Electroabsorbtion in Multiple Quantum Well Structures

We studied electroabsorption in GaAs/ AlGaAs Multiple Quantum Well (MQW) structures. The waveguide used for this study was designed as a phase modulator with an active layer containing six 75 A GaAs quantum wells and 80 A Al_35Ga_55As barriers. Electroabsorption in MQW structures is dependent on, among other things , the light polarization, the orientation of the crystal optical axes, and the direction of the applied electric field. In the first study we measured the spectral dependence of the quadratic electrooptic coefficient for TM polarized light ( s33 near the absorption band edge. The light hole exciton resonance occurs at 820 nm for the particular device we used. The spectral range studied was from 840 nm to 880 nm. In this region, the quadratic coefficient demonstrated large values (1.2 x 10s cm /kV at 840 nm) near the band edge. This is about two orders of magnitude larger than that observed in bulk material. It also falls rapidly as the wavelength increases further away form the band edge. A knowledge of this material parameter permits the calculation of the change in refractive index with an applied field, which is very useful in phase modulator design. This along with the change in the absorption coefficient as a function of the applied field (also measured) determines the operational characteristics of the phase modulator at a given wavelength. In the second study, we investigated the effect of the crystal optical axes orientation on electroabsorption in MQW's. We reported, for the first time, a difference in the change in electroabsorption by almost a factor of 2 between [-1,1,0] and [1,1,0] oriented structures. This makes the former orientation better suited for intensity modulator / switching devices as one gets larger change in the absorption coefficient with lesser applied voltage. We attributed that orientation dependence in the absorption coefficient between the two orientations to the anisotropy of the valence subband structure brought about by band mixing due to the quantum wells periodic potential. In the third study, we devised a novel technique to extract the spectral dependence of the absorption coefficient in these structures from measurements of the transmitted light intensity as a function of the applied voltage at a single fixed wavelength. Results are in very good agreement with the measurement of the spectral dependence of the absorption coefficient by varying the wavelength. This technique is a rather simple one and does not require a coherent tunable light source which is very expensive and not readily available in every lab. DEDICATED TO MOM AND DAD

the phase modulator at a given wavelength.
In the second study, we investigated the effect of the crystal optical axes orientation on electroabsorption in MQW's. We reported, for the first time, a difference in the change in electroabsorption by almost a factor of 2 between [-1,1,0] and [1,1,0] oriented structures. This makes the former orientation better suited for intensity modulator / switching devices as one gets larger change in the absorption coefficient with lesser applied voltage.
We attributed that orientation dependence in the absorption coefficient between the two orientations to the anisotropy of the valence subband structure brought about by band mixing due to the quantum wells periodic potential.
In the third study, we devised a novel technique to extract the spectral dependence of the absorption coefficient in these structures from measurements of the transmitted light intensity as a function of the applied voltage at a single fixed wavelength.
Results are in very good agreement with the measurement of the spectral dependence of the absorption coefficient by varying the wavelength. This technique is a rather simple one and does not require a coherent tunable light source which is very expensive and not readily available in every lab.    ABSTRACT: We have measured the wavelength dependence, for TM polarization, of the quadratic electro-optic coefficient s 33 for 75A GaAs/ Al. 35 Ga. 65 As structures near the absorption edge. Waveguides oriented along the [I, 1,OJ and [1,1,0] directions were used for measurement. Results show a rapid decrease in s 33 (cm 2 /kV 2 ) as >. increased further away from the absorption edge. We have also plotted the quadratic electrooptic coefficient versus the detuning energy from the exciton fundamental absorption edge in me V for TM polarized light 8 33 and compare it with 8 13 for TE polarization. The change in refractive index for TM polarization as a function of applied field and wavelength is also presented as an aid to device design.

Introduction
In electro-optic devices it is sometimes necessary to determine the change in refractive index due to an applied field. The change in the refractive index is a function of three factors which are: 1. The direction of the polarizing field ( E electrical), 2. the orientation of the crystal optical axes, 3. and the polarization of the optical electric field.
The behavior of the refractive index under the influence of an applied field is governed largely by the linear and quadratic electro-optic coefficients of the material. These coefficients are traditionally defined as the expansion coefficients of the change in the dielectric impermeability tensor T/ij = Eo ( C 1 )ii as a function of the external field or where Ei are the components of the electric field and Tijk and Sijkl are the elements of the 3rd rank linear ·electro-optic tensor and the 4th rank quadratic electrooptic tensor respectively. The higher order expansions terms have been neglected and summation over repeated indices is assumed. The dielectric impermeability tensor TJ relates the optical electric field to the optical displacement vector Eo When diagonalized, the elements of the tensor TJ are related to the refractive indices by where Tli are the diagonal elements and the ni are the refractive indices along the principal axes of the crystal. Again the summation over repeated indices has been assumed.
The electro-optic coefficients for bulk GaAs and other crystals have been extensively studied and cataloged [1], [2]. In quantum well structures, however, the research has been relatively limited, mainly due to the relative newness of the field.
In this paper we present the quadratic electro-optic coefficient, as a function of wavelength, for a GaAs/ Al. 35 Ga. 65 As quantum well (QW) structure, near its band gap. The electric field was applied parallel to the [1,0,0] direction which, in the case of our 'fri-n structure, corresponded to a field perpendicular to the layers. The polarization of the optical wave was perpendicular to the QW layers , i. e., in the TM mode. For such a configuration there is one coefficient, the s 33 , that is involved (see appendix A).
Where an ambiguity could arise between the coefficients of the quantum well and those of the bulk, the former will be referred to as rq and sq, where Q signifies the quantum well, and the bulk coefficients will be referred to as TB and SB· From the measurements, we also calculate the change in the refractive index. This is helpful in device design and determination of its operational characteristics.

Experimental setup
The experimental set-up is divided, for the purposes of description, into three parts. The first is the source, which covers the lasers and the polarizing elements. The second deals with the interferometer itself, and the last deals with the detector arrangements. Figure (1.2) depicts the experimental setup.

