Comparative analysis of the performance limits of Franz-Keldysh effect and quantum-confined Stark effect electroabsorption waveguide modulators

M.K. Chin

Indexing terms: Waveguides, Modulation, Quanfum Electronics

Abstract: The author systematically analyses and compares the performance limits of quantum- confined Stark effect (QCSE) and Franz-Keldysh effect (FKE) electroabsorption waveguide modula- tors, in terms of insertion loss, contrast ratio, drive power and bandwidth (or bit-rate). The author first derives the universal material figures of merit for eiectroabsorption modulators which form the basis for comparison. In addition to the magni- tude of electroabsorption Au, the critical material parameters are Aa/uo and Au/F2, where U, is the onstate residual absorption, and F is the applied electric field. The author proposes a waveguide design which will meet the insertion loss and con- trast ratio requirements while minimising the available powerbandwidth ( P J A f ) ratio. The author shows that, while satisfying the same inser- tion loss requirement, a QCSE modulator employing the optimum quantum well structure can have, in principle, an order of magnitude better performance in terms of Pac/Af than one based on FKE. Correspondingly, the Aa/F2 is an order of magnitude larger in QCSE than in FKE, principally because of the much larger absorption change that is possible with QCSE.

1 Introduction

The requirement of large bandwidth and low drive power, and the potential for optoelectronic integration, have prompted an intensive study of 111-V compound semiconductor modulators and switches [l]. Most of these semiconductor modulators are based on the elec- troabsorption (EA) effect near the fundamental absorp- tion edge. Two major physical mechanisms that give rise to the electroabsorption effect are the Franz-Keldysh effect (FKE) and the quantum-confined Stark effect (QCSE). The Franz-Keldysh effect [2] describes the broadening of the absorption bandedge in a bulk semi- conductor due to tunneling-assisted absorption in the presence of an applied electric field. The quantum- confined Stark effect [3] describes the energy shifts of the quantised energy levels in a semiconductor quantum well heterostructure in the presence of an electric field. Yet

0 IEE, 1995 Paper 1661J (El l , E13), first received 10th June and in revised form 6th October 1994 The author is with the School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 2263, Republic of Singa- pore

IEE Proc.-Optoelectron.. Vol. 142, No. 2, April 199s

another mechanism that shares the features of both FKE and QCSE is the Stark-Wannier localisation in a super- lattice material [4]. All these effects have been demon- strated and utilised for modulator applications.

The purpose of this paper is to systematically analyse and compare the theoretical performance of QCSE and FKE electroabsorption waveguide modulators. This helps to decide which is the more appropriate technology for further development. Previous attempts [SI to compare the performance of these two modulators were carried out without a clear definition of the criteria for comparison, and without suggesting ways to optimise the performance. To remedy this situation we will first define the performance criteria and derive a minimum set of material and waveguide figures of merit needed to char- acterise the total performance of the modulator. On this basis we will then derive a simple design approach to optimise the performance. Based on this approach, we will compare the relative performance of the two types of modulators. We will focus only on semiconductor material systems of importance to optical communication at 1.3-1.5 pm wavelength region.

2 Criteria of performance

For either analogue or digital modulators, one of the primary measures of performance is the drive power per unit bandwidth, P,/AJ or the switching power per bit- rate, PJB. [6] They are proportional to CAV’ (the switching energy), where AV is the relevant voltage swing [7]. In addition, the insertion loss is an important con- sideration for any waveguide modulator, and the contrast ratio is also important for large-signal on-off modulation. These four parameters are the most basic for any (digital or on-off) modulator.

Comparison of the performance of various types of modulators in terms of the drive power and the band- width has been given before [SI. In Fig. 1, such a com- parison is reproduced and extended to include more recent published results. The modulators include those based on the linear electrooptic effect [9] the Franz- Keldysh effect [lo, 111, and both the electroabsorption [12-161 and the electrorefraction [lq of the quantum- confined Stark effect. The drive power is defined as Pae = V2/8R, where, for the electroabsorption modulators, V = V,, dn, the peak to peak voltage swing required to give a 10 dB on-off ratio, and for the electrooptic modu- lators, V = V,, the voltage required to give a x phase shift. The bandwidth is defined here as Af = 1/2nRC, where C is the intrinsic capacitance of the device. R is the drive impedance, which is assumed to be 50 Q.

