Analysis and design of vertical cavity binary-phase modulators

G.Clarici, M.P.Y.Desmulliez and B.S.Wherrett

Abstract: The Fabry-Perot equation used to model asymmetric Fabry-Perot cavity reflection mode modulators can be considered as a bilinear transformation between the amplitude reflectivity of the modulator and the optical parameters of the cavity. Conformal mapping of this transformation into a suitable plane is used to visualise the operation of such a device working as a binary-phase modulator and to evaluate the tolerance of points of operation towards changes in the physical or operational parameters. Operation away from cavity resonance is shown to be less sensitive to parameter variations than conventional on-resonance operation.

1 Introduction

Optical beam-steering plays an important role in free-space optical interconnects for computing, network switching and phased array radar processing. A two-dimensional (2-D) array of optical modulators can be electrically or optically addressed to produce sets of diffractive gratings. By steering the reflected or transmitted light from such an array onto different detectors, a reconfigurable optical interconnection pattern can be achieved. Up to now, beam-steering applications have been marred by the low frequency of their operating devices. Ferroelectric Liquid Crystal (FLC) spatial light modulators (SLM) allow recon- figuration times of 10-100 p s [I]. Beam-steering arrays consisting of reflective 111-V semiconductor intensity modulators, capable of up to 40 GHz operating frequency at the device level, can potentially be operated at frequen- cies required by applications such as packet switching [2]. However, owing to the pure amplitude modulation, a large part of the power is wasted in the DC component of the diffraction pattern and in its off-state elements.

A semiconductor modulator based on the Asymmetric Fabry-Perot Multiple Quantum Well (ASFP MQW) device is shown in Fig. 1. Used as intensity modulators, a high contrast ratio is achievable owing to the destructive inter- ference between light reflected by the top and bottom mirrors; an adjustable absorption within the MQW cavity enables switching between a high reflecting state and a state of very low reflectivity. An electric field applied across the cavity region is used to produce the absorption change. In 1994 Trezza and coworkers proposed and fabricated an ASFP modulator that allows pure binary phase modulation capable of up to l00GHz operating frequency [3]. By altering the absorption within the cavity the device switches between dominance of the top mirror reflection and that of the bottom mirror. An appro-

,o IEE, 2000 IEE Proceedings online no. 20000805 DOI: 10. 1049/ip-opt:20000805 Paper first received 20th April and in revised form 5th September 2000 The authors are with the Heriot-Watt University, Riccarton, Edinburgh EH14 4AS, United Kingdom

IEE Proc.-Optoelectron., Vol. 147, No. 6, December 2000

priate choice of top and bottom mirror reflectivities as well as of the cavity length leads to equal power reflectivity at two points of operation (partner-points) for which the phases of the reflected beam differ by n (Fig. 2). Arrays of such devices provide binary phase gratings for beam- steering applications [4].

"b

top mirror

MQW cavity

bottom mirror

substrate

Fig. 1 Vertical asymmetric Fabiy-Perot cavity modulator

R

top mirror dominates

dominates

absorptivity

Fig. 2 power reflectivity R and phase change on reflection 4

Reflectiviv characteristic of an ASFP modulator

377

The operating wavelength of Trezza's phase modulator was chosen to be at a Fabry-Perot resonance for both points of operation. This means that the round-trip phase term of the cavity is real. Although the assumption simplifies the analysis of the device, it restricts the selection of the points of operation by requiring that the applied field produces a change in absorption but not in refractive index. The resulting poor tolerance caused by device parameter varia- tions or nonuniformity across arrays of devices has restricted the subsequent exploitation of the original design. There is an alternative choice of operating points that also simplifies the analysis [ 5 ] . If the absorption is very high, ideally infinite, at one point of operation, then the device reflectivity is equal to the top mirror reflectivity and independent of the cavity length. At the partner point of operation the cavity must be on resonance in order to produce the n phase change. Unfortunately, typical achiev- able absorption changes prohibit operation of such a device with low insertion loss [6]. In summary, the points of operation investigated to date have been restricted to certain special cases. These cases require that the device be operated at resonance, which is well known to lead to susceptibility to deviations. Here, we investigate the improved device tolerance achieved by relaxing the reso- nance requirement.

To implement such a generalised analysis, in which the round-trip phase change is arbitrary, the relation between the device reflectivity and the cavity parameters is visua- lised, using a technique similar to that employed in micro- wave filter design [7]. In particular, permissible regions of operation and certain distinguished points can easily be displayed. Furthermore this method can be applied to intensity ASFP modulators as well as grey-scale ASFP phase modulators [8].