The source
The source of tunable radiation for this experiment was a Coherent 599-01 Dye Laser, with Styryl 9M as the lasing dye. The useful lasing range of this dye was 800-920 nm. The linewidth of the output radiation was about 27 GHz. The laser was run at a power output of 80 mW, though the maximum output power was about 500 mW.
The pump for the dye laser was a Coherent Innova 70-4 Argon Ion Laser, with a maximum power output of 6 W. The laser produced a multiline output which ranged in wavelength from 488-514 nm. About 3 W was required to produce the required 80 mW from the dye laser.
The output from the dye laser was guided to the interferometer using beamsteering elements. A polarizer and an attenuator (neutral density filter) were used to ensure the desired polarization and intensity respectively.
Not shown in the diagram is a beam-splitter, located just before the polarizer, which sampled a portion of the beam and directed it to a SPEX 1702 spectrometer for wavelength measurement. Also not shown, is a He-Ne laser which was used for the alignment of the interferometer. It was arranged such that the output of the dye laser and the He-Ne beam would reach the interferometer collinearly.

The interferometer
The operation of this interferometer is simple. Light from the source is split into two parts by a beamsplitter. One beam travels along one arm and the other travels along the other arm of the interferometer. These two beams meet again and interfere. Normally, concentric or parallel fringes are seen on the screen.
These fringe patterns move as the difference in optical path length between the two arms of the interferometer changes.
If the alignment between the two arms is very good, then the size of the fringes grow large enough such that the entire image on the screen increases or decreases in brightness as the optical path length changes. Achieving this is very important as the the measurements were done using a photodiode, which only measures average intensities. If the fringes were small, even though the fringes move, no change in average intensity would be observed.
The mirror located in the reference arm was equipped with a piezo-electric translation positioner to adjust the difference in path length. This positioner could move distances as small as 0.1 µm. The optical path length depends on the physical length as well as the refractive index of the path. The positioner changes the optical path length by changing the physical length, while the phase modulator changes it by changing the refractive index. On application of a voltage, the phase modulator changes the path length of this arm and a change in interference pattern is measured by the photodetector. This is the basic raw data used to evaluate the electro-optic coefficient s 33 •

The detection system
The detection system consists of an infra-red TV camera and a silicon photodiode. The camera is used for alignment of the interferometer, and the photodiode is used for the phase vs. voltage measurements.
The photodiode is connected in series with a 100 kO resistor and a voltage source. has the tendency to drift slowly, with a time period of the order of a couple of minutes. This is probably caused by a slow drift in the laser line due to heating g effects. This means that, with a 1 s measurement cycle, the data will be very reliable as the drift will be negligible. Another advantage of high speed is that several cycles can be performed within a short period, and the results averaged to produce data curves. In a typical measurement, about 25 measurement cycles are performed in about half a minute. These measurements are averaged to produce the final data curves.
For each wavelength, the entire measurement procedure consists of three steps.
The first is the measurement of the intensity of the reference arm, which is done with the device arm blocked. The device arm is the arm of the interferometer which contains the phase modulator. Next, the voltage dependence of the device arm intensity is measured, with the reference arm blocked. This is done as described in the previous paragraph, with each measurement cycle taking about 1 s. The change in intensity of light passing through the device is due to the change in absorption coefficient in the modulator with applied field. The last step is the measurement of the interference change with voltage, with both arms unblocked. The result of this measurement is labelled F(V) , and the change in device arm intensity is labelled f (V). These two quantities, along with the reference arm intensity R, are used to calculate the electro-optic coefficients.

Phase change versus voltage measurements
The phase change produced in the light guided through the device was measured using the Mach-Zehnder interferometer described above. The device to be tested was placed in one arm of the interferometer. The light was launched into, and collected from the device using a pair of microscope objectives. The polarization was perpendicular to the layers, i.e., TM. The output microscope objective was positioned such that it produced an image of the device near-field on the IR TV camera. A pair of similar objectives was also placed in the reference arm in order to focus the light onto the image of the device near-field in the camera and also to compensate for the change in path length introduced in the device arm due to the presence of the device-arm microscope objectives.
The interference pattern was imaged on the IR TV camera as well as on a photodiode. The camera helped in alignment, while the photodiode was used in the actual measurement.
The application of a reverse bias causes the refractive index of the guiding layers in the device to change, resulting in a phase shift of the guided wave. This shift can be observed as a change in intensity of the interference pattern. If the interferometer was not well aligned then a shift in the fringe pattern would be seen. In our case, the two arms were aligned so that the phase shift was seen only as an intensity change. This intensity change labeled F(V), measured by the photodiode, is shown in Figure ( 1.3,1.4) for two different wavelengths.
Moving from peak to trough on the plot corresponds to a phase shift of 180°.
These figures also show the measured intensity modulation, labeled f (V), caused by the device due to electro-absorption. This was measured with the reference arm of the interferometer blocked. The next step was to extract the phase versus voltage relations from the measured data. It can be shown that the phase change as a function of voltage <f>(V), can be expressed using the measured data as [3] Aocos where F(V) is the measured interference curve while f(V) is the measured intensity variation with one arm of the interferometer blocked Figure ( as a quadratic we can proceed to evaluate the quadratic electro-optic coefficient

