Robustness of spatial-coherence multiplexing under receiver misalignment

Transcription

1 Robustess of spatial-coherece multiplexig uder receiver misaligmet Lawrece J. Pelz ad Betty Lise Aderso It has bee show previously that the spatial coherece of a source ca be modulated ad demodulated; hece it ca be used as the basis for a ew dimesio of multiplexig i high-speed optical commuicatio liks. We address the sesitivity of such a system to misaligmets of the receiver with respect to the beam ad examie how chagig trasverse modes affect the spatial coherece i the lateral directio. Specifically, we show that such a system is surprisigly robust for both lateral offsets, i which the receiver is ot properly aliged o the beam ceter, ad rotatioal offsets, i which the receiver is tilted with respect to the plae of the spatial coherece modulatio. The presece of higher-order trasverse modes or chages i the trasverse-mode structure are also show to have little effect o the system operatio Optical Society of America OCIS codes: , , Itroductio There is always a eed for icreased speed ad capacity for optical commuicatio liks. Oe approach to icreasig lik capacity is to superimpose various orthogoal types of multiplexig, for example, combiig itesity modulatio with wavelegth-divisio multiplexig to use the full capacity of a chael. Photoic systems ca also use polarizatio ad coherece multiplexig. Coherece multiplexig ca use temporal coherece, 1 4 ad recetly a scheme for exploitig the spatial coherece was proposed. 5 The spatial coherece is modulated by varyig of the spatial modes of a light source. I this paper we examie the effects of receiver misaligmet o the reliability of a spatial-coherece multiplexed lik, as well as the effect of ay uwated or chagig modes i the directio perpedicular to that of the modulatio. For misaligmet the effect of lateral offset of the receiver with respect to the beam ceter is examied, as well as the impact of rotatio of the receiver iput plae with respect to the trasmitter s modulatio plae. The we study the effects of the existece of higher-order modes i a The authors are with the Departmet of Electrical Egieerig, Ohio State Uiversity, 205 Dreese Laboratory, 2015 Neil Aveue, Columbus, Ohio Received 2 Jauary 1997; revised mauscript received 13 Jue $ Optical Society of America directio perpedicular to the directio of iterest: For example, if the x-depedet lateral modes are the oes that are modulated, what is the impact if the y-depedet trasverse modes are also chaged? We begi with a brief summary of the spatialcoherece modulatio techique i Sectio 2. The attedig theory is reviewed i Sectio 3. I Sectios 4 ad 5 we address the effects of lateral ad rotatioal offsets. I Sectio 6 we discuss the effects of the higher-order trasverse modes. We summarize our coclusios i Sectio Spatial-Coherece Modulatio The spatial-coherece properties of a laser beam are determied directly by the trasverse-mode structure composig that beam. To see the effect of the spatial-mode structure o the spatial coherece of a light beam, cosider a laser or waveguide carryig moochromatic light emittig i a sigle spatial mode. There is uity perfect spatial coherece everywhere across the beam. Whe oe or more higher-order spatial modes are also preset, the spatial-coherece fuctio SCF exhibits maxima ad miima at predictable locatios. 6,7 The effect of the spatial modes o the spatial coherece ca be exploited. It follows that, by itetioally modulatig the spatial-mode structure of a source, oe ca also modulate the SCF. A spatial-mode modulator was proposed to perform this fuctio. Iformatio ecoded oto the spatial-coherece state of the beam is demodulated with a simple iterferometer. 5 The trasmitter or modulator was take i Ref February 1998 Vol. 37, No. 5 APPLIED OPTICS 815

2 Fig. 1. Electro-optic modulator used to chage the umber of spatial modes of a beam. to be a electro-optic waveguidig device. I oe cofiguratio a sigle passive waveguide is flaked o either side by adjacet electro-optically cotrolled waveguides, as show i Fig. 1. Whe o voltage is applied, the light effectively reaches oly the ceter passive waveguide, which supports a sigle spatial mode. Whe the outrigger waveguides are activated, their refractive idices chage to match that of the passive cetral waveguide, makig the effective waveguide wider ad able to support the ext higherorder mode. Oe measures the spatial coherece by iterferig the electric fields at two poits o the beam x, x chose symmetrically about the beam ceter. The visibility of the iterferece friges is a direct measure of the spatial coherece if the beam is symmetric. 