsP MQW modulators through design optimization.

An adequate theoretical model of the basic electroabsorption properties in
InGaAsP quantum wells is required as a basis for this optimization. The quantum confined Stark effect was first described in the GaAs/ AlGaAs system by Miller et al. 4,5 Since that time the theoretical treatment of the electroabsorption in that material system has Leer. extensively treated 7-17 and the best models describe the experimentally observed behavior quite well. In contrast, there have been few experimental measurements of InGaAsP MQW electroabsorption [18][19][20][21] and less work to theoretically Vl explain performance or to seek to optimize device performance.6,22-23 Qualitatively, the electroabsorption characteristics of InGaAsP MQWs are quite different from those of GaAs/AlGaAs MQWs and further from optimum. Exciton absorption peaks in InGaAsP are less prominent and fade or broaden quickly with field. Exciton linewidths are increased, giving a less sharp absorption edge. Modulator performance depends on the ab~oq.>tion change which can be achieved with a given voltage. It is enhanced by prominent, sharp exciton peaks. The observed broadening and fading of InGaAsP excitons leads to reduced, though still usable, performance. It is therefore necessary to explore the applicability of those theoretical models developed for GaAs/ AlGaAs to the InGaAsP system and to investigate modifications necessary to account for differences between the two systems.
The objective of this work has been two-fold. The first objective has been to develop a theoretical model of electroabsorption in InGaAsP MQW modulators to serve as the basis for design optimization of such modulators. The goal has been to develop a model which balances the requirement for accurate reproduction of experimental measurements of the exciton peak and absorption edge, with the requirement for comp1~tational simplicity to enable multi-parameter device design optimization. The basic approach has been to make use of previous theoretical· work done for the GaAs/ AlGaAs system, extending or modifying it as necessary to account for differences with the InGaAsP system. Results have been compared to several published experimental studies of lnGaAsP devices and unpublished measurements available from Siemens researchers.
The second objective is to use the theoretical model as part of a multiple parameter device design optimization. To date there has been very little published work to optimize device design and that which has appeared has made use of overly simplistic models for the electroabsorption.6, 16,[22][23][24][25] In addition, previous device optimization studies have focused on a limited number of performance parameters, usually some combination of bandwidth, drive voltage, loss, and extinction ratio. It is common to form composite vu figures of merit from some of these performance parameters and to optimize the design for these figures of merit However, when the figures of merit leave out key performance parameters or place undo emphasis on some parameters, the implications of the design optimization become misleading. Two examples are illustrative. The use of fiber optic amplifiers allows higher modulator insertion loss in some important applications. Figures of merit which over-emphasize low loss may lead to wrong design choices. Also, there has been no published study which considered the modulator chirp performance as part of the device optimization, yet chirp performance is a key parameter in long distance, high bit rate communications and is the primary reason driving the use of external modulators in the first place. The work presented herein considers device optimization across a full range of performance parameters, including chirp performance.
Chapter 1 presents a theoretical description of the unbounded conduction and valence band states and the exciton states in a quantum well (QW). Chapter 2 develops the theory for optical absorption associated with electron transitions between these states. Both continuum absorption due to unbounded states and exciton absorption is treated.
where V uc is the unit cell volume.

QW Single Particle States
In the QW structure, the particles are partially confined and their wavefunctions localized. The individual particles are confined in the growth (henceforward the z) direction by the potential difference between the well and barrier, giving localization in 1 the z dirrction. The localized functions may be described by a linear combination of the Bloch wave functions. In the general case of full three dimensional localization: The various D 8 (k) are the coefficients of this expansion.
When localization is only in the z direction, the functions may be expressed as: where r 11 and k 11 are the position vector and wavevector in the x,y plane of the QW layer, respectively. In the effective mass or envelope function approximation, a major assumption is that the Bloch functions are not strong functions of k and can be approximated by their value at the band edge k=O. Thus: The full wavefunction is described as the product of the band edge Bloch functions and the envelope wavefunction F 8 ( r). The envelope function is the solution of the effective mass equation rather than the full Schrodinger equation : Since absorption near the band edge is of primary interest, parabolic dispersion relationships are usually assumed for each band. n2 Here m~ is the effective mass for band B, E 8 • 0 is the band edge energy, and Vis the macroscopic potential (which may include the potential due to any applied field but doesn't include the lattice potential). For the MQW, the envelope function 'l'~(z) in the z direction is just the solution of the one dimensional "particle in a box" problem. Vis the confining potential of the well defined by the bandgap difference between barrier and well and the splitting of that difference between conduction and valence band. It may also include any field applied in the z direction. The splitting in InGaAsP is often assumed to 2 be 60%:40% between valence and conduction band, respectively. The envelope function is labeled with the band index Band the subband index n labeling the various solutions of the "particle in a box" problem.
The InGaAsP valence band structure is complicated by a light and heavy hole band degenerate at zone center, and the split off band as shown in Figure 1.1. The exciton binding energy is much less than the energy gap to the split off band and this band is usually ignored. The reduction in symmetry in the QW and the band discontinuities lifts the degeneracy of the light and heavy hole bands at zone center. Provided the wells are sufficiently narrow, the subband separation is greater than the exciton binding energy and there is very little coupling between bands. In this case the heavy and light hole bands can be treated separately. Several authors have included band coupling in more complicated theoretical treatments. [26][27][28] Their results showed that including such effects gave only small corrections, these corrections becoming negligible as field was applied.
In what follows, coupling between bands is ignored and each band can be solved separately.

Resonant Tunneling Method (RTM)
It is necessary to calculate the electron and hole· energy subband levels in the one dimensional quantum wells both with and without applied field. Associated with this is the task of determining the electron and hole z direction envelope wavefunctions.  4 The resonant tunneling method is used within the effective mass approximation.
The sloping potential due to an applied field is approximated by small steps as in Figure   1.2. In each step the particle envelope wavefunction is written in terms of plane wave states: where k is the complex wave number: (1.9) m* is the particle effective mass, Vo is the potential, and E is the total particle energy. A transfer matrix is derived to describe transmission across a potential step by requiring that the wavefunction and probability flux on each side matches at the interface. At x=p: This leads to the transfer matrix equation: Y th e coefficients (A,B) at each step in the structure give the particle one energ , dimensional wavefunction as in Figure 1.4. Finally, the method also gives the broadening of the energy level due to the limited lifetime of the particle in the well due to tunneling. The resonant peak is approximately Lorentzian in shape as seen in Figure 1.5.
This shape corresponds to the tunneling induced broadening of the energy level. As the barrier thickness is reduced and tunneling is increased, the resonance becomes broader. This method may be utilized for multiple wells and coupling between wells is well described. In the rest of this work, however, only calculations with single wells will be employed. It is assumed that coupling is sufficiently low that the MQW characteristics can be described by calculations for a single well. The tunneling linewidth will serve as a check on this assumption.