Evaluation of s 33
The electro-optic coefficient SQ for the quantum well device is obtained using a technique similar to that used in [3], namely, a least-squares fit to the measured data </>(V). The main difference from [3] is that for TM polarized light, symme- can be written as (1.5) where the right hand side represents the linear terms of a Taylor's expansion.
As the Ans are small, this approximation is regarded as adequate. The above equation can be written more compactly as (1.6) where Ei is the field in the ith layer due to an applied voltage V. The Ani(Ei) is the change in the refractive index of the ith layer due to a field Ei. In order to get Eq. (1.6) into a more desirable form, it is divided into two sums as (1. 7) where the first sum covers only the GaAs quantum wells and M 1 is the number of wells. The second sum represents the other layers which are not wells and obviously M 1 + M 2 = M. The first sum in the equation contains the term An;(E;) which is the change in refractive index of the quantum well. This can be related to the electro-optic coefficients by (1.8) where the subscript Q refers to the quantum well and nQ is the refractive index.
The s does not carry the subscript j as they are assumed to be the same for all j. The ~nfc is the change in the refractive index due to the presence of free carriers. This is important if any of the layers were not fully depleted, as could occur if the doping were too high in the i-layers. This was important in our case as the unintentional doping in the i-layer was about 7x 10 16 cm-3 • The expression used for ~nc was [4] (1.9) Where Nd is the concentration of free carriers removed by the field. Values for n 8 were obtained using the equations developed by Afromovitz [5]. Of course, we also need to know the refractive index nQ, of the quantum wells, as a function of wavelength. For this, we used the analytic expression used by Sonek et. al. [6] for the dielectric constant Continuing, using Eq. (1.8) in Eq.(1.7) we get ~/3(V) = C1sQ + C2 = </>(V)/ L, (1.11) where (1.13) We are now in a position to solve for sq of the quantum well. Using Eq. (1.11), a linear least-square error fit to the data <f>(V) yields the desired quantities.

Results
Figure (1.5) shows the calculated quadratic electro-optic coefficient s 33 as a function of wavelength for both the [-1,1,0] and the [1,1,0] orientations. We see a difference between the two orientations and that the difference decreases as the wavelength increases. It is clear that s 33 is large compared to that of bulk materials based devices by roughly more than one order of magnitude. This makes the quantum well based devices more attractive. Even though the absorption may be high in this spectral region(~ SOmeV from edge) the high s 33 allows for very short modulator lengths which might make the effects of absorption tolerable. This is especially important for integration of the laser and modulator on the same wafer. In the simpler schemes, the laser and modulator would have the same absorption edge and generally the lasing would be about 40 me V below the edge, which might be a wavelength where a sufficiently short phase modulator could function without too much loss or intensity modulation. This will however, be limited to digital modulation schemes due to the quadratic nature of the modulation. Also, we note the rapid decrease of s 33 as we go away from the absorption edge.
We have also plotted the quadratic electrooptic coefficient versus detuning from the exciton fundamental absorption edge in me V for both polarizations on the same graph for the sake of comparison, figure (1.6) shows the result. For TE polarized light, the quadratic electrooptic coefficient involved is the 813 (see appendix A), while for TM polarized light 8 33 is the coefficient involved as previously indicated. 8 13 was measured in [3], 8  Also, the change in refractive index as a function of the electric field for different wavelengths was calculated. Figure (1.7) shows the result. The calculation of ~n(>., E) is based on (1.14)

7 Conclusion
The wavelength dependence of the quadratic electro-optic coefficient s 33 has been measured near the band edge for QW structures for a TM polarized light. This was obtained by measurement of the phase changes induced in a waveguide structure by an applied electric field using interferometric techniques. The results indicates that the S33 shows a dramatic increase near the edge.
We have discussed the utilization of the large quadratic coefficient for digital phase modulator integration with a laser.
We have also shown a comparison between the values of the quadratic electrooptic coefficient in both TE and TM polarizations. Results show that s 13 is larger than S33 , and that it decreases faster .
We have also presented a set of data curves for the .ti.n as a function of applied field with wavelength as parameter as an aid to modulator design and operational characteristics evaluation.

Appendix A
In this Appendix we determine t he non zero elements of t he electro-optic tensor that are involved in our measurements.

The electro-optic effect in zincblend type crystals
In certain types of crystals, the application of an electric field results in a change in both the dimensions and orientation of the index ellipsoid. This is refered to as the electro-optic effect.
The propagation of optical radiation in a crystal can be described completely in terms of the impermeability tensor TJii (TJ = € 0 C 1 ). The two directions of polarization as well as their corresponding indices of refraction can be found by using the index ellipsoid which assumes its simplest form in the principal coordinate system: (1.15) where x, y, z are the principal axes, that is the directions in the crystal along ( 1.16) where E is the applied electric field, riik is the linear (or Pockels) electro-optic coefficient and Sijkl is the quadratic (or Kerr) electro-optic coefficient.

Zincblend Crystal Structure
It can be viewed as two fee structures displaced from each other by 1/4 of a body diagonal. About each atom, there are 4 equally distant atoms of the opposite kind arranged at the corners of a regular tetrahedron. In crystallography, different structures are grouped into families. The zincblend type crystals belong to the family 43m. Very shortly we explain what it means and how this grouping helps identify the non zero elements in the linear and quadratic electro-optic coefficient-tensors.

