Loss-induced phase-sensitive spatial light modulator with a wide field of view Man-Fang Huang and Elsa Garmire We investigate the feasibility of achieving practical phase modulation by changing only the absorption of the device. We present a spatial light modulator that consists of a three-mirror cavity with lossy slabs inserted between the mirrors. When only the absorption of these slabs is changed, this device can have a gradual change in phase, but no intensity modulation. Theoretical analysis by the use of numbers appropriate to multiple quantum wells shows that the total phase-tuning range can be as large as 27rwith an absorption coefficient change of less than 9000 cm-'; the length of each multiple-quantum-well slab is -1.5 pum. This gives an expected 16° field of view with the nanosecond speed typical of semiconductors. 1. Introduction Optical phase modulators are of interest for applica- tions such as in either single-element or few-element geometries for fiber-optic communications or in a highly parallel multielement geometry for image pro- cessing. 1 - 3 Conventionally, the phase of an optical beam is tuned by changing the optical path length. As the electro-optical effects are usually weak, such phase modulators use waveguide geometries or rather long lengths. However, for some applications such as image processing and optical computing, phase modulators operate on light normal to the plane of the device rather than in a waveguide configuration. 4 Phase modulation usually requires a long path length. Recently, a binary-phase modulator in a vertical geometry that used a Fabry-Perot cavity was pre- sented. 5 However, this deviceis unsuitable for appli- cations in which a gradual change in phase is re- quired. 6 A phase modulator that can give a continuous change in phase consists of a three-mirror cavity with lossy slabs inserted between mirrors. The configura- tion is shown in Fig. 1; the lengths of the lossy slabs are Li and L 2 ; r, r 2 , and r 3 are the electric-field reflection coefficients of mirrors 1, 2, and 3, respec- tively. The idea is to control the phase of the reflected light from all three mirrors by varying only The authors are with the Center for Laser Studies, University of Southern California, University Park, DRB 17, Los Angeles, California 90089-1112. Received 30 July 1993; revised manuscript received 12 October 1993. 0003-6935/94/142856-05$06.00/0. c 1994 Optical Society of America. the absorption of the slabs, holding the reflection coefficients of the mirrors fixed. As we show below, the phase of the reflected light can be tuned by changing only the absorption coefficients within the slabs; the intensity of the reflected light can be maintained constant if the absorption of the two slabs is varied in a prescribed way, determined by numeri- cal calculation, as we discuss below. The optimization of such a phase modulator de- pends on several parameters. To find where in the parameter space the modulator will operate most efficiently, we can use an approximate phasor analy- sis to estimate values of the relevant parameters. These approximate solutions are then used as initial values for the exact Fabry-Perot resonance analysis. In this paper we present an analysis of this phase- sensitive spatial light modulator (SLM)by using both the approximate phasor and the exact Fabry-Perot resonance approaches. We show that there is a trade-off between overall reflectivity and the amount of required absorption change. We find that, by using numbers typical of semiconductor multiple quantum wells (MQW's), we can achieve the modula- tor with Act= 9000 cm-' and an overall reflectivity of 1%. Higher reflectivity can be achieved if larger Aix is available (provided that the values of the absorp- tion coefficients are within the possible range of the semiconductor material utilized). In a real QW, there is usually an index change An whenever there is an absorption change A.7 This index change can be taken into account by the proper cavity design. It is not the purpose of this paper to present a complete design for such a MQW-SLM,but only to investigate the feasibility of our initial concept: can absorption changes alone be sufficient to achieve 2856 APPLIED OPTICS / Vol. 33, No. 14 / 10 May 1994
Transcript
Loss-induced phase-sensitive spatiallight modulator with a wide field of view
Man-Fang Huang and Elsa Garmire
We investigate the feasibility of achieving practical phase modulation by changing only the absorption ofthe device. We present a spatial light modulator that consists of a three-mirror cavity with lossy slabsinserted between the mirrors. When only the absorption of these slabs is changed, this device can have agradual change in phase, but no intensity modulation. Theoretical analysis by the use of numbersappropriate to multiple quantum wells shows that the total phase-tuning range can be as large as 27r withan absorption coefficient change of less than 9000 cm-'; the length of each multiple-quantum-well slab is-1.5 pum. This gives an expected 16° field of view with the nanosecond speed typical of semiconductors.