QW Exciton States
Excitons are bound pairs of electron and hole states. The exciton states are thus two particle states. In the exciton, the wavefunctions are further localized in the x,y direction by their mutual interaction. In the effective mass approximation the Schrodinger-like equation for the envelope wavefunction of the exciton is written by combining the equations for electron and hole and including the Coulomb interaction. 32 As noted before, the heavy and light holes will be treated as separate exciton systems. It is assumed that the barriers are thick enough that coupling is negligible between neighboring wells and the MQW can be described by considering the action of a single well. For the electron: Ve <1Bd v h are the confining potentials of the wells defined by the band gap difference between barrier and well and the splitting of that difference between conduction and valence band. The field applied perpendicular to the MQW layer is FJ..
Combining (1.15) and (1. 16), and including the Coulomb interaction, Ve-h: (HKEe +Ve -eFJ.ze + HKEh + vh + eFJ.zh + ve-h(re,rh))'I'(re,rh) = E'I'(re,rh) (1. 18) where 'I'(r.,rh) is the two particle envelope wavefunction for the exciton. The coulomb tenn couples the electron and hole coordinates and the problem is no longer separable. It is useful to transfonn to center of mass coordinates in the plane of the layer.  ( 1.20) It should be noted that this represents a change in notation. The relative position vector r in the plane of the layer should not be confused with the previous general three dimensional one particle position vector r. It can be shown that:  where: A common simplification5 is to ignore the coulomb effect in the z direction, aswming it to be insignificant in comparison to the confining potential of the well. The single particle z direction envelope wavefunctions are then used unchanged. The variational minimization is performed on only one variational parameter in the plane of the layer. A separable trial wavefunction of the following form is used: The motion of the center of mass is ignored in the following. The photon momentum is too small to significantly affect it. The z direction wavefunctions '¥ e and 'Ph are given by the solution of the one dimensional single particle equations: Assuming this form for the wavefunction in effect assumes that the coulomb interaction in the z oirection is insignificant in comparison to the well confinement and does not alter the wavefunctions from those of the single particle case. The coulomb interaction is only ted for in the x-y plane. The radial envelope wavefunction is a solution of the two accoun ·onal hydrogenic problem, labeled by index n. In the following only the 1-S like dimens1 state is considered: The binding energy is a combination of the kinetic energy of the relative electronhole motion in the layer and the Coulomb potential of the electron-hole motion: (1. 31) where: tz2 £ :::: (<t>IH KE.r l<t>) :::: 2µ/l.2 XE.r (1.32) E :::: ('f'!Ve-hl'f') PE.r The potential energy may be written as: The integral over 8 is trivial and the integral over r can be handled separately. Equation (1. 34) where: (1.38) The first and zeroth order Bessel function of the first kind and the zeroth order Bessel function of the second kind are found from power series expansions.
The calculation of E,E, is modified from that of Miller et al.5 in that the electron and hole one dimensional wavefunctions from the resonant tunneling calculation,'¥ e and 15 'l'b, are utilized rather than approximate analytic forms used by Miller et al. The double integral over Ze and Zb is evaluated numerically using Simpson's rule.
As field is applied, the electron and hole are pulled to opposite sides of the well and the exciton is less strongly bound. This effect enters the calculation through the use of the resonant tunneling electron and hole wavefunctions. As a result, the exciton lateral sire is increased. This effect gives a small correction on the exciton energies. More importantly, the exciton size will be found to be important for determining the inhomogeneous linewidth broadening and the oscillator strength.

Absorption Coefficient
The absorption coefficient for the QW material is derived using first order time where Ao is the amplitude and e v is the unit polarization vector of the electromagnetic vector potential given by: The absorption coefficient is given by: where U is the energy density of the electric field and (c/n)U is the energy flux. The amplitude of the electric field is obtained by equating the energy in the field to 1im, finally obtaining: 2 a= m 0 c:(c/n) ~~ ~ IUlei~ .. ev · Pli)j2 8( E 1 -Ei -1im) (2.4) £ l ·s the dielectric constant and n is the volume. where The MQW absorption spectra exhibit both sharp resonances due to excitons and a broad continuum absorption due to band to band transitions between unbounded valence and conduction band states. The above equation may be used for evaluating both the continuum absorption due to the unbounded electron and hole states and the exciton absorption due to the exciton states. It is necessary to use the correct forms for the initial and final states and determine the matrix elements for the specific case of interest

QW Continuum Absorption
As shown in Chapter 1, the total wavefunction for either conduction or valence band states may be expressed as: where r 11 is the in-plane position vector and the periodic Bloch function has been split into separate functions with z and x,y dependence.
The matrix element is separable as: e ev pt -z e ev·Pz tz J11e ev·P11~1 (2.6) The in-layer part is handled first. An electron transitions from a valence band state to the conduction band in the process of absorbing a photon.
Next the z dependent part is solved.
The same procedure is followed as with the in-layer part and it can be shown that: Finally, equations 2.4, 2.6, 2.8, and 2.10 are combined. The sum over initial and final states becomes a sum over the various conduction and valence subbands, over . and over all k 11 in the Brillouin zone. spin, Here the two in parenthesis is due to spin. The optical matrix element Pcv comes from combining the z and in-plane matrix elements as: Pcv = (uc,z lev · Pzluv,z)(uc,11 lev · Puluv,11) (2.12) Next, the summation over k 11 is converted into an integral and cylindrical coordinates are used. Given that the main interest is in the lowest energy transitions near zone center, the bands may be assumed to have approximately parabolic dispersion relations: Finally: where the reduced mass is given by: To study the absorption near the band edge only a limited number of valence and conduction subbands will be considered in practice.