Perturbed Index Ellipsoid
Applying an electric field E to a crystal changes the dimensions and the orientation of the index ellipsoid and makes it to deviate from equation (1.15) due to the appearance of mixed terms. The most general form of an index ellipsoid in the presence of an applied electric field is given by the quadratic form: ( 1.17) which can be written as: in the above equations i, j = 1, 2 and 3 refer to the x, y, and z axes respectively.
In this case, x, y and z are not the principal axes any more. It can be easily shown [7] that the dielectric tensor Ei; is a symmetric tensor provided that the medium is lossless and optically inactive, i.e. Ei; = E;i. According to the definition of Tlii we conclude that Tli; must also be a symmetric tensor, hence Tlii = TJ;i· Consequently, the indices i and j can be permuted and ( 1.18 ) can be written as: ( 1.19) Note that when the electric field vanishes, the index ellipsoid (1.19) reduces to (1.15). We are now going to use the standard contracted index notation which is the replacement of the double subscript by a single one. It is important to remember that the contraction of indices is just a matter of convenience. In this customary notation: TijkEk + SijklEkEl can be written in the form: As mentioned earlier, the zinc blend type crystals belong to the group of point symmetry 43m. This group has four-fold (90°) rotation axis with inversion symmetry which is, by strict convention, taken as the z-axis, as well as two mutually orthogonal three-fold (120°) rotation axes of symmetry that lie in the plane normal to z. These are designated as the x and y axes. From these two rotation axes, MQW structures based on III-V material compounds (which have a zincblend type structure), because of the quantum wells, the symmetry is reduced in the z-direction due to the non infinite extent of the crystal lattice in that direction.
This reduced symmetry makes the two mutually orthogonal 3-fold rotation axes no longer 3-fold but rather 2-fold (180°) or diad axes, hence, the group will be called (42m) instead of (43m).
If we denote the axes of rotation x, y and z by x 1 , x 2 and x 3 , the 2-fold or diad axes could be x 1 or x 2 and the 4-fold inversion rotation axis is always x 3 • Diad means rotation by ¥ = 180°, it is easy to see that if we rotate by 180° around the axis x 1 , the coordinates of a general point Pz 1 .z~.za will be Pz 1 ,-z~,-z 3 , or simply: If we use this rule in (1.23) to operate on the subscripts of the linear electrooptic coefficient Tifk, and using the fact that if rifk = -ri;k, then this means that (i. e. ) we are left with 8 non zero elements so far and we have: In the previous part, there was a quicker way to deduce (1.25) from (1.23) by inspection. Since the diad axis was xi, then all elements in (1.23) that have either: no l's or two l's will be zero elements.

b. Rotation around the 4-fold inversion axis II xs
We use the fact that the 4-fold inversion axis x 3 includes 2-fold or diad axis. We can follow a formal procedure similar to that of the previous section regarding the rotation around the diad axis. Namely, Pz 1 ,z 2 ,z 3 will be P -z 1 ,-z 2 ,z 3 after 180° rotation around x 3 • Alternatively, we use a quick rule analogous to that of the previous section, all elements that have either no 3's or two 3's will be zero elements. If we look at (1.25) one finds that: (1.26) now we will be left with: Furthermore, when the losses are neglected we have £1; = f.;1, this means r1;k = rkij, so we have r 231 = r 312 . In the contracted index notation 23 = 32 = 4, 31 = 13 = 5 and 12 = 21 = 6, so r 231 = r 41 and r 312 = r 52 , and hence we have (l.29) differs from the bulk case (43m point group symmetry). In the bulk case, ru = r63· In other words, any of the three principal axes can be chosen as the z-axis and the resulting change in the refractive index would be the same.
This is obviously not true for quantum well structures. An applied field parallel to the layers would produce results dramatically different from one applied perpendicular to them.

The Quadratic Electro-optic Coefficient [ Sijki]
Su S12 S13 S14 S 15 S15 S 5 1 S52 S53 S54 S55 S 55 S1111 S1122 S1133 S1123 S1131 S1112 S1211 S1222 S1233 S1223 S1231 S1212 a. Rotation around the diad axis 11 x1 We can follow the same procedure outlined earlier for determining the [r;;] tensor, however we can use a quicker technique to identify the zero elements by inspection. For the [s;;kz] with diad rotation around xi, all terms that have one 1 or three 1 's are zero elements so we will end up with S1111 S1122 S1133 S1123 Since the 4-fold inversion axis has diad axis included in it, we use the same rule as above: all terms that either have one 3 or three 3's are zero elements, so we will have: S1111 S1122 S1133 It is easy to figure out the transformation rule for a point Pz 1 ,z 2 ,z 3 that rotates around a 4-fold inversion axis II x 3 as we have done that for a diad axis. We find that: or simply: So finally we end up with [sii ] in the contracted index notation as: Su S12 S13 0 0 0 S12 Su S13 Again this differs from the bulk case. It can be shown that in the bulk case we have, S33 = su, s 66 = s 44 , and s13 = S31 = s12.
Note that ~ = ~ so we will call both ~- We now need to diagonalize (1.38). Due to the symmetrical appearance of x and y in (1.38) we chose the transform: 1 or x -:-::-2 + r 63 z + y -:=-2 -r53 z + ~ = nz nz nz If we put: with -:\-, + , and +, given by (1.42). If we assume that the change in the n:z:' n 11 , nz' refractive index is small compared to n 0 , we can use the differential relation: apply (1.44) where ~nx'' ~n 11 1, ~nz' are given by (1.45). In our configuration, for TM polarization, the optical field vector is polarized along the [0,0,1 J direction, i.e. normal to the QW layers. Therefore, the corresponding ~n = ~nz'· Excellent review of the basic material discussed in this appendix can be found in [7J, [8], and [9J. ,,......

Introduction
In recent years the optical properties of multiple quantum wells (MQW) have been the subject of increasing interest. Because the excitonic absorption resonances are retained at room temperature in a Q W, they have the potential for practical applications in electrooptic devices.
When an electric field is applied perpendicularly to the quantum well layers, the band edge shifts to lower photon energies while still retaining their excitonic features. This electroabsorptive effect has been called the Quantum Confined Stark effect [1], it is therefore particularly interesting both from a practical and fundamental viewpoint as it forms the basis of a variety of modulators and optical switching devices.
In the theoretical analysis of multiple quantum well structures (MQW's) [2], the wavefunction is usually separated into a Z-dependent and a transverse part, with the Z-part containing the features of the potential well-and its modification by the applied electric field.
A loss in the oscillator strength of interband absorption is observed as the electric field is increased. In the simplest approximation [ 3], if one may neglect any perturbation of the electron and hole wavefunctions in the plane of the layers by the electric field; the oscillator strength, and hence interband absorption, is proportional to the overlap integral of the wavefunctions normal to the layers.
This overlap integral decreases as the electrons and holes are pulled to opposite sides of the well by the electric field leading to an increase in the change in electroabsorption.
In many device applications, it is desirable to achieve the largest change in absorption with the smallest possible applied voltage. Our measurements show that at a given wavelength and applied voltage, the change in absorption is larger in the [-1,1,0] oriented devices than in the [1,1,0] ones for both TE and TM polarizations.