We see from Fig. 1 that, in terms of Pa, and AA the QCSE absorption modulators (solid circles) are seen to

109

be the best at present. However, the experimental per- formance given in this Figure, based only on Pa, and Af (for a given contrast ratio CR), can be misleading. The

/*

l o o 0 l /'

011 1 10 100

Fig. 1 Summary of the required RF drive power, P, , versus the RC- limited bandwidth, AJ for different types of modulators, including ( i ) a L i N W , LEO absorption 191, (ii) Franz-Keldysh ( F K ) absorption modula- tors [ I O , 111, and MQW QCSE absorption [12-16] and phase modula- tors [17J for comparison, the dotted line is the theoretically predicted performance far a QCSE electroabsorption modulator based on the opti- mised quantum well structure and the small-y waveguide design discussed in the text, and the dot-dashed line isfor the optimised FK modulator with the same waveguide design

total performance of these modulators are, in fact, much less impressive. By total performance we mean per- formance as measured not only by Pat, Af and CR, but also by the insertion loss. When this additional require- ment is imposed, the performance in terms of P,, and Af will be compromised, regardless of the material system or the physical mechanism utilised by these modulators. All demonstrations of device so far have tried only to push the limits in one or the other of these performance parameters, but not the total performance as a whole [6] . Likewise, considerations for optimisation have focused only on some aspects of performance [lS]. Although for specific applications some parameters are more impor- tant than others, it is nevertheless useful to understand the tradeoffs involved, and to have some idea of the limits of total performance that can be achieved in principle. For example, the theoretical performance limits for QCSE modulator, derived later, is represented by the dashed line in Fig. 1. This gives the combinations of Pa, and Afthat can be achieved under the conditions that the total insertion loss (neglecting reflection loss) is smaller than 4.5 dB, and the contrast ratio is 10 dB (10 dB is used for consistency with other results in Fig. 1). Under the same conditions, the theoretical limits for FKE modula- tors are substantially inferior, as shown by the dot- dashed line. These theoretical limits suggest the kind and range of total performance that can in principle be achieved with these modulators. In particular, the theo- retical curve of Fig. 1 implies that there is significant room for improvement to all present day QCSE modula- tors whether by the narrow or the broad definition of performance. The approach discussed below, besides yielding the theoretical limits, also inherently brings out a scheme for optimising the device design to exploit more fully the potential of these modulators.

110

bandwidth, GHz

3

In this paper, we consider a high-performance modulator as one which has the smallest drive-powerbandwidth ratio, and which at the same time satisfies the specified insertion loss and contrast ratio. The insertion loss is taken to consist of the coupling loss to, and from, a plane-cut single-mode optical fibre, and the propagation loss through the waveguide. The optimum design will depend on the specific requirements. Our basic approach for optimising the modulator is to minimise CAV' (or PaC/Aj ) while simultaneously meeting the minimum required insertion loss and on-off ratio.

The fundamental material figure of merit for electro- absorption modulator is, of course, the magnitude of the electroabsorption, Au. If Au is too small, no efficient modulators can be obtained. On the other hand, even if Au is sufficiently large, additional conditions must be satisfied before any high-performance modulator can be achieved. These conditions lead to two new figures of merit, and are discussed below.

(i) To minimise the static propagation loss, the residual absorption (U,) in the ON state must be small. Since the propagation loss (L,), in dB, is given by L, = 4.343(uOyL), and the contrast ratio (CR) is given by C R = 4.343yAuL (dB), where y is the optical filling factor, Aa is the electro- absorption, and L is the device length, a useful figure of merit which is independent of y and L is the ratio m = Au/u, . This parameter must satisfy the condition

Universal figures of merit for electroabsorption modulators

m 2 CR/L , = m, (1) if a given combination of C R (dB) and L, (dB) is to be achieved. The physical length of the modulator is deter- mined once yAu or yu, is known, and is given by

L = C R (dB)/(4.343yAu) = L, (dB)/(4.343yu0). (2) Both conditions, eqns. 1 and 2, are required to satisfy simultaneously the specified C R and L, . If L is shorter, the C R will be too small; if L is longer, the propagation loss will be too large.