QW Exciton Absorption
(2.13) (2.14) ( Actually start with an electron in valence band: Combining (2.17) and (2.19) to get the initial state: Converting to center of mass coordinates in the x,y plane: Te Ze Th zh Ui1,cl ' 11,vuz,cuz,v As before, the center of mass part will be ignored in the following.
The final exciton state is expressed as: '11,vuz.cuz ,v The matrix element is again separable as: The photon momentum term was dropped because, as was seen in the continuum case, it does not affect the results of interest.
Considering the z problem first: However, the transition from a hole in the conduction band and electron in the valence band to an electron in the conduction band and a hole in the valence band is the same thing as an electron transitioning from valence to conduction band.
(fz lev · Pzli. ) = ('1' e ( Ze )uz,c lev · P.1'1': Following the same derivation as used earlier for the continuum, it can be shown that: 26) 1 . dering the part in the layer: Next, cons {Ju ~v. Pul~i):: (<I> 11 di mensional hydrogen-like wave function in the x-y plane is next expanded in Tue two f th e in-plane "ii plane wave states: Finally, combining (2.26) and (2.31): where in the last statement attention is restricted to near zone center where the coupling is assumed independent of k.
Finally, in determining the absorption, the sum over initial and final states becomes a sum over the various conduction and valence subbands, over spin, and over index n. (3) Gaussian inhomogeneous broadening due to random alloy compositional broadening, . both wells and barriers, and (4) inhomogeneous broadening due to random disorder Ill variations in well width. The first two are intrinsic, while the second two will be sample dependent.

Tunneling Broadening
It has already been shown in Chapter 1 that the resonant tunneling method well deseribes the linewidth broadening due to tunneling. Figure 1.5 indicates that this broadening is well described by a Lorentzian lineshape.

Thermal Broadening
Measurements of exciton linewidth versus temperature were reported by Chemla et aI. 35 for GaAs/ AIGaAs multi-quantum wells and by Sugawara et al. 33,36 for InGaAsP/InP. Their results were well described as broadening due to exciton ionization due to interaction with longitudinal optical (LO) phonons. The linewidth is proportional to the density of LO phonons, described by Bose-Einstein statistics as:35 where C is the new concentration and Co is the average concentration. Actually the wavefunction is a two particle wavefunction and it is appropriate to consider separately 24 mposition affects the electron and hole. Recalling the previous form of the how the co f tion from equation 1.27 the following is obtained: For a purely random alloy or one with random clustering, the probability of a composition fluctuation ..1C occuning over a volume V is governed by binomial statistics.
This probability has been derived by both Schubert et al.38 and Singh and Bajaj39. Schubert notes that the binomial distribution can be approximated by a Gaussian distribution, the necessary condition being met throughout the composition range of interest. When this is done the full width at half maximum (FWHM) of such a distribution is given by Singh and Bajaj as40: where Ve is the cluster volume or the minimum volume over which a concentration fluctuation can occur. For a purely random alloy without clustering, the cluster volume is the volume per cation or anion. Combining the above: The exciton linewidth is found by considering all possible volumes V to find those configurations which maximize the shift LIB.
Next, cylindrical volumes are considered, with variable height ~ (not related to photon wavevector ~) and radius r with variable parameter y such that : r= rll V-: r.:·f ll2{3 Using such a volume and the separability of the wavefunctions, the linewidth is given by: First consider the optimization over the plane of the layers. The wavefunction is given by: Thus: This is maximized for y=0.9 at which the value is 0.59685. This is the QW version of the quantum mechanical correction factor first introduced for bulk semiconductors by Singh and Bajaj.40 This has previously been wrongly omitted by other authors.34 Incorporating this factor into equati_ on 3.9: Two methods were used to evaluate equation 3.12. The first utilizes the RTM to analyze the energy shifts when composition is changed in regions of various height, ~.
Because the method is computationally intensive and must be done numerous times (left barrier, right barrier, well; x and y; three exciton types; multiple fields) it is too involved for 05 e in a modulator design optimization model. The second model is a first order approximation. Both will be described and the results compared.

RTM Method
The RTM is capable of handling an arbitrarily complex stack of semiconductor layers. It is thus possible to alter the composition slightly in a small region of either the well or barrier and recalculate the electron and hole energy levels. Comparison with the levels from the normal configuration gives the shift associated with the composition change. It is noted that this is not a first order approximation because the RTM completely recalculates the problem, determining both new subband levels and new wavefunctions. It takes into account changes in both bandgap and effective mass associated with the composition change. The RTM automatically takes into account the wavefunction weighting of the perturbation: rforrned for the case in which the entire well composition is changed. This is levels 1s pe the top diagram of figure 3.1. This is repeated for both a composition change shown as in x and for a y change. The amount of composition shift for the RTM calculation was varied to ensure that the derivative converged. The final program altered the composition b 2 percent in x or y. The subband level shifts for the various regions are calculated as: Because the fluctuations involved are small, the binding energy and exciton size/... are assumed to be unaffected by the composition perturbations and are not recalculated.