Measurements and results
A tunable dye laser, using the Dye Styryl 9M, was employed as the light source.
The light was chopped at a reference frequency, launched into and collected from the device using a pair of microscope objectives. The light out from the device was imaged on the IR TV camera as well as the photodiode. The camera helped in the alignment by looking at the fundamental mode profile and insuring that it was guided under the ridge. The photodiode was used in the actual measurements of the transmitted light intensity.
A DC programmable power supply applied the reverse bias to the device, from 0 to 6 volts in steps of 0.1 volts, while two computer controlled lock-in amplifiers measured both the transmitted light intensity and photocurrent. The photocurrent (electron hole pairs generated due to the absorption of light photons) and the current of the photodiode were converted to voltage using two transimpedance amplifiers. The transimpedance amplifiers were made up of a dual operational amplifier chip and feed back resistance that defined the current to a voltage gain.  Where Pie is the coupled power in the device, r is the confinement factor and L is the device length, hence one can write the change in absorption ~a(V, >.) The voltage independent part of the detected light, which we shall call Pieak, should be subtracted from the measured light intensity by the lock-in amplifier.
So equation (2.2) is modified to We have also carried out the same measurements, ~a(V, >.), for double heterostructure (DH) waveguide modulators. Figure (2. 7) shows a cross section of the double heterostructure sample used. Figure (2

Results and explanation
We believe that the orientation dependence in the electroabsorption is attributed to the anisotropy [4] of the valence sub band structure of the two dimensional hole system in the Q W. This is equivalent to different hole effective masses in different directions.
The conduction-and valence subband energies and the binding energies of the excitons in the GaAs/ AlGaAs MQW structures in the presence of an electric field has been calculated by several groups [2]. In these calculations, it is assumed that the heavy and light hole subbands are decoupled and hence Dingle's [5] simple particle in a box model is valid. The results of these calculations are in qualitative agreement with the experimental results.
Recently, it has been pointed out [6] [7] that the valence subband states are more complicated and far from being trivially parabolic even at the Brillouinzone center. This is because the QW periodic potential mixes the bulk heavy hole and light hole states of both parities giving rise to dipole allowed transitions between all pairs of valence and conduction subbands. This in turn affects the energy versus momentum dispersion curves and the optical matrix elements, i.e. the electronic and optical properties of the semiconductor Q W. Strong exciton structures corresponding to "forbidden" transitions have been observed in luminescence studies [8] and have been reproduced in theoretically calculated absorption spectra taking valence band hybridization into account [9].
It is clear that if one wants to account for this electroabsorption orientation dependence, one has to start from first principles to calculate the absorption coefficient as the simple particle in a box model will not be adequate.
The free carrier (electron, hole) states are to be solved within the effective mass approximation, where the CB an VB states are expressed in terms of the zone center Bloch functions for bulk GaAs. Unlike the case for the conduction band, the valence band in bulk GaAs, in the limit of infinite spin-orbit splitting, is four fold degenerate at the band edge with total spin of J=3/2. The heavy hole (Jz = 3/2, -3/2) and the light hole (Jz = 1/2, -1/2) subband states are thus strongly hybridized and hence the effective mass hole Hamiltonian will be a matrix ( 4 x 4). This matrix Hamiltonian is a function of k11 (kz and k 11 ). Due to the lack of inversion symmetry in the appearance of ( kz and k 11 ) as entries to hole matrix Hamiltonian in the off diagonal terms, it is orientation dependent as we will shortly show (Appendix A).
The implication of this orientation dependence in the hole matrix Hamiltonian is the anisotropy of the hole sub band in plane dispersion which has been recognized by several groups. For every kz and k 11 , the solution of the hole multiband matrix Hamiltonian will render a different E(k;;) and a different hole eigenstate. These matrix element and then the value of the absorption coefficient. Consequently, the absorption coefficient, which is an explicit function of the electron / hole energies and the optical matrix element (see Appendix A), will be different along different orientations.

Transverse sample measurements
We have done the same measurements on a transverse probe as opposed to the waveguide sample we measured earlier. Figure  10-14 cm-3 and around 10-11 cm-3 in the P and N layers so that the applied voltage was dropped entirely across the i-layer. The input face (P-AlGaAs) was AR coated using an etch-tuning method to prevent unwanted Fabry-Perot interference effects.
The light was incident normal to the quantum well layers and the electric field vector was rotated using a polarizer and a half wave plate. In this configuration, one can not investigate of course the TM polarization. We were not able to see any significant change in the electroabsorption as the electric field vector was rotated in that particular arrangement. The reason for that might be the short interaction length between the incident optical radiation and the quantum wells (0.525 µm) for this transverse sample.
To elaborate more on that, from the geometry one can write for the transmitted light intensity Pt(.X, V) in terms of the incident power Pin : It is clear that ratios between the transmitted light intensity at an applied voltage of 5 volts and at zero voltage are almost one for both orientations (this means that the transverse sample's interaction length is not big enough to see an appreciable change in the transmitted light intensity). Also, the difference between the two orientations is very small to resolve considering the inherent background polarization sensitivity in the optics involved in the setup (lenses, half wave plate). This explains why it was not possible to see any significant change in the electroabsorption as we rotated the electric field vector in the transverse sample measurements arrangement.

Effect of stress/strain on the QW electronic states and optical properties
The introduction of stress in a solid produces changes in the lattice parameter and in some cases, in the symmetry of the material. These in turn produce significant changes in the electronic band structure. All configuration of stress can be divided into two classes: the isotropic class, which give rise to a volume change without disturbing the crystal symmetry, and the anisotropic class, which in general reduces the symmetry present in the strain free lattice, alter the electronic states and energy gaps, and in some cases, remove the degeneracies. Also, effective masses will be affected by the variations in energy gaps as well as by variations in the interband optical matrix elements.