2 Modelling

The parallel layers of semiconductor found in the vertical cavity modulator are analogous to a sequence of transmis- sion lines with different characteristic impedances connected in series. For such arrangements a number of well-established design tools exist. One tool is the Smith Chart [7], which allows the analysis and solution of transmission-line problems in a very intuitive graphical way. Here, a diagram similar to the Smith Chart is devel- oped to visualise the operation of the vertical cavity modulator. The analytical treatment of ASFP devices is usually based on the Fabry-Perot (FP) equation [9]. This equation, originally derived for a three-media structure, can be applied to a distributed mirror-cavity-distributed mirror combination if the top mirror is a loss-less quarter- wave mirror [9]. The overall amplitude reflectivity of the modulator structure, r, is related to operational and physi- cal parameters by the FP equation

with rtop and Yhot,om being the amplitude reflectivities of the top and bottom mirror respectively, 0 = nLc(2n//2) - j aL , is the single pass phase factor of the cavity, n is the real refractive index, CI the absorption coefficient, L , the length of the cavity and 2 is the operating wavelength. The top mirror is operated at the Bragg wavelength so that rtrl,, is real and typically negative corresponding to a n phase change on reflection. The phase-term and the bottom mirror reflectivity can be combined into one parameter z = rbottome-2j@, which represents the bottom mirror ampli-

378

tude reflectivity as seen from the top of the cavity. Defining p = rto,, , the FP-equation takes the form:

P + Z

1 - t p z r = --

For a given p # k 1 the FP equation corresponds to a bilinear transformation between the complex quantities z and Y [lo]. For a device to be designed and fabricated the overall power reflectivity lrI2 is usually specified. Mirror and cavity parameters have then to be calculated.

Solving eqn. 2 for z, gives the mapping of the r-plane onto the z-plane

( 3 )

In the z-plane, assuming that the bottom mirror reflectivity is constant, each point represents one pair of absorption coefficient and real refractive index values. The absorption coefficient determines the magnitude of z while the refrac- tive index determines its argument. Fig. 3 shows the polar mesh of the r-plane mapped onto the z-plane, for a chosen p value of 0.7. From the diagram, reflectivity values can be related to their corresponding z-values and hence to their respective absorptionirefractive index combinations. The solid circles are contours of constant power reflectivity. The dashed curves are circles for which the phase of the reflected field is constant in the top half-plane (9) and equal to (cp + n) in the bottom half-plane. The two opera- tional points of the binary-phase modulator, partner points, therefore need to be at the intersections of a given solid circle with a given dashed circle.

Partner points z and z" must obey the relation r(5) = - ~ ( z ) , hence

- i k + 1 z + k

z = (4)

with

Eqn. 4, which is also a bilinear transformation, gives for any z the associated (partner) f as a function o f p . Once a pair of operational points has been determined, the magni- tude of the reflectivity and the absolute phase are obtained by using eqn. 2.

Fig. 3 solid ~ constant power reflectivity circler:, dashed - constant phase). 0 . . . origin, M . . .perfect matching condition, K . . .partner point of 0, z and i sample pair of points of operation, U , . . mapped infinity point and C, mapped imaginary axis

IEE Proc.-Optoelectron., Vol. 147, No. 6, December 2000

r-plane mapped onto the s-plane

The real axis of the r-plane maps onto the real axis of the z-plane. The imaginary axis maps onto a circle C,, which is centred on the real z-axis and intersects with it at the points M (z= - p ) and U (z= - l/p). The point M is the mapped origin of the r-plane (zero reflectivity, also known as the perfect matching condition [3]). In the case of the intensity modulator, M is the ideal point of operation for the off-state. The point U corresponds to infinity in the r- -plane (infinite reflectivity). All device operating points must lie within the outer full circle shown in Fig. 3, which corresponds to 100% reflectivity. Within the circle C, the polar mesh of the r-plane is strongly compressed indicating that small deviations in z generate large varia- tions in r. Outside C, the polar mesh is wider, indicating that in this region r is less sensitive to deviations in z. However, as it can be seen from the intersections of the phase and amplitude circles in Fig. 3, every point outside the circle C, has a partner point inside the circle. Therefore the lower-tolerance region cannot be avoided.