First Order Approximation
The linewidths may be approximated by assuming that the change in energy levels is directly related to the energy gap change alone:33 When changes in x composition are considered, the less-accurate, two-way interpolation formula is used: Levels from the two equations cannot be compared without errors. Thus, when the RTM program determines level shifts due to x fluctuations it is necessary to compare levels from the unperturbed case derived using 3. 18  the two methods for this case. The variation between the RTM method and the first order approximation for the heavy hole runs from 2% to 7% with fields from 0 to 120 kV/cm. Agreement is good for the light and second heavy hole excitons as well. Given the limited accuracy in determining exciton linewidths and the lesser importance in comphlison with other broadening factors, this level of accuracy is sufficient. The first order approximation was therefore used in the modulator design optimization. 2.1e-3 .,,.,, ...
where Leff is "an effective length to which the exciton wave function penetrates into the barrier and has to be calculated numerically". This definition indicates that they did not consider the composition effects on electron and hole separately. This is further indicated because they considered both barriers together rather than separately. This is mappropriate. The two barriers are not equivalent except at zero field. As field is applied, the electron shifts into one barrier while the holes shift into the opposite barrier. The linewidth contributions will differ from one barrier to the other. This study has treated barriers separately. A comparison of equation 3.21 with 3.12 indicates that they the tWO _ 1 ,.,,. failed to include the 0.59685 quantum mechanical correction factor. have iW)V The difference in linewidth contribution from one banier to another and from to hole can be seen by looking at just the sub band shifts that result when the electron . bam·er is changed in composition, one banier at a time. This was done for the enure previous example of the 7.5 nm well and 10 nm banier. The result is shown in Figure   3 . 3 . The subband shift is greatest for the electron since it penetrates further into the banier. At zero applied field the shifts due to left or right banier changes are equivalent.
As field is applied the electron moves into the right barrier and its contribution to subband shift dominates. Likewise, the hole moves into the left banier and its contribution grows.
In the barrier, the wavefunctions generally fall off in an exponential manner. For this reason, the regions of altered composition start at the well/banier interface and different thicknesses are used. The regions used are depicted in Figure 3.4 for the conduction band. An RTM calculation is done to determine the subband level shifts for each of four beta thicknesses. To generate smaller ~ steps, a curve of subband shift vs.
Pis piecewise quadratically fit to the four RTM calculated cases and used to generate the shifts for ~ regions in between. The ~ regions were calculated with steps of single monolayers, 2.88 Angstroms. Again, the calculations must be performed for composition changes in both x and y.
The valence band is similar.
The barrier calculation is very computationally intensive. Separate RTM subband level calculations must be done for the following: It is for this reason that this method was not considered appropriate for the design optimization program and a simpler approximation was sought. The first order approximation was applied by using equations 3.16 and 3.12 with the same barrier ~ regions as described above. Results of the two methods are shown in Figure 3.5. They

37
. · ly well when it is considered that the first order approximation effectively agree surpnsing antum well confinement nature of the problem. The agreement is also jgnores the qu . th t, given the small linewidth contribution relative to other broadening suftietent a the estimate is acceptable for use in the modulator design optimization. S ition broadening will decline and the barrier composition broadening well cornpo inCrease· Sugawara et al. did not consider the application of field, but inclusion of the a1><>ve factors also captures the field effect of pushing the excitons into the barriers.

Bulk Composition Fluctuations
As mentioned above, the linewidth models will later be compared to experimental data that includes quantum wells of various thicknesses and bulk semiconductors.
Therefore it is necessary to derive the composition broadening for bulk semiconductors.
The derivation of Singh and Bajaj40 is presented. Returning to equation 3.3: As before: ..JV It is assumed that : The three dimensional exciton has a hydrogenic wavefunction: al over the wavefunction is calculated: 'lbe maximum over V is found: The maximum over g is found to occur at g=l. 16   Here the linewidth increases with field. The field pulls the electron and hole wavefunctions toward the barrier, increasing the wavefunction at the interface. This linewidth is also highly dependent upon well width because the wavefunctions are higher at the interface for narrower wells. The above equation may also be used to describe the broadening from random variation in well width from well to well within the multiquantum well stack. Some fabrication error in well width is to be expected.

Comparison of Linewidth Theory with Published Experimental Data
The existing theoretical work on exciton linewidth is largely unverified. Many of the broadening mechanisms appear to be of similar magnitude and it is difficult to sort out  Well Given the usual lack of detailed data on such quality factors, and the quantum · Po~bility that they vary from manufacturer to manufacturer or even from wafer to wafer, these parameters may be considered fitting parameters. In the composition broadening, the cluster volume is undetermined. Practical devices may exhibit clustering     46 ent has an associated variable parameter: cluster volume for random Eaehcompon . · 0 broadening, FWHMcomp for macroscopic composition broadening, and compos1uo fWHMisland for well width broadening. These were chosen to give the best fit of the the<>retical linewidth to the data The result is that the narrow well data indicates the well width broadening, the additional broadening in the mid-well-size region indicates the random composition broadening, and the bulk data indicates the macroscopic composition broadening. Resulting cluster volumes were 2.5 and 1.5 times the volume per cation for the y=l.0 and 0.6 cases respectively. The volume per cation is given by 00 3 /4 where ao, the lattice constant, is that of the InP to which the MQW is lattice matched. Resulting values for FWHMisland are 0.19 and 0.73 monolayers for y=l.0 and y::0.6 respectively using a value of 2.9344 angstroms per monolayer.
The random composition broadening is seen to increase with reduced well width due to the associated shrinkage of the exciton volume. The smaller wells confine the electron and hole closer to each other, increasing the binding energy and resulting in a smaller exciton size. It is more likely to realize a composition fluctuation across a smaller volume. At very small well widths the exciton penetrates into the barrier and the random well composition broadening drops. The well width broadening is non existent for the bulk which has no interfaces. It is still very low for large well widths but rises rapidly at small well widths because the wavefunction at the interface is greater in such wells. In the next chapter, comparison of a full absorption spectra model with experimental data from Zucker et al. 19 and unpublished data from Siemens will be presented. The linewidths are quite different between the Sugawara, 18 Zucker,19 and Siemens data. Both the Zucker and the Siemens devices demonstrate much worse lin~width. This is more than can be accounted for by thermal broadening in the room temperature spectra. The Zucker and Siemens data were also quite different, with the Zucker linewidth remaining essentially unchanged with applied field while the Siemens linewidth increased with field. It appears that at the present stage of development of InGaAsP devices, there is great variability in the process control and accuracy of fabrication achieved. Different manufacturers may have quite different levels of the various linewidth factors. Accuracy· of linewidth measurements is also somewhat questionable. While MQW stacks of many layers have been used in measurements with the GaAs/ AIGaAs system, most measurements with InGaAsP have used far fewer quantum wells)8,l9 The total absorption length is thus reduced, leading to poorer signal to noise ratio in the spectral absorption measurements.
In conclusion for this chapter, a number of line broadening effects have been theoretically modeled. The composition broadening model was derived based on a model by Hong and Singh34 but including a quantum mechanical correction factor and 1 nted through the resonant tunneling method for the first time. A simplified iJDP eme rnethod was also developed based on assuming that the change in energy levels is directly related to the energy gap change alone. Results for the two models were compared and the simplified model was shown to be sufficiently accurate for use in device design optimization. A barrier composition broadening model was also developed, which for the first time correctly treats the exciton as a two particle system with the electron and a(tzm, F) = j('P. l'P ht q.xL( tuv, Ecv( F)-Eb( F)) + f l('P e I 'I' ht NqconK ( E', Ecv(F)