Lattice mismatch between quantum well layers grown along the [001] direction
creates stress which introduces strain with a uniaxial component along that direction. The relative lattice mismatch ~ between the host materials of hetao erolayers varies from almost zero in (GaAs/Al 0 . 3 Gao.1As) to values as large as several percentages (7 per cent between InAs and GaAs, for example). This fact makes us believe that the possibility of unintentionally introducing stress/strain in the wafer from which the samples were made is a rather slim one.
It is still possible that these stresses might have been introduced externally during the phase of preparation/mounting of the devices. If this was the case, we wonder why did the [-1,1,0] and the [1,1,0] oriented devices showed consistently the same behavior when measurements were performed on several devices ? Furthermore, the effects of external stress along several directions on the electronic structure and optical properties of various III-V based quantum wells have been investigated both experimentally [13] and theoretically [14] for example. In

7 Effect of refractive index change due to the linear electrooptic effect on the confinement factor r
One last point remains to be addressed so as to complete the picture. From (2.3) we see that ~a(V, >.) depends on the confinement factor r. If for example we consider a [-1,1,0] oriented device subject to an electric field normal to the layers, for TE polarization, the change in the refractive index ~n(E) due to the applied electric field E is given by [15]

Summary and conclusion
We polarization. This makes the [-1,1,0] orientation more suitable even for electroabsorptive device applications. To our knowledge, orientation dependence in the change in electroabsorption for MQW's has not been reported to date. We also gave a qualitative explanation to support our observations and correlate them with quantization and excitonic effects in quantum wells.

Appendix A
This appendix is intended to give a qualitative explanation of the orientation dependence in electroabsorption between the [-1,1,0] and [1,1,0] oriented devices which we have experimentally observed. In order to do that, one has to start from first principles to calculate the absorption coefficient in MQW structures with band mixing and excitonic effects considered. This has to be done so as to realistically calculate the electronic states, and hence the energy versus momentum dispersion curves, in MQW structures. They both appear explicitly in the expression to be derived for the absorption coefficient, and also affect the calculated value of the optical matrix element which appears explicitly in the expression for the absorption coefficient as well.
In what follows, we calculate the absorption coefficient in MQW structures using envelope function description of electronic states within the frame of the multiband effective mass scheme. The absorption coefficient is defined as the energy absorbed per unit volume per unit time divided by the incident energy per unit area per unit time.

.1 Absorption coefficient calculation
Let us represent the incident light by the vector potential A(r,t)  (2.19) using the facts that c 2 = 1/ µ 0 e 0 , q = nw / c , and IEl 2 = E; = w 2 A;, one can write 1 hence, the absorption coefficient is To determine W ( w), we need to solve the SchrOdinger wave equation for an electron in the presence of the perturbing potential due to the optical radiation represented by the vector potential A 1 .
[ -+ V(r) + -p.A + -A 2 ]w = ihW 2mo mo 2mo (2.23) or [Ho+ H]w = ih4' (2.24) Pj;(k) =< )·, k jP jj,k > (2.37) In the above expression it was assumed that all states in the jth band are occupied and all states in the ] are empty. If this was not so, the occupational probabilities of the states have to be included.
So the calculation of a actually amounts to the calculation of the electron and hole states and their associated energies as a function of the k. We will point out how to calculate them using the effective mass envelope scheme.

Envelope function description of electronic states in the multiband effective mass scheme
There are some rather stringent requirements for a satisfactory procedure to compute electronic states in heterostructures:  Where the index n and m label the subband, the index u and v label the z component of the spin, e and h refer to electrons and holes, uo(r) is the zone center periodic part of the Bloch functions (we will discuss this further shortly), kll = kxx + kvfJ, p =xx+ yfJ, and f~(ze) and g ':n(k11,zh) are the envelope functions for electrons and holes respectively. the parallel wave vector k 11 is a good quantum number because of the translational symmetry along the x and y directions (the plane of the quantum well layers).
In many III-V based heterostructures, the band edges relevant to optical and transport properties have the symmetries of the r 6 (electron bands) ' rs ( degenerate heavy hole and light hole bands), and r 7 valence band which is split by the spin orbit coupling .6. from the rs edge. In GaAs, .6. = 340 mev, it is a fair approximation to take only the r 6 and rs edges into consideration when building up a quantum well state from a bulk state. This is the infinite spin orbit splitting approximation and is exploited to limit the sum to 2 bands (r 6 , up and down spins) for electrons in (2.38), (indeed the two bands are identical as they are decoupled since the electron Hamiltonian is diagonal) and to limit the sum to 4 bands (rs heavy and light holes, up and down spins) for holes (2.39 In the effective mass scheme, to find an electron/ hole state wavefunction w~. 1:n we must solve the Schrooinger equation (2.41) Rather than using the usual Bloch functions In, k > corresponding to H 0 , (2.42) The scheme uses a slightly modified Bloch functions to expand w~::n,. (2.43) Un,o(r) is the periodic part of the Bloch functions calculated at the zone center.
We assert that these modified basis functions form a complete orthonormal set if the Bloch functions do [10].
The reason for this choice of basis is to avoid the difficulties associated with the variation the Un,k(r) with wave vector k from one band to another when build- reproduce the correct energy dispersion relations for the bands). It is a sensible approximation to neglect the difference between the / parameters of GaAs and AlGaAs due to the similarity of their band structure.
When kz = k 11 = 0, the off diagonal terms in Tv,;, vanish, consequently (2.48) will he diagonal, and (2.47) will split into two sets of single component equations.
In other words, the valence subbands would decouple into a purely heavy hole (Jz = ±3/2) subband with effective mass in the z-direction =2 and a purely The solution of (3. 7) and (2. 4 7) can be obtained variationally [9] by expanding fn(ze) and g~(k; ;, zh) as sums of 10 even and 10 odd Gaussian type orbitals of the form e-f3z 2 and ze-f3z 2 , where the exponents {3 are chosen to cover a broad physical range. Substituting these expansions for the envelope functions into the effective mass equations (3.7) and (2.47), reduces the problem to a generalized eigenvalue problem to be solved for the subband energies E~(k;; ) and E~(k;1) and the envelope function expansion coefficients.
Remains the calculation of the optical matrix element Pnm ( k 11) as defined m IS> is completely symmetric in x,y, and z like Jx 2 + y 2 + z 2 , IX> is antisymmetric in x but symmetric in y and z like xJx2 + y2 + z2, IY > is antisymmetric in y but symmetric in x and z like yJx2 + y2 + z2, IZ > is antisymmetric in z but symmetric in x and y like zJ x 2 + y 2 + z2. Electrons in the conduction band have completely symmetric wavefunctions IS >. The spin up and spin down u~=±l/ 2 (r) are written as a two component spinor