8 Figure 2 shows the calculated effect of the chagig spatial modes o the SCF. 6 Whe a sigle spatial mode is preset, the spatial coherece is uity everywhere, i.e., the iterferometer would produce high-visibility friges. Whe the ext-higher-order mode is preset as well, the SCF goes to zero at some particular value of x. The plot shows this poit as a fuctio of the spot size, which is the 1 e 2 itesity poit of the fudametal mode. The locatio of that zero is determied uiquely by the proportio of the total eergy carried i each of the spatial modes. The receiver therefore is a iterferometer that is desiged to combie the fields from the particular x Fig. 3. Electro-optic demodulator device as a iterferometer that measures the visibility correspodig to the SCF x, x. at which the SCF will go to zero. As metioed above, that locatio depeds o the modal weights. It has bee show that a 50:50 ratio of the fudametal to the first higher-order lateral mode is optimum for commuicatio purposes because, at the locatios of zeros of the SCF for this particular modal distributio, the itesity of the beam does ot chage whether there is oe mode or two preset. 5 A electro-optic SCF demodulator or receiver is show i Fig. 3. The receiver is a waveguide versio of Youg s classical two-pihole iterferometer. 9 The two arms sample the beam at x. For measurig the spatial coherece some optical path differece must be itroduced to produce costructive ad destructive iterferece. Oe ca accomplish this by electro-optically varyig the refractive idex of oe arm. As oe beam is delayed with respect to the other, the iterferometer output itesity goes through miima ad maxima. We refer to these peaks ad valleys as friges, although they are ot maifested i space but rather i time, as a fuctio of the applied voltage. The amplitude of the friges directly measures the state of coherece, whereas the itesity of the beam remais uchaged. Operatio of the lik is as follows: Whe the modulator is OFF, that is, o voltage is applied, the waveguide supports a sigle spatial let us say, lat- Fig. 2. SCF x, x as a fuctio of x, the distace from the beam ceter. Solid lie: Oly oe spatial mode is preset. Dashed curve: Two modes are preset i equal weights. The spot size is measured at the 1 e 2 itesity poit of the fudametal mode. Fig. 4. Sigals of a spatial-coherece-multiplexed lik. The voltage applied to the receiver icreases liearly, so that friges top sigal appear at the receiver output whe there is coherece first half of SCF sigal. The itesity varies from high to low ot zero, as show i the secod lie of the figure. The resultig detected sigal shows bright ad dim friges whe the SCF equals 1, ad it shows bright ad dim light with o friges whe the SCF equals zero. 816 APPLIED OPTICS Vol. 37, No February 1998

3 eral mode. The beam is icidet o the receiver, which samples it at x. The fields at x propagate through the iterferometer ad are combied at the detector. A ramp voltage is applied to oe arm, however, so the refractive idex of the material i that arm chages liearly, thus itroducig a phase delay for that part of the field. The frequecy of the friges is determied by the slope of the ramp ad the electro-optic coefficiet, ad it ca thus be extremely fast. Whe voltage is applied to the modulator, two lateral modes are supported, the SCF goes to zero at x, x, ad o friges are observed at the detector sice there is o loger ay coherece betwee the fields at these two poits. Examples of the sigals of a spatial-coherece multiplexed lik are show i poit x, y, which is the sum of the fields of each of the modes, i the form E x, y, t a a m f x f m y exp i t t m t, (1) where a,m is the amplitude of the th x-depedet ad mth y-depedet mode, f x is the fuctioal form of the th lateral mode, f m y is the fuctioal form of the mth trasverse mode, is the cetral agular frequecy, ad,m t are the phase depedecies. I geeral, there will be a time delay betwee the fields arrivig from the two arms of the iterferometer. The total itesity measured at the detector is I r 1, r 2 E x 1, y 1, t E x 2, y 2, t E x 1, y 1, t E x 2, y 2, t ( a a m f x 1 f m y 1 exp i t t m t a a m f x 2 f m y 2 exp i t t m t )c.c. a 2 a m 2 f 2 x 1 f m 2 y 1 a 2 a m 2 f 2 x 2 f m 2 y 2 2Re a 2 a m 2 f x 1 f x 2 f m y 1 f m y 2 exp i, (2) Fig. 4. We assume that the modal weights have bee chose properly such that the itesities at the poits x do ot chage. Therefore the itesity ca