Two additional limiting circles can be drawn in the z- .plane. One circle is defined by the maximum achievable ,absorption (a=a,,,) and the other. is for the minimal absorption (a = amin). Both circles and their transforms further restrict the areas permitted for the points of opera- tion (Fig. 4). The reachable operating points of the device lie on a curve, z(@, that terminates on the a,,,,, and ~l~~~

{circles. The precise form of this of z(E) is determined by the dependence on the static electric field E of the cavity #absorption coefficient and refractive index. For each point Ion the curve there is a partner point Z(E), which be tcalculated using eqn. 4, but which in general is not reach- able physically. A partn_er point is reachable only if there is a field E, for which Z(E) lies on the reachable .(E) curve. .Binary phase modulation is only possible if z(E) and Z(E) .mtersect.

If operation is restricted to partner points on the real axis (of the z-plane, then the Trezza model [3] is regained; this icorresponds to operation at resonance. In the infinite absorption model [5] one point of operation is at the origin of the z-plane and the other point of operation at the point K (z = - lik). This model can be seen as a special case of the Trezza model. However, only the K-point requires operation at resonance. As binary phase modula- lors only require a 7c-phase change between the two states but do not restrict the absolute phase, operation can be away from the z-plane real axis.

Fig. 4 Escluded areas

Light grey due to residual absorption, dark grey due to finite absorption. White is the area of operation

IEE Proc.-Optoelectron., Vol. 147, No. 6, December 2000

3 Device design

Given knowledge of the field-dependencies of the cavity absorption and refractive index, &(E) and n(E), the techni- que described in Section 2 can be used to determine whether a given choice of device structure can operate as a binary-phase modulator. In this section the procedure necessary to achieve binary-phase modulation by modify- ing a failed design, or to improve the tolerances of a working design, is discussed.

The free parameters in the device design are the top and bottom mirrors ( p and rbottom) and the length of the cavity (L,). The aim of the design procedure is to find points of operation (values for these parameters and for the operat- ing voltages), which exhibit 7c-phase change, have a speci- fied value of the power reflectivity and are located in a region of specified tolerance. Considering the complex z- plane, if the characteristics z(E) and F(E) discussed in Section 2 do intersect then by definition the intersections lie on a circle of constant power reflectivity in the z-plane (constant Irl). In device design the intersections have to be adjusted to fall on the correct lrl-circle. The intersections also fall by definition on one of the circles around which the device phase change on reflection is constant (q) in the upper half of the z-plane and (9 + n) in the lower half of the plane. The novelty of the present design is that there is no restriction on the particular circle (q can take any value); this distinguishes the present design from previous schemes. The device tolerance is improved if that partner point within the C,-circle (Fig. 3) is away from the horizontal axis where the mesh of the circles is dense; the other partner point will be well to the other side of the axis. By implication achievement of high tolerance requires that the imaginary part of z has a strong depen- dence on the field E. In addition to improved tolerance, the operation at off-resonance eases the design restrictions that were identified in [4].

The first stage of the design procedure is to choose the top-mirror reflectivity, p , variation of which changes the location of the circles in the z-plane. p must be greater than IrI but should be chosen as close as possible to it because this improves the device tolerance, as described in Section 4.

Variation of rbvtton, and L, alters the shape and position of z(E) but does not alter the circles in the z-plane. rbottom

may in general be complex because it is not essential to work at the Bragg-wavelength of the bottom mirror. (i) A phase change in rb,,,,, leads to a simple rotation of z(E) around the origin, by arg(rbottom). (ii) ( z I is proportional to IrbuttomI. Hence an amplitude-change in rbOtfonl leads to a stretch or compression of z(E) and the effect is most pronounced in regions of high IzI. (iii) Variation of L, produces both radial-translation and rotation in the z-plane for each point of z(E). The largest amplitude changes in z occur in the middle of the IzI range; while the change in phase depends on the according n value of each point of z(E). The manipulation of z(E) by adjusting rboftvnl or L, is summarised in Table 1.

The dependence of lm-z on E increases with Lc for which a large value should therefore be chosen. However, this leads to large device absorptivity values, which must be compensated for by setting the amplitude of the bottom mirror reflectivity close to unity. Arg(rbotlom) should then be adjusted such that z(E) intersects with the real-axis in the z-plane. Finally I'botfomI and L , should be adjusted, using the rules described above, to produce intersections with the required lrl-circle, and a single phase-circle, as illustrated in Fig. 5. As design tips: if z(E) does not reach

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Table 1: Effects of changes in L, and r,,,, on points of the characteristic

Operation

increase of cavity length

decrease of cavity length

change of magnitude of bottom mirror reflectivity

change of argument of bottom mirror reflecivity

Effect on points of z ( 0

moved radially closer to 0 aod turned clockwise

moved radially away from 0 and turned counter-clockwise

moved radially away from 0 (increase) or closer to 0 (decrease)

angular movement

close enough to the z-plane origin then p should be increased; if z(E) does not reach close enough to the unit-circle, then Lc must be reduced and arg(rbottom) adjusted to compensate for the phase change.