(ii) The third figure of merit is deduced from the opti- misation of CV', which leads to the important expression C61

C V z /wd:\ 1 (3)

where w is the electrode width, di is the 'intrinsic' layer thickness (in a p i - n structure), cs is the dielectric per- mittivity, and C R = exp (yAaL). Eqn. 3 contains the design guidelines for minimising Pu/Af for a given CR. Besides the waveguide parameter, wd,/y, it states that, to minimise CV', one should maximise the material param- eter, Aa/(AF)'. Therefore, the optimum material is one which not only satisfies the required Au and Aa/a,, but also maximises Au/(AF)'. Note that, if for some reason it is more important to minimise the drive voltage per unit bandwidth [19], then the relevant parameter is Au/AF instead of Au/(AF)'.

In summary, the material figures of merit for electro- absorption modulators are Au, Aa/u,, and Aa/(AF)'. The relative merits of different materials or device structures, and of different physical effects, such as QCSE and FKE, can all be compared and evaluated in terms of these parameters.

To minimise CAV', the waveguide design must also be considered. According to eqn. 3, a waveguide

I E E Proc.-Optoelectron., Vol. 142, No. 2, April 1995

with minimum wd,/y is required, but this is not desirable from the insertion loss consideration as a small wd,/y implies a small waveguide mode area and hence poor mode matching. However, a compromise is possible pro- vided Aa/(AF)2 is sufficiently large, in which case it is possible to preserve a reasonably small CAV’ even with fairly large wdi/y. Also, we must maintain a small d , in order to keep the drive voltage low. Such a waveguide, with a fairly large wdi/y but a small di, implies that y is small, and can be obtained by making the core thickness much larger than the active quantum-well intrinsic layer thickness, i.e. d, B d , . The value of d, is determined by the desired coupling efficiency. This large-core, mostly passive waveguide has been described elsewhere. [6] We merely mention two points relevant to the following dis- cussion: (i) In the small-y limit, y is linearly proportional to d , , and eqn. 3 can be simplified as

(4) which does not depend on di and L (we have written y = gd,, where g depends primarily on d , and w). (ii) The smaller y is, the larger Aa will have to be to satisfy a given bandwidth. Therefore, the small-y design is viable only if a sufficiently large Aa is available. Otherwise the achievable bandwidth will be limited.

4 Theoretical performance limits for FKE and QCSE modulators

In this section, the intrinsic efficiencies of FKE and QCSE as competing electroabsorption mechanisms are evaluated based on the material figures of merit discussed above. The design approach is first given, followed by a specific example based on empirical data wherever pos- sible. In comparing the power-bandwidth performance, the other parameters such as insertion loss and contrast ratio are kept the same, so the waveguide structures are assumed to be identical for both types of modulators. For comparison of bandwidth, it is adequate to consider the intrinsic bandwidth, ignoring any parasitics or extra- neous effects. 4.1 Franz-Keldysh Effect The analysis of FKE modulators is based on the assump- tion that, first, the optical absorption in the presence of an electric field is given by the analytic expression [20]

CAV2 = E,(w/g) In (CR)/(Aa/F2)

= a, exp (- [ d , ) ( 5 ) where ap is the peak absorption coefficient, E, is the bandgap energy, hv is the photon energy, F, = (V + Vbi)/d, is the internal field, K i is the built-in poten- tial, and

2m ( E , - hv)3/2 [ = ; J(G) e(v + Vbi) ’

and secondly, the static absorption coefficient at zero bias, ao, is approximated by an Urbach tail with the expression [21] .