additioally be modulated idepedetly of the spatial coherece. I this way a extra degree of multiplexig is achieved. For example, as show i Fig. 4, data could be modulated oto the itesity ad address iformatio could be ecoded oto the spatial-coherece state. For iformatio carried o the beam itesity to be distiguished completely from iformatio carried o the spatial coherece, the two arms of the receiver iterferometer must be at exactly the correct locatios x, x. If the receiver is misaliged, cross talk will occur. The sesitivity of such a system to various types of misaligmet is the subject of this paper. We begi i Sectio 3 by reviewig briefly the equatios goverig this system; the i the subsequet sectios we perturb those equatios to allow for misaligmet. 3. Brief Review of the Theory The relatio betwee spatial coherece ad the spatial modes has bee developed i Ref. 6, ad expressios for predictig the locatios of the zeros are give i Ref. 7. Usig the coordiate system show i Fig. 1, let us write the total electric field at some where r 1 ad r 2 represet poits x 1, y 1 ad x 2, y 2, respectively. I the last step we assumed the time averagig to be over a sufficietly log time so that terms of the form t t average to zero. The first two terms i the last lie of Eq. 2 are the itesities i arms 1 ad 2, respectively, of the iterferometer. The last term ca be writte i the form 8 2 I 1 I 2 r 1, r 2 cos, (3) where I 1 ad I 2 are the itesities at r 1 ad r 2 ad is the magitude of the complex degree of spatial coherece. The cosie fuctio is the origi of the friges that appear as the time delay is varied, while the amplitude of these friges depeds o. Assumig the fields are symmetric about the beam ceter, we ca the defie the magitude of the degree of spatial coherece as r 1, r 2 a 2 a 2 m f x 1 f x 2 f m y 1 f m y a 2 a 2 m f 2 x 1 f 2 m y 1 a 2 a 2 m f 2 x 2 f 2 m y 2 (4) The detectio circuitry will ot measure the state of spatial coherece directly but rather will measure 10 February 1998 Vol. 37, No. 5 APPLIED OPTICS 817

4 Fig. 5. the stregth of the sigal at the frige frequecy. Whe the coherece is high the amplitude of the friges is high, ad whe the coherece is zero the eergy at the frige frequecy is zero, eve though there is light preset. By measurig the frige amplitudes relative to the total power, the receiver i effect measures the frige visibility V, give by V I max I mi 2 I 1 r 1 I 2 r 2 r 1, r 2. (5) I max I mi I 1 r 1 I 2 r 2 4. Effects of Traslatioal Offset The receiver ca be offset from the beam ceter owig to a traslatioal misaligmet. We assume for this discussio that the modal structure uder modulatio is the lateral directio; the trasverse modes, whatever they are, do ot chage. The receiver iterferometer arms are i the x z plae. First, cosider a traslatioal offset of the receiver i the y directio oly, such that the samplig poits are x 1, y 1 ad x 1, y 1 istead of x 1,0 ad x 1,0. For these samplig poits the spatial coherece give i Eq. 4 becomes x 1, y 1 ; x 1, y 1 Geometry for lateral misaligmet of the receiver. a 2 f x 1 f x 1 m a 2 f 2 x a 2 f 2 x a 2 f x 1 f x 1 a 2 f 2 x a m 2 f m 2 y 1 a m 2 f m 2 y (6) a 2 f 2 x 1 I other words, the spatial coherece i the x directio is idepedet of the choice of y. This result follows directly from the assumptio that the x ad y depedecies are separable, as is the case, for example, with Hermite Gaussia modes. This result cofers the advatage that several iterferometers of differet arm spacigs ca be stacked i the y directio, ad each looks for zeros i the x directio at differet places o the same beam. I other words, they look for differet modal weights for the codig of complex addresses. 5 For offsets i the x directio, as show i Fig. 5, we evaluate Eq. 4 at the poits x 0 x, y 0, x x 0, y 0 for the two cases of iterest: a sigle lateral mode ad two lateral modes of equal modal weight. Whe there is a sigle lateral mode Eq. 4 tells us there is o differece i the spatial coherece; it is uity for ay two samplig poits x 1, x 2. However, the visibility of the friges does chage, accordig to Eq. 5, because the itesities at the two iterferometers arms are ot equal whe they are offset laterally from the beam ceter. Figure 6 a shows the coherece ad the visibility for the sigle-mode case call it a logic 1 as a fuctio of lateral misaligmet i uits of, the itesity s 1 e 2 beam radius, ad the assumptio of Hermite Gaussia modes. For the twomode case the spatial coherece ad the visibility are show i Fig. 6 b. The visibility reaches a maximum ad the decreases