)L( nm, E')dE'
(4.1) E,,,, where L is a Lorentzian function representing homogeneous broadening, Ecv is the energy separation of the electron and hole subband levels, Eb is the binding energy, qexc is the exciton oscillator strength, N is the density of states, and qcon is the continuum oscillator strength which was fit to the data. K is the Sommerfeld factor which is given by: where Ry is the three dimensional Rydberg constant: The Sommerfeld factor represents the enhancement in the continuum due to .d nn· g the coulomb interaction. It was first derived for the three dimensional cons1 e . by Elliot SO Unlike the MQW case, the three dimensional and the pure two exciton . ·,.,.nal exciton problems can be solved analytically within the effective mass diJD'OSh.
l ·mation In doing so, Elliot showed that the solution was a series of exciton lines. approx · 'JbeSe became more closely spaced and dropped in intensity as the bandedge was approached, forming a quasi-continuum whic~ blended smoothly into the true continuum above the band edge. In chapter two we derived the absorption between unbounded states in a MQW, neglecting any Coulomb interaction between the electrons and holes. It is straightforward to derive a similar result for the three dimensional bulk semiconductor.
The ratio of Elliot's absorption to that with the no-interaction derivation gives the three dimensional Sommerfeld factor. Shinada and Sugano51 derived the absorption for the pure two dimensional exciton. Their Sommerfeld factor is given as equation 4.2. It gives an enhancement by a factor of two at the band edge and fades to no enhancement far above the band edge. Calculations by Chan52 suggest that for a MQW the Sommerfeld factor may be less than two. With this in mind a variable Sommerfeld factor has been used: with values of 11 between zero and one.
A major drawback of the work of Stephens et al. 16

and Bandyopadhyay and
Basu 2 3 is that they used a fixed ratio for the oscillator strengths: This value for the ratio is between the values for a pure three dimensional or pure two dimensional case. They maintain the same ratio, even when the applied field changes.
This i~ equivalent to assuming that the exciton does not change size or binding energy with field. 51 I I U ·o of oscillator strengths is directly available from equations (2.14) and Tite ra (2.33).
Nqcot1 (4.6) 1be ratio is seen to be quite sensitive to the exciton size. Thus, this ratio may be the most .. e measure of lambda available because the binding energy is small and only sens1uv }ineafly dependent upon lambda. Exciton size varies with applied field and this should have important impact on the above ratio. For these reasons, the following form for the absorption spectra was initially used in this study: The normalized Lorentzian function is: where <1hom is the homogeneous linewidth due to tunneling. J(ramers-Kronig calculation of index change will be used to determine the In adctuon. a -' rmance of the modulators. In such a calculation, a light hole absorption cbitP penO featUJ'C at higher energy contributes significantly to the index change at lower energies. It js therefore important to include the light hole exciton. The le:2hh exciton was found to fall at almost the same energy as the le:lh. It is initially a forbidden transition at zero field. The electron wavefunction is symmetric and the second heavy hole wavefunction is assymetric, giving zero for the overlap integral. As field is applied, however, this is disturbed and the overlap integral grows so that the le:2hh exciton is a significant contribution to the spectrum at high fields.

55
The expression in equation 4.7 is convolved with a Gaussian lineshape . g the inhomogeneous and thermal broadening. The linewidth of the Gaussian representm 'oination of thermal, random composition broadening, macroscopic composition jsacom d ru ·ng and well width broadening. The well width broadening can represent either broa e • interface roughness or variation in well thickness in a MQW stack. The thermal broadening is not expected to vary much from manufacturer to manufacturer. Sugawara etal.36 found a thermal FWHM of 8.9 to 9.2 meV and values of9 to 9.5 meV were used in this study. The fitting parameters for the other broadening factors were varied in order to fmd a combination which fit the experimental zero-field spectra, and continued to fit as the linewidth changed with applied field.
Typical calculated spectra illustrate the importance of using a full absorption spectrum to evaluate any theoretical model. Many researchers have not compared the full spectra, but have tried to read the exciton shifts, or exciton heights, or linewidths from measureJ data.  A key feature of the Zucker data is that the linewidth remains approximately unchanged with applied field. In addition, the linewidth is greater than that of Sugawara's devices,18,33 even when thermal broadening is accounted for. It was necessruy to increase some of the inhomogeneous broadening components relative to that found for Sugawara's data and to find a combination yielding constant linewidth. Well composition broadening is the only linewidth component which decreases with field and hence is capable of offsetting tunneling and well width broadening which increase. The fitting factors which best fit the data are given in Table 4.1.
Strain applied in the plane of the layers through lattice mismatch, either intentional or unintentional, has as one of its effects the shifting of the light hole exciton relative to the heavy hole. In figure 4.5 to 4.9 a strain shift of+ 10 me V was incorporated to better match the separation of light and heavy hole in the data.
The major problem with the fit is that the model spectrum falls too much with field. As mentioned previously, references (16) and (23)   This simplification was justified in the past in that it gave fairly accurate results.
This problem would have been less of a problem for the GaAs/ AlGaAs system for which most of the theoretical work was done, because the well depth for electrons is significantly greater than that in InGaAsP. It may be that other simplifications, such as .
. the ).-2 dependence, masked the deficiencies in predicting the absorption 1gnonng . de It is also noted that the magnitude is the characteristic most sensitive to such roagn1tu .
A 20% error in binding energy is of minimal impact to the exciton shifts but an error.
. substantial change in magnitude through the A -2 dependence.
gives a 'Theoretical approaches which include the z direction interaction were investigated.
One The second possible explanation is the uncertainty in the splitting ratio. While a 40% to 60% conduction band to valence band split has been used so far, various researchers have found a broad range of splitting ratios. This ratio was treated as an additional fitting factor in the method already described. ..   The fitting factors which best fit the data across the range of applied fields are given in Table 4

74
Model Data {rorn rough interfaces or from variations in well thickness from well to well. It is also d th at tunneling is significantly higher than for the Zucker et aI.19 devices. This is a note difeCt consequence of the smaller composition difference between the wells and barriers, giving a shallower well.