(6) Aa is then given by a(V) - a , . Thus, Aa (and Aa/FZ as well, where F = V/d,) is a function of E, - hv, V and di. The above approximations are well supported by experi- ment [22] as discussed below.

Recently, an interesting FKE waveguide modulator based on GaAs/AlGaAs [23] has been demonstrated in which the value of di is designed to maximise the effective

a, = A exp [ - (E, - hv)/Eb]

IEE Proc.-Optoelectron., Vol. 142, No. 2, April 1995

modal absorption coefficient, antode, = ya cl di exp ( - [d i ) , which is maximised for d , = l/[. For example, for GaAs, the optimal absorbing layer thickness (d,) is 0.18 pm for V = 1OV and E, - hv = 100meV. In the present context, instead of ya, it is more appropriate to maximise Aa/F2 with respect to d , . The optimal d , (in pm) in this case is given by

(7) where (E, - hv) is given in meV and V in volts. Conse- quently,

(8)

di(pm) = 2/1 N 36(V + &,)(E, - h ~ ) - ~ ”

Aa = a, exp ( - 2 ) - a, = 0.135aP - a,,

and

F = V/di = [ V / 2 N 0.O28(Eg - hV)3/2[V/(v + Vb,)] (9) The same results for Aa and F would be obtained had we maximised Aa/F2 with respect to V instead of d , . This is because the expression for Aa/F’ is symmetric under exchange between V and di. Similarly, we could have maximised Aa with respect to (E, - hv) for a fixed d i /V ratio. The optimum (E, - hv) thus obtained, however, will generally be too small and the insertion loss too large. In practice, the actual detuning energy is deter- mined by the specified acceptable propagation loss, as discussed further below.

The core of our design approach is thus to first maxi- mise Aa/FZ with respect to di, thereby making the optimum value of d , dependent jointly on the detuning energy (E, - hv) and the operating voltage. Also Au is now pegged essentially to a fixed value equal to a frac- tion of the peak absorption coefficient. This means we can increase Aa/F’ only by minimising F, which implies that the drive voltage should be made as small as pos- sible (cf. eqn. 9). In practice, however, we have to satisfy certain minimum bandwidth requirement, which can be achieved only if the voltage is not too small.

To design d i , the detuning energy must first be deter- mined. If maximising Aa/F2 were the only consideration, then a small detuning energy, AE E, - hv, would be desirable. However, one will quickly reach a limit set by excessive propagation loss owing to the bandtail absorp- tion which increases rapidly at lower detuning energies. Thus the choice of the detuning energy is constrained by the operational requirement that a specified combination of the CR and the propagation loss L, be satisfied, i.e. Aa/a, 2 CR/L , = m, (cf. eqn. 1). This requirement leads to the constraint

AE 2 E, In (A/0.135ap) + Eb In (mo + 1) (10) where we have used eqns. 6 and 8. This equation gives the required detuning energy in terms of m,, i.e. in terms of the desired combination of CR and L , . Once AE is thus chosen, d , is then determined by the available or required drive voltage, according to eqn. 6. With di known, the length (related to d, through y) is given by the required CR or L, according to eqn. 2, where y is related to d , in some fashion. Finally, the intrinsic bandwidth of the modulator is determined once d , and L are known. The procedure can be iterated for a different value of V if the required bandwidth is not satisfied.

To give a specific example based on the InGaAsP/InP material system, we make use of the data of Mak [22] (Fig. 2), which has given the best performance for an FKE modulator (at 1.3 pm wavelength) so far, in order to determine the values of a,, V,,, A and Eb . These data are assumed to be representative of the material system. The data can be applied to eqns. 4 and 5, yielding the values

111

up U 2220 cm-', Qi 1 1.3 V, A % 55000 cn-' and Eb N

6 meV, where we have assumed y = 0.5 for the structure in Reference 22. These parameter values are valid for

Fig. 2 Available m, = Aula. = C R / L , as a function of the detuning energy (E, - hv) for nn F K E modulator. For a propagation loss ( L J of 1.5 dB, the corresponding contrast ratios ( C R ) are given on the right axis. The detuning energies corresponding to C R = 10 and 20 dB are indicated by dotted lines

detuning energies in the range of interest. Based on these values, AE is given by eqn. 10 to be AE(meV) = 31.3 + 6 In (mo + I), which is plotted in Fig. 2. Therefore, for consistency with the QCSE results to be discussed later, we will assume that the propagation loss is restricted to L, = 1.5 dB, so that C R = 1.5m0 dB. We then calculate, for two typical values of C R (10 dB and 20 dB) and as a function of V, the values of di (using eqn. 7) and L (using eqn. 2) required to satisfy the specified CR. The results are shown in Fig. 3.