agai with icreasig misaligmet, eve though the coherece costatly icreases. This is due to the fact that the itesities at the two iterferometer arms differ icreasigly from each other as the receiver is icreasigly misaliged laterally. The extictio ratio ca be defied as the sigal stregth of a logic 0 divided by the sigal stregth of a logic 1. Here the sigal is the visibility, ad the extictio ratio is V 0, (7) V 1 where V 0 ad V 1 are the visibilities of the friges for the logic 0 ad logic 1 cases, respectively. For a ideally aliged system the extictio ratio is 0. Figure 6 c shows the chage i extictio ratio as a fuctio of lateral offset, x. The extictio ratio usually varies betwee 0 ad 1, but here it is possible for the sigal stregth of a logic 0 to exceed the sigal stregth of a logic 1. That is, for some lateral offsets the visibility of the friges for the two-mode case actually exceeds the visibility for the oe-mode case, a clearly udesirable result. If we set a extictio ratio of 0.1 as a arbitrary tolerable limit, the lateral offset ca be up to 0.21, or 21% of the beam radius. 5. Effects of Rotatioal Misaligmet The geometry of the rotatioal misaligmet is show i Fig. 7. Here we address the questio of what happes whe the iterferometer arms do ot lie precisely i the x z plae. If the rotatio agle is, the iteded measuremet poits x 1,0 become x 2, y 2 ad x 2, y 2, where x 2 x 1 cos, (8) y 2 x 1 si. (9) I this case the problem becomes more complicated, sice we must take ito accout the y variatio i the field as well as the x. 818 APPLIED OPTICS Vol. 37, No February 1998

5 Fig. 7. Rotatioal misaligmet geometry. Ope circles idicate the locatios of iteded measuremets; filled circles idicate the actual measuremet locatios. Fig. 6. Effect of lateral offsets o the spatial coherece ad visibility a whe oe lateral mode is preset ad b whe two lateral modes are preset i equal weights. c The resultig extictio ratio. The extictio ratio exceeds 1 for some misaligmets because the visibility for logic 0 ca exceed the visibility for logic 1 for some misaligmets. Ulike the case of traslatioal offset, rotatio has o effect o the measured visibility, sice uder rotatio the samplig poits are still symmetric about the beam ceter. For the sigle lateral-mode case, the visibility remais uity at all agles of misaligmet. For the case whe two lateral modes are preset, as usual i equal weights, the coherece ad visibility do chage. Figure 8 shows the resultig chage i extictio ratio as a fuctio of rotatioal misaligmet, agai assumig Hermite Gaussia modes. We assume there is always a sigle trasverse mode. At 10 of tilt the extictio ratio is approximately 0.01, ad it does ot reach 0.1, our arbitrary limit of acceptability, util the tilt agle is 27, a very serious misaligmet ideed. 6. Effect of Higher-Order Trasverse Modes Because the modulator device is likely to be less tha ideal, we should cosider the possibility that, whe the voltage is applied to permit higher-order lateral modes, the waveguide may also support some small amout of eergy i a higher-order trasverse y mode. We assume, as usual, Hermite Gaussia modes. We are expectig to modulate the source beam betwee the two states: 1 all eergy i TEM 00 ad 2 eergy equally divided betwee TEM 00 ad TEM 10. If we assume that the eergy is equal i the two lateral modes for the secod state but that some eergy is also ow preset i TEM 01, what is the effect o the commuicatio lik? We have already show i Sectio 4 that the spatial coherece, whe measured i the x directio, is idepedet of the modal structure i the y directio. This is true whether or ot there is a lateral offset. Uder rotatio misaligmet, however, the trasverse modes start to play a role. Figure 9 shows the chagig SCF i the eighborhood of oe of the zeros as the stregth of the TEM 01 mode is icreased, keepig TEM 00 ad TEM 10 i equal stregths. The curves are plotted for a agular misaligmet of 10. The left-most curve is for the case i which there is o eergy i the higher-order trasverse mode, ad the subsequet curves to the right are for 10%, 20%, 30%, Fig. 8. Effect of tilt misaligmet o the extictio ratio. 10 February 1998 Vol. 37, No. 5 APPLIED OPTICS 819