JS>
For the valence band, the situation is more complex. An electron moving in a strong electric field sees an effective magnetic field due to relativistic effects even if there were no external magnetic fields. This is known as the spin orbit interaction. It modifies the spin up and spin down u~(r) in a non trivial way as follows [11] v= 3/ 2 _ _ 1 ( It can be shown that due to the symmetry of the functions JS >, JX >, JY >, and JZ >mentioned earlier, only the following integrals are non zero< SIPzlX > = < S IP 11 IY > = < S IPz lZ > =Po, where P 0 is a constant and can be estimated [12] using It has been pointed out [6] that for quantum wells, the approximation Pnm(k;;) as L = -2v'3')' 3 (-kzik 11 )kz instead. Obviously, that will make the energy vs momentum, the hole electronic states, and the optical matrix elements different for both directions. Consequently, the absorption coefficient will also differ for both orientations.
In other words, if one wants to accurately assess the orientation dependence of the absorption coefficient, the isotropic approximation, (used by some people for the sake of simplicity), can not be used due to the anisotropy of the valence subband dispersion, (which have been reported by several groups as previously indicated), and direct.ion dependence of the optical matrix element Pnm ( k 11). Both appear explicitly in the expression for the absorption coefficient. This makes the integration over k not a simple 1-D integration. and [1,1,0]

Introduction
When an electric field is applied perpendicularly to the layers of a MQ W structure, the band edge shifts to lower photon energies while still retaining its excitonic features. This effect is referred to as the Quantum Confined Stark Effect (QCSE) [1]. It is this steep rise of the absorption edge, coupled with the very high excitonic peak, and its shift with the applied field, that make this effect particularly interesting both from a practical and fundamental viewpoint as it forms the basis of a variety of modulators and optical switching devices.
For the design and development of optical light modulators utilizing the QCSE, one needs to know the value of absorption for the electron-heavy hole (e-hh) fundamental edge as a function of wavelength and applied electric field. The value of a(F, >.) can be experimentally measured for a given material composition and device geometry. This measurement requires a spectrally variable light source which can be scanned over a relatively wide range of wavelengths. Furthermore, this range of wavelengths needs to be scanned for each value of the applied electric field.
In the method we describe shortly, we use a source em1Lt ing at a properly chosen wavelength >-z, and measure the transmitted light intensity as a function of the applied electric field F. From this measurement we can deduce a(F, >.) at any>.
shorter than >-z, and any value of F.
The structure of this paper is as follows: In section II we describe the sample used, the experimental s~tup, and show results of the measurement. In section III we outline the technique used in processing the measured data. Section IV discusses the results, we t hen conclude with a summary in sect ion V .  when the e-hh and e-lh transition energy decreases due to the applied electric field until it matches the incident photon energy. If >.x was higher, those features would have occured at higher voltages. This is because the incident photon energy will be smaller and one needs more applied electric field to decrease the e-hh (or e-lh) transition energy so as to match the photon energy.

Sample
Before we explain the proposed technique, let us quickly verify the position of the e-hh exciton peak. For the particular sample used, calculations and measurements [2] indicate that the e-hh exciton peak at zero applied electric field occurs around 1465 meV. If Ax = 860 nm ( i.e. Ex = 1441.9 meV ), then the energy difference between the heavy hole exciton energy at zero field Ehh(O) and the incident photon energy Ex is 23.1 me V. The electric field F that will cause Ehh(O) to decrease by that very same difference ( i.e. 23.1 meV) can be readily calculated and is found to be about 103 kV / cm which corresponds to an applied voltage of about 9.3 V for this particular sample. Looking at Figure ( 3.3) it is clearly seen that the e-hh exciton peak truly occurs at 9.3 V.