1. Introduction
Optical phase modulators are of interest for applica-tions such as in either single-element or few-elementgeometries for fiber-optic communications or in ahighly parallel multielement geometry for image pro-cessing.1-3 Conventionally, the phase of an opticalbeam is tuned by changing the optical path length.As the electro-optical effects are usually weak, suchphase modulators use waveguide geometries or ratherlong lengths. However, for some applications suchas image processing and optical computing, phasemodulators operate on light normal to the plane ofthe device rather than in a waveguide configuration.4Phase modulation usually requires a long path length.Recently, a binary-phase modulator in a verticalgeometry that used a Fabry-Perot cavity was pre-sented.5 However, this device is unsuitable for appli-cations in which a gradual change in phase is re-quired. 6
A phase modulator that can give a continuouschange in phase consists of a three-mirror cavity withlossy slabs inserted between mirrors. The configura-tion is shown in Fig. 1; the lengths of the lossy slabsare Li and L2; r, r2, and r3 are the electric-fieldreflection coefficients of mirrors 1, 2, and 3, respec-tively. The idea is to control the phase of thereflected light from all three mirrors by varying only
The authors are with the Center for Laser Studies, University ofSouthern California, University Park, DRB 17, Los Angeles,California 90089-1112.
Received 30 July 1993; revised manuscript received 12 October1993.
0003-6935/94/142856-05$06.00/0.c 1994 Optical Society of America.
the absorption of the slabs, holding the reflectioncoefficients of the mirrors fixed. As we show below,the phase of the reflected light can be tuned bychanging only the absorption coefficients within theslabs; the intensity of the reflected light can bemaintained constant if the absorption of the two slabsis varied in a prescribed way, determined by numeri-cal calculation, as we discuss below.
The optimization of such a phase modulator de-pends on several parameters. To find where in theparameter space the modulator will operate mostefficiently, we can use an approximate phasor analy-sis to estimate values of the relevant parameters.These approximate solutions are then used as initialvalues for the exact Fabry-Perot resonance analysis.In this paper we present an analysis of this phase-sensitive spatial light modulator (SLM) by using boththe approximate phasor and the exact Fabry-Perotresonance approaches. We show that there is atrade-off between overall reflectivity and the amountof required absorption change. We find that, byusing numbers typical of semiconductor multiplequantum wells (MQW's), we can achieve the modula-tor with Act = 9000 cm-' and an overall reflectivity of1%. Higher reflectivity can be achieved if larger Aixis available (provided that the values of the absorp-tion coefficients are within the possible range of thesemiconductor material utilized).
In a real QW, there is usually an index change Anwhenever there is an absorption change A.7 Thisindex change can be taken into account by the propercavity design. It is not the purpose of this paper topresent a complete design for such a MQW-SLM, butonly to investigate the feasibility of our initial concept:can absorption changes alone be sufficient to achieve
practical phase modulation? We find, finally, thatthe fabrication tolerances for this device are verystringent on Fabry-Perot length.
2. Approximate Theoretical Analysis
We first demonstrate how to choose relevant param-eters to achieve a full 2 phase-sensitive SLM byusing an approximate phasor analysis. In this ap-proach, we consider only first-order reflected light El,E2, and E3 , which are the fields reflected from mirrors1, 2, and 3, respectively. The electric fields of thereflected waves can be treated as three phasors andexpressed as follows:
where 4), and 4)2 are the round-trip phases, and Al andA2 are the round-trip absorptions of the first and thesecond slabs, respectively, and
Fig. 2. Phase diagram that explains the concept of the three-mirror-cavity phase modulator.
where Er and 4) are the amplitude and the phase of thereflected light, respectively. Solving Eq. (5) gives
Er = ([Pl + P2 cOs(yl) + P3 COS(Y2)]2
+ [P2 sin(yl) + P3 sin(y2)]11/2, (6)
Er cOs c) = P, + P2 cosy, + P3 cosy 2 ,
Er sin 4) = P2 sin y, + P3 sin Y2-
(7)
(8)
To ensure no intensity modulation, let E be aconstant, and then solve Eq. (6) for P3 as a function ofP2. However, parameters Er, Pl, yl, and Y2 must beproperly chosen so that a 2 phase shift can beobtained.
Equations (7) and (8) indicate that only wheny, andY2 are located in quadrants I and III, II and III, or IIand IV, is it possible that cos 4) and sin 4) can be eithersmaller or larger than zero. Only then is it possible
Ai = atiLiri, i = 1, 2, (4)
where oti is the intensity absorption coefficient of thecorresponding lossy medium, and r is the fillingfactor of the lossy medium within the slab (thefraction of the slab that contains, for example, a QW).These equations are valid in the limit that multiplereflections can be ignored.