The device lengths are calculated in the small-y approximation, assuming y = 0.3di for a waveguide with d, = w = 3 pm (this approximation is valid for di < 1 pm). These waveguide dimensions are designed to give a coupling loss of 3 dB. The inverse slopes of the di /V curves are equal to the applied electric fields, which are (for large V) approximately 8 and 9 V/pn for C R = 10 and 20dB, respectively. From these we can determine that Aa/F2 are approximately 4.1 x lo-' and 3.4 x lo-' pmV-*, respectively, when V is large. Finally, the intrinsic bandwidth and the power-bandwidth ratio are calculated as a function of the drive voltage, as shown in Fig. 4. Note that for drive voltage limited to 5 V, the achievable intrinsic bandwidth is less than 20 GHz for a C R of 10 dB, and much smaller for C R = 20 dB. Thus, if a larger bandwidth is required, then a much larger voltage is necessary. The tradeoff between V and Af is such that the greater the drive voltage, the larger Pm/Af (or CV') becomes. Pac/Af is not independent of voltage because Aa/F2 decreases with increasing V according to eqn. 9. However, it can be seen that the bandwidth is approximately a quadratic function of V , so P,, plotted as a function of Af should be approximately linear. Typical values of PJAf are large, about 1 to 10mW/ GHz depending on the CR. The case for C R = 10 dB is indicated in Fig. 1. as the dot-dashed line. All these results are significantly worse than those for the QCSE modulators, discussed below.

These performance limitations arise primarily from the relatively small Au values, and as a consequence, the use

112

of small-y waveguide design limits the available band- width for a given drive voltage.

4.2 Quantum-confined Stark effect The optimisation of QCSE modulators has been dis- cussed in detail elsewhere [SI. Here we merely present the

0 1 0 0 2 4 6 8 1 0 1 2

drive voltage, V

Fig. 3 function of applied bias V , for C R = 10 d B and 20 dB

Optimal d u e s of d,, and the corresponding values of L, as a

10

8

N

$ 6 5 E r 0 2 4

n

5 X i

U O . L U

C

drive voltage, V Fig. 4 drive voltage for F K E modulators

Bandwidth and the power-bandwidth ratio as a function of

main results. In some material systems such as In,Ga, -,,As, P , -&nP, the optimisation of the quantum-well design consists of determining the optimum combination of well width (L,) and material composition x (y is determined by x) that maximises Au/F2 while satisfying a specific minimum required Aalci,. For a fixed photon energy, the combination of L,

IEE Proc.-Optoelectron., Vol. f42, No. 2, April f995

and x also determine the detuning energy E,, - hv, where E,, is the exciton resonance energy. Generally, with decreasing detuning energy (up to some small value), Aa/FZ increases but Aa/a, decreases, so the optimum detuning energy is the smallest detuning that still yields the required Aa/a,, . However, a desired detuning energy can be given by many combinations of L, and x. We have used a semiempirical model adapted from one which was developed for GaAs/AlGaAs [24] to calculate Aa/F2 and Aa/a, as a function of L, and x for the InGaAsP/InP material system [SI. Important differences about this material system, such as the smaller peak absorption, the broader linewidth, and the faster line- width broadening, compared with GaAs/AlGaAs quantum wells, are incorporated in this model in order to give realistic results. The results, as illustrated in Fig. 5,

100-

80

N .