Technique
From the geometry of t he sample one can write for the transmitted light intensity Pt(>., V) in terms of the input coup led power Pc as : (3.1) where R is the reflection coefficient of the output face t (the AlGaAs/ air interface) and dis the sum over the GaAs well widt hs (.525 µm) , hence The coupled power can be written as Pc = fJc Pin, where fJc is t he coupling efficiency and Pin is the incident power. fJc was assumed to be one since the light was sharply focused onto the sample which had a large aperture. Alternatively, one may use optical fibers to couple t he light in and out of the sample. The input fiber should have a small diameter to insure that all the light is coupled into the fiber, and the output fiber should have a bigger diameter to insure that all the light emerging from the output face of the device was collected.
Figure (3.4) shows the absolute value of a(>.z = 860 nm ,F) in cm-1 using equation (3.2) with the solution of Poisson's equation employed to convert the horizontal axes from voltage to electric field.
The optical absorption coefficient can be written as a product of the excitonic absorption peak ahh and a line shape function [3]. Optical light modulators operate on the long wavelength side of the fundamental absorption edge which fits extremely well a Lorentzian line shape [4], so a(F, hv) can be written as :  (3.4) This constant of proportionality is easily calculated using the measured value of ahh(F) and the calculated value of the overlap integral between the electron and hole wavefunctions at the same field F.
Ehh(F) is the sum of the electron and hole energies in the z-direction minus the binding energy of the electron hole pair which has two components: the electron hole pair kinetic energy in the plane of the Q W layers, and the potential energy due to their Coulomb interaction. We have used perturbation methods [4] to calculate the electron and heavy hole energy as well as the perturbed wavefunctions as a function of the applied field F. Details of the computation of the perturbed wavefunctions, (using perturbation methods), and the exciton binding energy, (using variational methods), as a function of the applied electric field is outlined in Appendix A.
We differ from [5] in the method used to calculate the exciton binding energy. We did not use variational techniques to calculate the perturbed electron and hole wavefunctions, we instead used perturbation techniques which rendered more realistic electron and hole wavefunctions of finite penetration into the barriers as opposed to using an effective well thickness beyond which the electron and hole wavefunctions identically vanish. Figure ( 3.5) and (3.6) show the calculated Ehh(F) and the oscillator strength OSCatr(F) versus the applied electric field F in kV /cm respectively. Now the HWHM of the exciton line shape r hh (F) in me V remains to be determined . This line shape is Lorentzian, it broadens as the electric field is increased.
To determine this field dependence of rhh(F) we use equation ( Figure   For waveguide type probes, one definitely needs a tunable coherent light source which is expensive, and may not be readily available in every lab. Here comes the strength and practical significance of the method we proposed, which uses a fixed wavelength coherent source and a programmable DC power supply which are more readily available in practically every lab, hence can be used at an advantage to measure the spectral dependence of a in waveguide structures made up of any material composition. Of course, for waveguide structures one will need to determine the confinement factor in both the transverse and lateral directions which can be easily calculated for a given device material and geometry.

Conclusion
We have proposed, for the first time, a useful and a rather simple technique that can be used to calculate the spectral dependence of electroabsorption in MQW based structures by measuring the transmitted light intensity as a function of the applied electric field. The technique does not require a tunable light source to characterize the device but rather a readily available programmable power supply.
This technique can be used for MQW devices of any material composition, and

7 Appendix A
In this appendix we describe in detail how we computed the electron-heavy hole exciton binding energy as a function of the applied electric field in GaAs/ AlGaAs Multiple Quantum Well structures.
Within the effective mass approximation, the electron-hole Hamiltonian can be written as: HKEZe + Ve(ze) + eFze + HKEZh + Vh(zh) -eFzh -HKEreh -Veh(r, Ze , zh) (3.8) where we have the following: 1. Ze and zh are the coordinates perpendicular to the plane QW layers for the electrons and holes respectively.
2. "r" is the relative position of the electron and the hole in the plane of the QW layer.
3. (3.9) are the kinetic energy operators for the electron and hole respectively in the z direction, with m;.l and mhl. being the effective masses of the electron and hole respectively in the z direction.
4. Ve(ze) and Vh(zh) are the built-in rectangular quantum-well potentials for the electron and hole respectively. 5. F is the applied electric field normal to the QW layers. are the effective masses of the electron and hole, respectively, in the plane of the layers). (3.11) is the Coulomb potential energy due to the electron-hole attraction.
To attempt a full solution of H'lj; = E'lj;, where His given by (3.8), we use a mix of perturbational and variational methods with the separable trial function: (3.12) where 1/Je(ze) and 1/Jh(zh) are the exact wavefunctions of the individual electrons and holes in the one-dimensional quantum wells with the applied field F [i. e. , the lowest eigenfunctions of HKEZe + Ve(ze) -eFze and the equivalent for holes], and (3.13) gg (i. e. , we chose a simple 1-s like orbital for the in-plane radial motion) with >., the amplitude diameter of the exciton orbit as a variational parameter.
Formally evaluating the expectation value of H we obtain: (3.14) where Em Ehz andEb are the energies of the individual electrons and holes kinetic energy in the z direction and Eb is the exciton binding energy we are interested in and is given by: Eb= EKEr + EPEr (3.15) where EKEr is the kinetic energy of the relative electron-hole motion in the QW layer plane and is given by: (3.16) and EPEr is the Coulomb potential energy of the electron-hole relative motion, is given by: (3.17) EPEr is of course dependent on the variational parameter >.. The calculation of (3.16) is straight forward n,2 a2 < <Peh I -2µ ar2 I <Peh > or (3.18) The evaluation of the integrals in (3.17) are more involved. The fact that the wavefunction t/J is separable is of less importance here because the potential Veh couples all variables. This integral in 3.17) will be carried out partly analytically (the integration over r) and partly numerically (the integration over Ze and zh)· Using (3.11), (3.12), (3.13) and (3.17) we have: The integral over () in (3.19) is trivial and the integral over r can be performed using integration tables, yielding:

G(t)
2r 21 00 re-T dr A r=O J1 2 + r 2 (3.20) where H1 ( u) is the first-order Struve function and Y 1 is the first order Bessel function of the second kind (Neumann function) and/= Ze -zh.
tPe(ze), tPh(zh) in (3.19) and Eez, Ehz in (3.14) are obtained by solving: (3.21) and the equivalent for holes. We use perturbation methods to solve {3.21). In this method, the potential of the applied field (eFz) is considered as a perturbation to the Hamiltonian and the perturbed wavefunctions (1/le(ze) and 1/Jh(zh)) are expressed as a linear combination of the unperturbed wavefunctions (1/l~(ze) and 1/1/: ( zh)), i.e. the bound states of the electrons and holes with no applied electric field (actually they are not truely bound states due to the finite height of the potential barriers), so we can write: The double integral over ze, zh: EPEr = -2 ;:A l:_ 00 l:-oo 1/!;(ze)1/J~(zh)G(r) dzedzh (3.25) is then evaluated numerically for every given A and applied electric field.