6 Fig. 9. Close-up of the effect of higher-order trasverse ydepedet modes o oe of the SCF zeros. The tilt agle is 10 ; the stregths of the TEM 00 ad TEM 10 modes are equal i all cases. The left-most curve is for the case i which oe of the eergy is i the TEM 01 mode. 40%, ad 50%, respectively, of the total beam eergy that is preset i TEM 01. If the iterferometer arms are set at x 0.5, where the SCF zero is expected for the ideal case, the the measured frige visibility will chage from the expected value of V 0 to oly V 0.02, eve whe as much as 50% of the total beam eergy is i a higher-order trasverse istead of the lateral mode. It is also iterestig to ote that, if the weights i TEM 00, TEM 10, ad TEM 01 are all equal 33% of the eergy i each, the the spatial coherece is ivariat with the rotatio agle. 7. Summary ad Coclusios We have examied the robustess of a proposed commuicatio lik based o modulatig the spatial-mode structure ad therefore the spatial coherece of a laser beam. The modulator is a waveguide that is electrooptically cotrolled to support either oe lateral mode spatial coherece is uity or two lateral modes spatial coherece is zero at certai positios o the beam. The state of coherece is detected by a electro-optic iterferometer that produces extremely high-speed friges temporal friges, ot spatial oes whe the modulator is i oe state ad o friges i the other state. The coherece is idepedet of the light itesity for properly chose operatig coditios, so the spatial coherece ca be modulated idepedetly of the itesity ad thus a sigle beam ca be made to carry two sigals istead of oe. We have show the lik to be completely isesitive to trasverse misaligmets whe the modes that are modulated are i the lateral directio. For lateral offsets, that is, i the same plae as the modulatio, there is a degradatio of the extictio ratio. If we set 0.10 as a upper limit of acceptability, the system ca tolerate a lateral displacemet of the iterferometer arms of up to 21% of the 1 e 2 beamspot size. If the iterferometer was rotated rather tha traslated, we foud that, with the same upper limit of 0.1 for the extictio ratio, the rotatio ca be as much as 27 from the correct plae. Fially, if higher-order trasverse modes are iadvertetly permitted i additio to the iteded chages i lateral-mode structure, we have show that there is o effect o the proper detectio of the coherece state for either a perfectly aliged receiver or for a traslatioal offset i either the lateral or the trasverse directio. The presece of higher-order trasverse modes affects the iteded measuremet if there is a rotatioal misaligmet; this effect was foud to be egligible at reasoable tilt agles 10, as log as the weights of the lateral modes are still kept equal. We have examied the effects of misaligmets oly for cases i which the lateral-mode structure is 50% of the eergy i each of the lowest two lateral modes. This particular coditio was examied because it is the optimum place to operate a spatial-coherecemodulated lik, as was demostrated i Ref. 5. The lik is surprisigly robust, particularly uder rotatioal as well as trasverse offsets, where the trasverse directio is defied as the directio perpedicular to the directio i which modes are modulated. The lik is also fairly isesitive to spurious higher-order spatial modes i the plae perpedicular to that uder modulatio. Because of these isesitivities ad the high speeds that are attaiable, spatial-coherece modulatio appears to be a attractive method for icreasig the iformatio-carryig capacity of optical free-space commuicatio systems. This study was supported i part by Natioal Sciece Foudatio grat ECS Refereces 1. J. L. Brooks, R. H. Wetworth, R. C. Yougquist, M. Tur, B. Y. Kim, ad H. J. Shaw, Coherece multiplexig of fiber-optic iterferometric sesors, J. Lightwave Techol. LT-3, K. W. Chu ad F. M. Dickey, Optical coherece multiplexig for iterprocessor commuicatios, Opt. Eg. 30, J.-P. Goedgebuer ad A. Hamel, Coherece multiplexig usig a parallel array of electrooptic modulators ad multimode semicoductor lasers, IEEE J. Quatum Electro. QE-23, J.-P. Goedgebuer, A. Hamel, H. Porte, ad N. Butterli, Aalysis of optical crosstalk i coherece multiplexed systems employig a short coherece laser diode with arbitrary power spectrum, IEEE J. Quatum Electro. 26, B. L. Aderso ad L. J. Pelz, Spatial coherece modulatio for free-space commuicatio, Appl. Opt. 34, P. Spao, Coectio betwee spatial coherece ad modal structure i optical fibers ad semicoductor lasers, Opt. Commu. 33, L. J. Pelz ad B. L. Aderso, Practical use of the spatial coherece fuctio for determiig laser trasverse mode structure, Opt. Eg. 34, M. Bor ad E. Wolf, Priciples of Optics, 6th ed. Pergamo, New York, 1980, Chap. 10, pp B. L. Aderso ad P. L. Fuhr, Twi-fiber iterferometric method for measurig spatial coherece, Opt. Eg. 32, APPLIED OPTICS Vol. 37, No February 1998