The phase diagram in Fig. 2 shows that, by fixinglength P, and varying lengths P2 and P3 of the twovectors describing reflection from r2 and r3 , we canobtain a total phase-tuning range of 2 r withoutchanging the amplitude of the resultant final vector.It indicates that we can have a pure phase modulatorby varying only absorptions Al and A2 (i.e., lengths ofvectors) in a prescribed way. The direction of eachvector is fixed as only absorption modulation isconsidered here.
The total reflected field for this model is given by
Er exp(i4)) = El + E2 + E3, (5)
o MQWl
A MQW2
1
0.5
a.
._ 0
-0.5
-10 0.2 0.4 0.6 0.8
Round-Trip Absorption1 1.2
Fig. 3. Total phase as a function of the round-trip absorption ineach slab region when a phasor analysis is used. The parameterschosen are r = 0.3, r2 = 0.46, r3 = 1, j = Y = 0.7;r, 2 = Y2 - Y =
Fig. 4. (a) Relation between round-trip absorptions Al and A2when the amplitude of the reflected light is maintained constant.(b) Total phase 4) as a function of round-trip absorption in each slabregion when Fabry-Perot resonances are included. The param-eters used are r = 0.3, r2 = 0.49, r3 = 1, A1 = 0.7 171r, 4)2 = 0.7177r,Er/Ej = 0.1.
that 4 can be located in any quadrant, i.e., an entirephase tuning range of 2 r is possible. Figure 2 showsan example when Yi is equal to 0.71r, which is inquadrant II, and Y2 is equal to 1.32rr, which is inquadrant III. Further analysis shows that the ratioEr/Pi must be less than unity in order to have a 27rphase change. This implies that the total reflectivityof the phase modulator must be smaller than thereflectivity of the first mirror.
The values of P2 and P3, which are calculated fromthe phasor analysis, can then be applied to thefollowing equations, which are derived from Eqs. (2)and (3), to give the values of absorptions Al and A2:
We intentionally set the reflection coefficient r3
(b)
Fig. 5. (a) Amplitude of the total reflected light as a function of Alwhen the length of each slab is longer than the desired length by 15A, (b) total phase 4 as a function of round-trip absorptions underthe same condition.
equal to unity to give the highest reflectivity possible.As r, is related to P1, r 2 can then be arbitrarily varieduntil practical values of A, and A3 are obtained.
Figure 3 shows that phase 4) can vary from - rr to mrwhen absorption Al is changed from 0.8 down to 0.1and back to 0.95 while A2 is changed from 0.6 to 1.1back down to 0.1 and back to 0.6. The parameterschosen were r1 = 0.3, r 2 = 0.46, r3 = 1, 4), = y, = 0.7w,4)2 = Y2 - Y1 = 0.62r, and Er/Ei = 0.1. Note thatwhen Al is changed, A2 is varied accordingly so thatthe intensity of the reflected light can be maintainedconstant.
3. Exact Analysis
Using the above first-order approximation, we havedemonstrated that phase modulation can be con-trolled while maintaining constant amplitude. How-ever, the phasor analysis is approximate because itignores multiple reflections. We now show that wecan still have a pure phase modulation by doing the
exact Fabry-Perot analysis including multiple reflec-tions.
For the three-mirror Fabry-Perot cavity shown inFig. 1, the overall reflection coefficient is
the shape of Fig. 4 will change. Also, l and 2 foroptimized performance will be somewhat different.This can be understood by imagining that phasors 2and 3 in Fig. 2 become somewhat tilted with field.
The procedure to solve this problem is similar tothe one that we discussed above. We use the resultsof Section 2 as initial guesses for the proper param-eter values. As the contribution coming from higher-order reflections is small compared with the first-order reflection, we can expect that the parameters tobe used in the exact result calculated from Eq. (11)will be close to those determined from the approxi-mate phasor analysis. To ensure no intensity modu-lation, let Er/Ei be a constant and then numericallysolve A2 as a function of Al by using Eq. (11). Theparameters that we came up with in the first-orderapproximation can be used as the initial values.Calculation shows that a total tuning range of 2rr canbe easily obtained by gradually changing 42 with 1fixed at its initial value. We can then vary r2 toderive operating conditions that use absorptions Aland A2, which are practical for fabrication. Figure4(a) shows the relation between Al and A2 when theintensity of the total reflected light is maintainedconstant. As shown in Fig. 4(b), a total phase-tuningrange of 27r is possible. In this particular design, Alvaries with a minimum of 0.17 to a maximum of 1.06,and A2 varies from 0.21 to 1.12 with parametersEr/E = 0.1, ri = 0.3, r 2 = 0.49, r3 = 1, 4k = 0.717Tr,and 42 = 0.717ir. To this point, the calculation hasbeen general, with parameters chosen to give reason-able maximum and minimum values of absorption.