Design Approach
The goal of all the material modeling thus far presented is to support the design optimization of optical modulator devices. It is necessary to take the absorption spectra vs field data of the material model and derive from these the modulator performance. It is first necessary to identify those performance parameters of interest and the design parameters which are available to define the design. Table 5.1 gives a possible list of each. Given the large number of design and performance parameters, careful thought b g iven to a basic optimization strategy. The few published works on optimization must e of MQW modulator design have usually relied on establishing figures of merit to combine several desired performance parameters into a single parameter. The most common are the device's bandwidth/drive voltage ratio or the ratio of the contrast to the drive voltage/length product. Figures of merit are inadequate when they leave out important performance characteristics or inappropriately weight the importance of the parameters.
With efficiency figures of merit it is also possible to drive the design to high efficiency and no performance, i.e. the design that gives the best bandwidth to drive voltage ratio may not give the bandwidth one desires. Given these problems, a figure of merit was not used and the performance parameters were simply presented in tabular form.
Performance cut-offs for the different parameters were established, and designs not meeting one or more of the requirements were not presented.
Several assumptions were made relative to the design process. Rather than designing with a variable drive voltage, it was assumed that a certain maximum drive voltage is available and the device must be designed for maximum performance with that driver. This is more commonly the situation in practice as drive power is severely limited at microwave frequencies. This eliminated drive voltage as either a design parameter or a performance parameter. A drive voltage of 2 volts and a built in voltage of 1.15 volts was used. In a few cases in Chapter 6, this fixed voltage was changed and the model recalculated to determine the effect of a lower drive voltage on the design process.
It was also assumed that fairly high insertion loss could be tolerated. This is certainly not true in all cases, but is true in many cases of current interest. The recent deveiopr'1ent and widespread implementation of fiber optic amplifiers at 1.55 µm or semiconductor optical amplifiers integrated with the modulators can change the constraints on system design. High modulator loss at a transmitter can be compensated by a power amplifier before transmission. The modulator's loss will not greatly impact m as long as the optical power level input into the amplifier is higher than the the syste P tical power level reached in the transmission amplifier chain. Thus, unlike past lowest o d . s this study has gone further in considering trading optical loss for other stu ie ' performance features. Finally, chirp was considered to be a major performance parameter. External modulation has mostly been considered over direct laser modulation due to the perception that chirp performance is improved with the former. External modulators, however, can exhibit appreciable chirp and the design must address minimizing it to meet system goals for dispersion in long distance communications.
Several parameters were fixed in the interest of minimizing the number of variable design parameters. A single bias voltage was used. Upon manufacture, a modulator's operation may be shifted relative to the design and bias voltage may be useful to tune the modulator's performance to the correct wavelength. It is not apparent, however, that it serves any other purpose. By comparing the Zucker and Siemens tunneling linewidths it is apparent that the barrier composition should be as close to InP as possible. The modulator efficiency is increased if the inactive barrier material is reduced by thinner barrier layers. Thinner barrier layers can only be used if the tunneling is kept low.
Siemens personnel indicated that they prefer a InGaAsP composition. There is always some GaAs diffusion and they have found that a small GaAs concentration is more accurately controlled. While it may thus be adviseable to maintain a small y in the barrier, it should be minimized. A fixed y of 0.1 was chosen.

Device Equations
This study addresses the performance of waveguide type modulators. The effective refractive index of the waveguide mode is given by: It can be shown that the device contrast ratio is given by: where Lis the device length. The absorption change at wavelength A. with maximum applied field F0ff is: Likewise !ia( Fb;,A) is the absorption change at A. with built in field Fb;· Fb; corresp011ds to the high transmission state of the modulator and includes both the device's built in voltage and any voltage applied in the "on" state to tune the modulator's performance to the correct wavelength. F 011 corresponds to the off state. It is noted that the "off' refers to the light being "off', not the applied field. The applied field is given by: The insertion loss is given by: The -3 dB bandwidth is given by: (5.11) th e terminating resistance, R0 , is 50 ohms. where

Index Change and Chirp
Associated with the modulator absorption change is a change in refractive index.
The change in refractive index can be found using a Kramers-Kronig integral58: It is necessary to calculate around a singularity at E'=E. The integral is devided as: In numerically calculating the index change it is not possible to evaluate the integral from 0 to infinite energy. The integral is started at an energy well below the exciton where the absorption change contribution is negligible. At the high energy end, the absorption spectrum model does not adequately represent the true spectrum because only the le:lhh, le:llh, and le:2hh excitons are included. Numerous additional transitions which have not been modeled also add to the absorption spectrum. This deficiency of the model does not have an appreciable effect upon the shape of the index change spectrum in the spectral region of interest near the absorption edge. This is because the primary contribution to the index change at a particular energy E is the absorption nearest to E.
The deficiency of the absorption spectra can have the effect of increasing or decreasing the entire index change spectrum. As field is applied the absorption curves shift to lower energy and the contributions of the light and heavy hole drop. The contribution of the second heavy hole exciton initially rises and eventually drops also. The dropping of the 81 I t ·on spectra decreases the total absorption contribution to the Kramers-Kronig absorp I integral and the entire index change spectrum is lowered. The most noticeable effect is that at low energy, far from the exciton, the index change is negative, instead of going to i.ero as expected. In reality, when applied field causes the hh and lh exciton oscillator strengths to decrease, the oscillator strengths of other transitions should increase to maintain the same oscillator strength sum. 8 This effect was approximated by integrating the calculated absorption change across the full range of the integral for each field. The absorption change spectra was then increased in a region well above the excitons as necessary to maintain the total absorption change constant. This eliminated the problem at iow energy without affecting the shape of the index change spectrum near the absorption edge.
The chirp performance of a modulator is often defined in terms of the chirp parameter a a , called the Henry factor: F is the field at which the absorption is increased by 3dB. This makes intuitive where 3dB Ul . that dispersion has little impact when the bit is off and there is no light to sense disperse. Both aH.on-o!f and aH.3dB were calculated, but the design effort focused on The phase shift was also calculated: The fmal design optimization program was comprised of two parts. The material modd program calculated the absorption change and index change spectra for a series of values of well width, well composition, and barrier width. For each combination it ran nine fields: 0, 14,28,42,56,70,84,98, and 120 kV/cm. The modulator's design wavelength of operation was chosen to be 1550 nm. The second program ran a series of lengths and well numbers. As a result, Fbi and Foff varied from trial to trial. It was necessary to interpolate from ~a and ~values from a minimal number of fields which still provided sufficient accuracy. In the first program the ~a and ~n values at 1550 nm were extracted for each field. Separate cubic splines were used to fit ~a vs. field and ~n vs. field and the spline parameters were stored for later use by the second program.  The third digit represents the well width: 12.5, 13.0, 13.5, 14.0, 14.5, or 15.0 nanometers. The fourth digit represents the barrier width: 4.0, 6.0, or 8.0 nanometers.
For quic!c reference these values are shown in Table 6.1. For each design, the bandwidth, contrast ratio, absorption loss with no applied field, 3 dB Henry factor a H,JdB, and the spectral offset or detuning of the exciton from the operating wavelength, both with and without applied field, were calculated.