> E 3 6 0 - : e

x . - N . 6 40- Q

20

-

-

InGoAsPl SnP

01 60 80 100 120 140 160

Fig. 5 Variation o/Aa/Fz with will width (L,)/or quantum well struc- tures ( L z , x) that have a constant detuning energy equal to 20, 25, or 30 meV

L Z , H

show that, for a given detuning energy, those com- binations with wider quantum wells, up to a point, tend to have larger Au/F2. The optimum material structures are those with L, 3 120 A.

There are two basic reasons: (i) the sensitivity of elec- troabsorption to field increases rapidly with the well width [25] and (ii) the figure of merit, Au/F2, gives more weight to the sensitivity to electric field. Note that the maximum calculated Aa/FZ, about 80 x p m T 2 , is an order of magnitude larger than those for the Franz- Keldysh effect given above. Because the waveguide struc- ture is the same, the power-bandwidth ratio, PJAL for the optimum QCSE modulator will also be an order of magnitude smaller than for the FKE modulator.

Fig. 6 shows the theoretical performance in terms of bandwidth and drive voltage for two different quantum well structures (Lz , x) optimised for two different values of CR. The waveguide in this case is designed to have a maximum insertion loss of 4.5 dB (i.e. 3 dB coupling loss and 1.5dB propagation loss), the same as in the FKE case. The V/Af curves again exhibit a quadratic depen- dence, implying that Poc/Af is a constant, given the C R , the material's Au/F2 value and the waveguide parameters

IEE Proc.-Optoebctron., Vol. 142, No. 2, April I995

w and d, . This is generally true for a small-y waveguide modulator (cf. eqn. 4). Similarly, for a given value of C Y 2 , various combinations of P,, and Af can be achieved by

'"1 80-

N

$ 6 0 -

drlve voltage. V Fig. 6 Maximum achieuable bandwidth as a function of drive voltage giuen by the optimal material structures corresponding to C R = 20 d B and C R = 10 d B (and propagation loss of I3 dB) in a small-y waueguide with dimensions d, and w designed to giue a coupling foss of3 d B

varying the values of di and L. [6] The achievable Pa, and Af for the case of C R = 10 dB is illustrated in Fig. 1 by the dashed line having a constant slope of -0.22 mW/ GHz, which is to be compared with the I-lOmW/GHz for the FKE modulators discussed above.

In summary, the results show that excellent total per- formance, with a large C R and a small insertion loss con- current with very large intrinsic bandwidths (< 100 GHz) and very small drive voltages (< 3 V), can in principle be achieved using a simple large-core waveguide structure, principally because of the much larger absorption change (and reasonably large Aa/a. and Aa/F2) that is possible with QCSE. Consequently, the small-y design is not a limitation, but a simple and robust approach for mini- mising the insertion loss with standard fibres without sig- nificantly compromising the modulation efficiency of the modulator. The same approach, however, is a limitation for the FKE modulators, as we have seen above.

Our analysis of the comparative merits of FKE- and QCSE-based EA modulators is based essentially on the criterion of Pa/Af (or C A V 2 ) for a given insertion loss and contrast ratio. In terms of other important criteria such as the polarisation sensitivity and the saturation properties, FKE modulators may be more favourable. However, these are not fundamental problems. Recently polarisation insensitive modulators [26, 271, and modula- tors with very large saturation intensity [28], have both been demonstrated without significant penalty on the electroabsorption efficiency. Finally, quantum well struc- tures have a lot more degrees of freedom for tailoring the optical properties which are yet to be explored. For example, we have not considered in this paper the effects of strain, or the use of nonsquare, asymmetric [29], shallow, or coupled quantum wells.