4 Sensitivity analysis

An important issue for the device design is the sensitivity of the optical response towards deviations of the physical and control parameters. The physical parameters include: the cavity length, the cavity structure, and the bottom mirror reflectivity. The control parameters are the applied field and in principle the ambient temperature. Any varia- tions of these parameters will result in a deviation of the z- parameter from its ideal value. The methodology estab- lished in the previous section allows a straightforward sensitivity treatment. Only the magnitude of the maximal (absolute) deviation of z, Az, has to be considered. The relative deviation Az/lzl is determined by the deviations of cavity length ALc , absorption coefficient Aa and real refractive index An. In the case of small deviations:

The key is that a bilinear transformation maps circles onto circles. Hence the circle centred around z with the radius Az maps onto a circular region D, in the r-plane with centre r, and radius Arc as shown in Fig. 6. Any nearby point z'

Fig. 5 The partner point of Z, 2, lies on the same amplitude and phase circles as Z. The next step is to modify z(E) to reach both points of operation

380

Derivation of the partner point in the z-plane

Im r

-

Fig. 6 Maximal amplitude and phme error

0.4

0.2

0

-0.2

-0.4

Re r

I 0.01

I I

-1 .o -0.8 -0.6 -0.4 -0.2 0 a

I / I I I

-1.0 ' -0.8 -0.6 -0.4 -0.2 0 b

Fig. 7 Upper limit for the magnitude of relative amplitude a (Arljrl) error b Phase error over the z-plane for p = 0.7 and A d l z l = 0.0 1

IEE Psoc.-Optoekctron., 6 1 . 147, No. 6, December 21100

which.satisfies the condition ) z - 2’) 5 Az maps to a point r’ for which (rc - J I 5 Arc.

From eqns. 2 and [ 10, 111

rc = (7) d. +z)(l +pz*) - p G

11 +pz12 -p2Az2

The deviated reflectivity r‘ = r(z’) is somewhere within the area D, but in genei-a1 not in the centre or on the radial diameter (Fig. 6). If the origin of the r-plane is in D, then the phase error can be up to f x which would render the device useless. In the z-plane this corresponds to the point z being too close to the point M. If (z + P I > Az, then the origin is outside D,. In this case the maximal magnitudes of the amplitude and phase error are AY,,, =2Arc and AWmax = 2sin-l (Arc/lrc\).

Figs. 7a and 7b, show the maximal magnitude of the amplitude (Fig. 7a) and phase (Fig. 7b) error for p = 0.7, assuming 1% deviation in 121. This figure reflects the density of full (Sa) and dashed (Sb) circles of Fig. 3. The phase is normalised to rc and the amplitude error is relative to the magnitude of the reflectivity, respectively. No contour lines have been drawn for values exceeding 0.2 (amplitude error) and 0.1 (phase error). All errors increase with increasing magnitude of z at arguments close to arg(-p). Values away from the axis are less susceptible to errors than those on the axis. The phase error also has a maximum around the point M of Fig. 3 as expected; around the point M a very small deviation causes a high phase error. The main conclusion from Figs. 7a and 7b is that it is actually beneficial to work at z-values away from the real axis (i.e. operate the cavity off-resonance) and avoid the proximity to the point M.

85 Conclusions

[n summary a new way of visualising the operation of a vertical cavity binary-phase modulator has been presented. Based on the fact that the Fabry-Perot equation is a bilinear l.ransformation, expressions have been given for the toler- ;mce of the points of operation with respect to changes in 1 he device physical or operational parameters. The opera-

tion of the modulator is not restricted to a cavity in resonance. Points chosen away from the cavity resonance exhibit a higher tolerance than points at cavity resonance. The worst tolerance occurs for points, which lie close to the perfect matching condition. Typically, the point of operation at low cavity absorption is significantly the more sensitive. The lower the insertion loss of the device the higher the sensitivity of this point of operation will be. Qualitative expressions have been given to estimate the influence of changes in the cavity length or the mirror reflectivities on the operation. These expressions can be used, in device design in order to produce binary-phase modulation andlor to modify an existing design towards higher tolerance.

6 Acknowledgments

G. Clarici acknowledges the financial support of Heriot- Watt University.

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References

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