Design Parameter Effects
It would be desirable to be able to distinguish the effects of changes in the design parameters. In general, however, this will not be possible because of the complexity of the interactions between the various parameters. It is useful to first understand the role each design parameter is playing and how it affects the modulator performance parameters.
The affect of the well composition parameter Yw is straight forward. Changing Yw primarily acts to change the exciton detuning without other major changes. The detuning is the spectral offset of the heavy hole exciton from the operating wavelength. This can 88 Iii be seen in Table 6.2. It is through the detuning that the contrast ratio, loss, and Henry ar e affected. The bandwidth is virtually unaffected. factor Modifying device length also has fairly simple effects. Table 6.3 presents a series of designs that illustrates changing the length. Changing the length has no effect on the shape of the absorption spectrum calculated for an individual well and doesn't affect the applied field. The detuning is thus unchanged. Adding length simply increases the loss and contrast ratio in direct linear proportion as expected. The bandwidth is affected directly through the increase in capacitance as the length increases.  The Henry factor is a material parameter and would normally not be expected to vary with a device parameter like length, which does not affect the individual well absorption spectra. The increase in the 3 dB Henry factor in Table 6.3 demonstrates one of the differences between the conventional Henry factor and the 3 dB Henry factor. The conventional Henry factor is based on intensity and phase change between an "on" and an "off' field, neither of which are affected by device length. The 3 dB Henry factor is based on an "on" field and a field which reduces the intensity by 3 dB. Because the device length affects how quickly 3 dB of loss is achieved versus applied field, the 3 dB Henry factor is modified as well.
The effect of the remaining parameters are not so simple, mostly because they alter the total MQW thickness, di, and hence the built-in field, applied field, and total field shift. Designs with small di lead to large field shifts between the bias field and the applied field. These shifts can be so large that instead of the applied field moving the operation up the leading edge of the first heavy hole exciton, it may move the operation well beyond, into the more complex spectral region of the second heavy hole and first 90 light hole excitons. The effects of these parameters are more likely to depend on the al Of the other parameters. Changing well thickness may have one effect when other v ues pararrieters are such that the applied field is still acting on the initial slope of the first heavy hole exciton. Changing well thickness will have quite another effect if the other parameters are such that the applied field operates in the region of the other excitons.

I
The number of wells is one of the parameters to have such an effect. Increasing the well number decreases the change in applied field. Depending on the spectral shape of the absorption, this might decrease or increase the contrast ratio. This is somewhat offset because the greater number of wells gives greater absorption per unit length, increasing loss and contrast ratio. The situation is further complicated because the built in field is also modified, modifying the initial detuning.' It was already shown above that this can modify all performance parameters except bandwidth. The effect on the 3 dB Henry factor is not straightforward because it depends in a complex way on the initial built in field, the shape of the absorption spectrum, and how quickly 3 dB loss is achieved versus applied field. The bandwidth is affected through the capacitance change with chu:lging di. Table 6.4 shows a particular example. The change in detuning at bias and the change in the magnitude of the total detuning shift is apparent. In this case, between 20 and 25 wells, the contrast ratio decreases with well number, indicating that the change in field shift is more important than the effect due to more absorption. In the 15 well case the applied field shifts the spectrum well beyond the first heavy hole exciton and results in decreased contrast ratio. The loss increases with well number, indicating that the increase in absorption due to extra wells is of greater significance than the change in detuning.
Little can be said a priori about the well width. Well width is another factor that changes the total thickness, di, and hence the bias and applied field. Larger well width allows greater exciton shift with applied field. However, it also reduces the exciton oscillator strength and can vary the exciton linewidth, all of which affect the shape of the absorption spectrum in a complex way. factor which changes the total thickness, and hence the built in and applied field.
Alternately, a reduction in barrier width can allow more wells without affecting di.