5 Conclusions

We have analysed the comparative merits of both FKE- and QCSE-based EA modulators for total performance

113

in terms of four criteria. One of the criteria is the require- ment of small insertion loss, which imposes a compro- mise on the bandwidth, drive power and contrast ratio performance, as well as a stringent constraint on the design of both material and waveguide structures. We propose a design scheme which will meet the insertion loss and contrast ratio requirements and at the same time minimise the power/bandwidth ratio achievable under the condition. We show that, while satisfying the same insertion loss and contrast ratio requirements, a QCSE modulator employing the optimum quantum well struc- ture can have, in principle, an order of magnitude larger Aa and Aa/F2 values and an order of magnitude better performance in terms of Pac/Af than one based on FKE. As a result, we can use the small-y approach to reduce the insertion loss. On the other hand, this approach may limit the available bandwidths in the case of FKE modu- lators. Ultimately, the advantage of QCSE, or any other electroabsorption mechanism, derives from having a large Aa, Aa/F2 and Aa/a, .

6 References

1 MILLER, S.E., and KAMINOW, LP.: ‘Optical fiber telecommuni- cation’ (Academic Press, 1988)

2 FRANZ, W.: 2. Naturforsch, 1958, 13, p. 484, L. Keldysh. ‘The effect of a strong electric field on the optical properties of insulating crys- tals’, Sou. Phys. J E P T , 1958.34, (7), pp. 788-790

3 MILLER, D.A.B., CHEMLA, D.S., DAMEN, T.C., GOSSARD, A., WIEGMANN, W., WOOD, T., and BURRUS, C.: ‘Electric field dependence of optical absorption near the bandgap of quantum-well struclures’, Phys. Rev., 1985, B32, pp. 1043-1060

4 MENDEZ, E., AGULLO-RUEDA, F., and HONG, J.M.: ‘Stark localization in GaAs-GaAIAs superlattice under an electric field’, Phys. Rev. Lett., 1988,60, p. 2426

5 TIPPING, A.K., PARRY, G., and CLAXTON, P.: ‘Comparison of the limits in performance of multiple quantum well and Fram- Keldysh InGaAsflnP electroabsorption modulators’, IEE Pr0c.J. 1989, pp. 205-208

6 CHIN, M.K., and CHANG, W.S.C.: ‘Theoretical design opti- misation of multiple-quantum well electroabsorption waveguide modulators’, IEEE J., 1993, QE-29, (9)

7 MILLER, D.A.B.: ‘Quantum well optoelectronic switching devices’, I s . J. ofHigh Speed Electron., 1990, I, pp. 19-46

8 COLDREN, L., MENDOZA-ALVAREZ, J., and YAN, R.: ‘Design of optimized high-speed depletion-edge-translation optical wave- guide modulators in 111-V semiconductors’, Appl. Phys. Lett., 1987, 51, (11). pp. 792-794

9 DOLFI, D., and RANGANATH, T. : ‘50 GHz velocity-matched, broad wavelength LiNbO, modulator with multimode active section’, Tech. L&p Integ. Photon. Res., Paper Pd2, 1992

10 SUZUKI, M., TANAKA, H., AKIBA, S., and KUSHIRO, Y.: ‘Elec- trical and optical interactions between integrated InGaAsP/InP DFB lasers and electroabsorption modulators’, J. Lightwave Technol., 1988, 6, (7), pp. 779-785

1 1 SODA, H., NAKAI, K., ISHIKAWA, H., and IMAI, H.: ‘High- speed GaInAsPflnP buried-heterostructure optical intensity modu- lator with semi-insulating InP burying layers’, Electron. Lett., 1987, 23, pp. 1232-1234

12 KOTAKA, I., WAKITA, K., MITOMI, O., ASAI, M., and KAWA- MURA, Y.: ‘High-speed lnGaAlAs/lnAlAs MQW optical modula- tors with bandwidths in excess of 20 GHz at 1.55 pm’, Photon. Tech. Letts., 1989, 1, p. 100

13 WAKITA, K., KOTAKA, I., MITOMI, 0.. ASAI, H., and KAWA- MURA, Y.: ‘High-speed InGaAsflnAIAs MQW optical modulators with bandwidth in excess of 40 GHz at 1.55 pm’. Proc. Conf. Lasers Electro-Opt., Tech. Dig. Ser., 1990,7, OSA, Washington, DC p. 70