Suppose that this device is operated with a MQW,as described by Lengyel et al. 8 To have practicalabsorption coefficients, we calculate the requiredlength of each MQW slab, assuming that it is com-posed of a 96-period 105-A GaAs well and a 96-period50-A Al0 32Ga0 68As barrier, and find a result ofL = 1.5pim for each slab. The required ranges of the absorp-tion coefficients are from 1705 to 10,360 cm-' forMQW1 and from 2076 to 11,000 cm-' for MQW2.These calculations assume a wavelength set at 853nm (- 0.012 eV to the long wavelength side of thepeak of the zero-field excitonic absorption) as indi-cated in Fig. 5 of Ref. 8. A suitable mirror betweentwo MQW slabs consists of five pairs ofA10.32Gao.6 8As/A]As (583 A/666 A) quarter-wave lay-ers, providing sufficient reflection.
4. DiscussionIn this design we have ignored the refractive-indexchange that occurs in the MQW when there is anabsorption change (i.e., electroreflection). If we con-sider this refractive-index change in the MQW, clearly
We could easily adjust our analysis to include thischange, but it would be device specific.
Rather than design a specific MQW structure asexactly as possible, we choose to investigate here howsensitive the design is to fabrication tolerances. Asan example, assume that the length of each MQWslab is longer than the designed length by 15 A. Thisfabrication error corresponds to an increase of 0.05,rfor both 4l and 42 If the relation between A andA2calculated from the original parameters is still fol-lowed, then the amplitude of the total reflected light,
1.4
1.2
1
A 2 0.8
0.6
0.4
0.2
1
0.5
In
a)0)cn
; -0.5
-1
... ............ . ....... l....l...
0 02 04 06 08
Al
(a)
o MQW1
AMQW2
F ~~~~~~~~~~~~~~~~~~~~._. ...... _-_.,: ... ... _,..,.. ,,
Fig. 6. (a) Relation between Al and A2 when round-trip phases $land 432 are different from that in Fig. 4 by 0.05r, (b) total phase 4 asa function of round-trip absorption under the same condition.
will no longer be constant. Figure 5 shows theamplitude of the total reflected light as a function ofAl and the total phase + as a function of round-tripabsorptions Al and A2 for this case. If the thicknessof each MQW slab can be measured, we can easily findthe new relation between Al and A2 to keep the totalintensity of the reflected light constant. Alterna-tively, the relation between Al and A2 to keep the totalintensity constant can be learned through a feedbackteaching circuit after the device is built. As a demon-stration of this adjustment, Fig. 6(a) shows how Alvaries with A2 when N1 and 4t2 change to 0.76rr, andFig. 6(b) shows total phase as a function of Al andA2. This is the limiting case because it requiresA 1 > .
Finally, the field of view of a phase modulatoroperating in reflection is given by the requirementthat
nkL(sec 0 - 1) << r. (12)
When the maximum index change is n, the tradi-tional phase modulator requires that L be equal toX/28n. So after a small angle simplification is made,inequality (12) becomes
/28n 1/20 << (3 / (13)
Often fast nonlinearities, such as the electro-opticeffect, have 8n of the order of 10-4, S 0 << 0.01 rad.The phase modulator proposed here has a total lengthof 3 gim, n 3.5, X 0.85 gim, so that the field ofview is given by 0.28 rad (160), or almost a 30-timesincrease in the field of view. The MQW nonlineari-ties can have nanosecond speeds, providing, for thefirst time, the possibility of fast, wide-field-of-viewphase modulators.
5. Conclusion
A wide-field-of-view phase modulator has been sug-gested by the use of the principle that large absorp-tion changes can be used in a three-mirror cavity to
achieve large phase changes. The proposed devicecan act purely as a phase modulator without intensitymodulation. We have shown that the total phasecan be tuned from --r to r gradually by changing theabsorption coefficients of two slabs within a three-mirror cavity. The proposed configuration could bea practical design as GaAs/AlGaAs MQW's requiredfor achieving the desired performance can be fabri-cated. However, it appears to be necessary to con-trol the tolerances closely.
This work was funded in part by the AdvancedResearch Projects Agency through the National Cen-ter for Integrated Photonic Technology and in part bythe National Science Foundation through its Light-wave Technology Program.
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