Modulator Design for Long Distance
The interplay of all of the effects leading to a full modulator design is now considered. The optimum modulator design will depend on the particular application, which will dictate the requirements for bandwidth, loss, contrast ratio, and chirp. For long distance applications, the chirp will be of major importance. Dorgeuille  achieved, but only if aH, 3 dB is reduced to -1.0. Figure 6.1 illustrates the loss and aH. 3 dB dependence on detuning for the case in Table 6.2.
There is a basic tradeoff between device loss and chirp performance. Optimum negative Henry factors as defined in (5.18) can be achieved by operating close to the exciton, but the drawback is greater device loss. It is of interest to see which design parameters can influence this tradeoff. Well width and barrier width can potentially influence the tradeoff through their effect on the absorption spectra. Figure 6.2 shows loss vs aH,Jd.B for designs in the 12xxx design series and shows that the resulting tradeoff appears independent of the well width. number of wells is clearly superior. However, only one of the 10 well designs achieved 10 dB contrast ratio and it didn't offer a low Henry factor.
In   enough wells to keep a favorable loss vs. chirp tradeoff, but enough to meet the contrast ratio and Henry factor requirement. The exact value will depend on the contrast ratio required by the system but will be about 15. The device is as short as possible for chirp minimization-about 50 microns. This has the added benefit of increased bandwidth. It should also have narrow barriers for.optimum loss/chirp performance.
The remaining design factor to be determined is the well width. The 12xlx design determined in the last paragraph was investigated further by running the model with smaller steps in Yw to determine the correct detuning to give a 3 dB Henry factor of -0.5.
Designs at different well widths were then compared when tuned in Yw for this same Henry factor. The device model was run with various drive voltages to determine the minimum drive voltage necessary to achieve 15 dB contrast ratio. These drive voltages and the design bandwidths are plotted in Figure 6.7. Unless bandwidth is critical, the plot would lead to a choice of 14.5 nm wells to minimize drive voltage. The optimum device is slightly detuned from case 12513 and will have a 3 dB Henr; factor of -0.5, a bandwidth of about 36 GHz, a contrast ratio around 15 dB, and a loss of around 18 dB. It is biased for a detuning of about 7.0 meV. Its drive voltage is 1.5 volts. This device loss is quite high compared with modulators which have been considered in the past. This work clearly indicates the tradeoff involved, and high losses will be necessary to achieve low chirp. Such a device could be acceptable if an amplifier is used after the modulator. It is possible to trade even higher loss for more bandwidth by choosing designs with thicker barriers. The 12534 design gives 42 GHz bandwidth but 22 dB loss.
The choice of well number is the primary difference from designs that seek to maximi:re bandwidth without regard for chirp performance. It will be seen in a later section that such designs use the maximum number of wells technically feasible.

Modulator Design for Short Distance
When designing a modulator for short distance applications, the chirp performance is much less relevant. When a requirement for negative Henry factor is not imposed, there is freedom to design the modulator to further maximize other performance parameters. In the previous section, the necessity to minimize the chirp guided the design choices quite strictly, offering few alternatives. This is not true for the short haul modulator. Further, it is necessary to determine what performance objectives the design is to mel?.t In some applications the required bandwidth is given and the design task becomes to maximize the contrast and minimize loss and drive power. In other applications maximum bandwidth will be sought. The full computer model will be useful regardless of the performance requirements sought. It is not possible, however, to plot out a design chart showing how to design any of these possible devices. The full device model has six inputs and four performance outputs and cannot be presented in a single plot or even set of plots. A three dimensional plot could show only one output versus two inputs. Other two dimensional plots could show several outputs relative to a single input. The most complicated plots or sets of plots cannot present the full complexity of the model. Any attempt to hold several parameters constant while varying others, results in a plot that is a cut of the full model. While it is useful for designs close to the frozen values it says nothing of designs away from those frozen values.
Despite these limitations, much can be understood from an example. For this example an arbitrary (but typical) design goal is considered. The goal in priority order will be: 1) maximize band width as highest priority, 2) require contrast ratio greater than 15 dB, 3) minimize drive power, 4) minimize loss.
TI1e model showed that bandwidth is maximized by choosing the shortest device and the most wells that are feasible. For this work these are designs with 50 µm length and 25 wells, designated 14xxx designs. Larger barrier widths also maximize the bandwidth by decreasing the device capacitance.
Bandwidth is also dependent upon well width through the capacitance. However, the difference is so small between the well widths considered that in most design situations the well width can be used to optimize the other performance parameters. The   The way the optimum well width changes with applied voltage may be a result of two opposing effects. If there is sufficient drive voltage, the lower well width will give less linewidth and greater exciton oscillator strength leading to better contrast ratio for lower loss. As drive voltage is reduced, greater well width will give greater exciton shift.
The contrast ratio enhancement due to the greater shift will play off against the oscillator strength and linewidth effects in such a way that different well widths are favored at different drive voltages.
If the low voltage option is chosen, the optimum device is slightly detuned from case 14633 and will have a bandwidth of about 56 GHz, a contrast ratio of 15 dB, and a loss oi 4.5 dB. Its drive voltage is 1 volt It is biased for a detuning of about 14.5 meV.
The design is not optimized for chirp as evidenced by its 3 dB Henry factor of 1.1.
In conclusion for this chapter, it has been shown that the MQW electroabsorption modulator model can be used to evaluate a large number of possible designs. It has been shown that design optimization is possible as long as what constitutes "optimization" is well defined. Different applications with different performance requirements can lead to distinctly different optimized device designs. For long distance applications where device chirp is important, the devices must be tuned closer to the exciton, with higher loss resulting. This loss can be tolerated if an optical amplifier is used prior to transmission.
Minimizing the loss led to a design with 15 wells. For short distance applications, chirp is no longer of concern, and the design may be concentrated on optimization of the rernair!ing performance parameters. If bandwidth is the top priority the design is optimized by the largest number of wells that is feasible. Well width is determined by the choice in a tradeoff between loss and drive voltage. This material model was then used to calculate the performance of MQW modulator devices as part of a multi-parameter design optimization. The performance factors of bandwidth, loss, contrast ratio, drive voltage, and chirp performance were all considered, rather than using figures of merit which ignore or inadequately prioritize these factors. The design of a modulator to give optimum chirp performance over long transmission distances was investigated. It was shown that by using present fiber or semiconductor optical amplifiers, higher device loss can be tolerated, allowing operation tuned closer to the exciton to optimize the chirp. The loss versus chirp tradeoff can be optimized through use of short device length, small well number, and small barrier thickness. Well width of 14.5 nm gave the lowest drive voltage requirement. The design of a modulator for typical short distance transmission was also considered. When chirp performance is ignored, it is possible to further optimize the remaining performance factors. The choice of a large well number is the main result to distinguish these designs from designs that optimize chirp performance. The choice of well width was found to involve a tradeoff of lower drive voltage versus higher loss.
The two design examples showed that there are numerous tradeoffs to be made between the various performance parameters. They also showed that performance can be significantly improved through optimized design. The proper prioritization of the various performance parameters will depend upon the specific application and such factors as the transmission distance, the loss budget, the ability to include an optical amplifier, the contrast ~atio requirements, the drive power available, and the bandwidth desired. Furthermore, the model is sufficiently complicated that the tradeoffs change with the design parameters and cannot be generalized. In such cases the judgement of a human designer who can weigh all the factors can be very valuable. This device model gives the designer the tool required to make those judgements.