14’WAKITA, K., KOTAKA, I., MITOMI, 0.. ASAI, M., KAWA- MURA, Y., and NAGANUMA, M.: ‘High-speed InGaAIAsnnAIAs multiple quantum well optical modulators’, J. Lightwaw Technol., 1990, LT-8, (9), p. 1027

15 KOTAKA, I., WAKITA, K., KAWANO, K., ASAI, M., and NAGANUMA, M.: ‘High-speed and lowdriving-voltage InGaAs/ InAlAs multiquantum well optical modulators’, Electron. Letts., 1991,27, pp. 2163-2164

16 DEVAUX, F., BIGAN, E., OUGAZZADEN, A., PIERRE, B., HUET, F., CARRE, M., and CARENCO, A.: ‘InGaAsPflnGaAsP multiple-quantum-well modulator with improved saturation inten- sity and bandwidth over 20 GHz’, Photon. Tech. Letts., 1992,4, (7, p. 720

17 WAKITA, K., MITOMI, O., KOTAKA, I., NOJIMA, S., and KAWAMURA, Y.: ‘High-speed electrooptic phase modulators using InGaAsflnAIAs multiple quantum well waveguides’, Photon. Tech. Letts., 1989, I, (S), p. 441

18 NOJIMA, S., and WAKITA, K.: ‘Optimization of quantum well materials and structures for excitonic electroabsorption effects’, Appl. Phys. Lett., 1988.53, (20), pp. 1958-1960

19 CHIN, M.K.: ‘On the figures of merit for electroabsorption wave- guide modulators’, Photon. Tech. Letts., 1992.4, (7), pp. 726-728

20 BOER, K.W., ‘Survey of semiconductor physics’ (Van Nostrand Reinhold, New York, 1990). p. 968

21 DOW, J., and REDFIELD, D.: ‘Toward a unified theory of Urbach’s rule and exponential absorption edge’, Phys. Rev., 1972, 86, p. 594

22 MAK, G., ROLLAND, C.C., FOX, K.E., and BLAAUW, V.: ‘High- speed InGaAsP-InP electroabsorption modulators with bandwidth in excess of 20 GHz’, Photon. Tech. Letts., 1990,2, (E), pp. 730-732

23 HAASE, M.A., MISEMER, D., OLSEN, L., and VERNSTROM, G. : ‘Separate confinement electroahsorption modulator for 633 nm light’, Appl. Phys. Lett., 1992,60, (17). pp. 2054-2056

24 LENGYEL, G., JELLEY, K.W., and ENGELMANN, R.W.H.: ‘A semi-empirical model for electroabsorption in GaAs/AlGaAs multi- ple quantum well modulator structures’, IEEE J., 1990, QEM, (2), p. 296

25 BASTARD, G., MENDEZ, E., CHANG, L., and ESAKI, L.: ‘Varia- tional calculations on a quantum well in an electric field‘, Phys. Rev., 1983, B?S, pp. 3241-3245

26 TADA, K., NISHIMURA, S., and ISHIKAWA, T.: ‘Polarisation- independent optical waveguide intensity switch with parabolic quantum wells’, Appf. Phys. Lett., 1991,59, (26). p. 2778

27 ZUCKER, J., JONES, K., CHIU, T.H., TELL, B., BROWN- GOEBELER, K. : ‘Polarisation-independent electro-optic waveguide switch using strained InGaAs/lnP quantum wells’. Tech. Dig Integ. Photon. Res., New Orleans, LA, Paper PW7.1992

28 WOOD, T.H., PASTALAN, J.Z., BURRUS, C., CHANG, T.Y., SAUER, N., and JOHNSON B.: ‘Thin AIGaInAs barriers for increased electroabsorption saturation intensities in GalnAs multi- ple quantum wells’. Prof. Conf. Lasers Electro-Opt., Tech. Dig. Ser. 1991.8, OSA., Washington, DC, p. 10

29 NOBUHIKO, S., and NAKAHARA, T.: ‘Enhancement of change in the absorption coefficient in an asymmetric quantum well: Appf. Phys. Lett., 1992.60, pp. 2324-2326

114 IEE Proc.-Optoelectron., Vol. 142, No. 2, April 1995