Design of Microchannel Free-Space Optical Interconnects Based on Vertical-Cavity Surface-Emitting Laser Arrays

Design of microchannel free-space optical interconnectsbased on vertical-cavity surface-emitting laser arrays

Rong Wang, Aleksandar D. Rakic, and Marian L. Majewski

We investigate the design of free-space optical interconnects �FSOIs� based on arrays of vertical-cavitysurface-emitting lasers �VCSELs�, microlenses, and photodetectors. We explain the effect of the modalstructure of a multimode VCSEL beam on the performance of a FSOI with microchannel architecture. AGaussian-beam diffraction model is used in combination with the experimentally obtained spectrallyresolved VCSEL beam profiles to determine the optical channel crosstalk and the signal-to-noise ratio�SNR� in the system. The dependence of the SNR on the feature parameters of a FSOI is investigated.We found that the presence of higher-order modes reduces the SNR and the maximum feasible inter-connect distance. We also found that the positioning of a VCSEL array relative to the transmittermicrolens has a significant impact on the SNR and the maximum feasible interconnect distance. Ouranalysis shows that the departure from the traditional confocal system yields several advantages in-cluding the extended interconnect distance and�or improved SNR. The results show that FSOIs basedon multimode VCSELs can be efficiently utilized in both chip-level and board-level interconnects.© 2002 Optical Society of America

OCIS codes: 200.2610, 200.4650.

1. Introduction

With the increase of speed of computers, networks,and switching systems, the requirement for high-speed and high-density data links between chips,multimodules, and boards, is steadily increasing.However, owing to the inherent inductance and ca-pacitance of copper wires, electrical interconnectshave encountered bottlenecks, such as pin conges-tion, clock skew, and bandwidth limitation. At thesame time, optics is becoming a mature technologyfor interconnects, for example, arrays of vertical-cavity surface-emitting lasers �VCSELs�, micro-lenses, and photodetectors are widely available now,and they have been applied to the design of VLSI,switching systems, and parallel optoelectronic pro-cessing systems.1–3 The integration of VCSELs andphotodetectors with electronic circuitry will allowhigh performance optical interconnects to be devel-oped that feature low energy consumption, widebandwidth, high speed, and high density.4,5

The authors are with The University of Queensland, School ofComputer Science & Electrical Engineering, Brisbane Qld 4072,Australia. A. D. Rakic’s e-mail address is [email protected].

Received 1 March 2001; revised manuscript received 2 January2002.

0003-6935�02�173469-10$15.00�0© 2002 Optical Society of America

Free-space optical interconnects �FSOIs� are seen aswell suited for the chip-level and board-level intercon-nects. Several interconnect architectures based onarrays of VCSELs have been recently proposed.6,7 Amajor concern of high-speed and high-density FSOIs isthe optical channel cross talk arising from the laserbeam diffraction and the misalignment betweenVCSELs, microlenses, and photodetectors. The issueof optical cross talk in interconnects has previouslybeen investigated by use of the Gaussian or Fraun-hofer approximation.8–10 In addition, the effect ofmisalignment between VCSELs, microlenses, andphotodetectors on the performance of FSOIs has alsobeen analyzed recently.11 However, the dependenceof an interconnect performance on the feature param-eters, such as VCSEL aperture, channel spacing, fillfactor of microlenses, and interconnect distance, hasnot been extensively explored. In addition, practicalVCSELs tend to operate in several transverse modessimultaneously.12 The impact of higher-order modeson the performance of interconnects has not, to ourknowledge, been investigated. In this paper, westudy the performance of FSOIs using the Gaussianbeam diffraction model and develop a generic model forthe design of FSOIs.

The integration of microlenses with VCSELs anddetectors has been investigated and a confocal sys-tem is employed in the design of FSOIs.13,14 Inthese designs the maximum interconnect distance

10 June 2002 � Vol. 41, No. 17 � APPLIED OPTICS 3469

is set to the Rayleigh range of a beam between thetransmitter Tx and the receiver Rx lenses. How-ever, the ultimate limiting factor for the maximuminterconnect distance of an interconnect is thesignal-to-noise ratio �SNR�. In addition, the rela-tionship between focal length �or the radius of cur-vature� of a lens and the location of the VCSEL hasa significant impact on the performance of an FSOI.Both issues, however, have not been investigated indetail thus far to our knowledge. In this paper weinvestigate the design of an FSOI’s system usingthe architecture introduced by Strzelecka et al.13,14

Our analysis can also be extended to some otherpreviously developed architectures.7,10 The Gaussiandiffraction model is described in Section 2 and Section3. The simulation results and analysis of SNR aregiven in Section 4. The design of FSOIs and the anal-ysis of maximum interconnect distance are describedin Section 5.

2. Gaussian Diffraction Model

SNR is a significant figure of merit of any optical orelectrical interconnect. In this section we describethe method for determining the SNR for a point-to-point microchannel optical interconnect based on amultimode VCSEL. We also analyze the impact ofthe modal contents of the beam on the performance ofthe FSOI. We assume here that the dominant noisesources in FSOIs are the photodetector–amplifierthermal noise determined by the noise-equivalentpower �NEP� and the optical channel cross talk re-sulting from the beam diffraction and misalignmentbetween the arrays of VCSELs, microlenses, and pho-todetectors. The SNR can be, therefore, defined asfollows15:

SNR � 10 log10

P0 S

P0 N � NEP�BdB, (1)

where P0 is the optical transmitter power per chan-nel, S is the normalized optical power received bythe corresponding photodetector, N is the normal-ized optical channel cross talk, and B is the channelbandwidth.15 A similar definition has been usedby Neifeld and Kostuk.10 They assumed, however,that the optical cross talk noise results from anypower that is not a signal power, or that S � N � 1.Their assumption, therefore, leads to the overesti-mation of the optical channel cross talk even if thereceiver lens array has a fill factor of one, becausethere would always be a fraction of the opticalpower that cannot be detected by the photodetec-tors. Our definition of optical channel cross talkproposed here is therefore better suited for practicalapplications.

First we describe the architecture of the intercon-nect and then outline the method for calculating theSNR in that particular system. Figure 1 shows aschematic of a microchannel FSOI using a lens ar-ray monolithically integrated with the VCSEL ar-ray �for the transmitter� Tx and with thephotodetector array �for the receiver� Rx as de-

scribed in previous publications.13,14 VCSEL ar-ray located at z � 0 is integrated with thetransmitter microlens array �the Tx lens� located atz � d1. The receiver microlens array �the Rx lens�located at z � d1 � d2 � d3 is integrated with thephotodetector array located at z � d1 � d2 � d3 �d4. The waist of the VCSEL beam at the emittingmirror is �0, the channel spacing is �, the diameterof the lens aperture is D, the radius of curvature ofthe lens is R1, and the fill factor of the lens array is�. Therefore, D � ��. �10 and �20 are the corre-sponding intermediate and final beam waists, and�1 and �2 are the beam spot sizes at the transmitterand receiver lenses, respectively.

To analyze the VCSEL beam propagation, theGaussian-beam diffraction model and the ABCD ma-trix have been employed.16,17

The Tx lens can be assumed to be a thick lens withone flat and the other curved interface with thicknessd1 and the radius of curvature R1.17 Therefore theprimary focal length of the Tx lens is given by18

f �n

n � 1R1, (2)

where n is the refractive index of the substrate. Us-ing the ABCD matrix formalism, the spot size at theplane z � d1 is given by

�1 � �0�1 � �d1�ZR�21�2, (3)

where ZR � n�02�� is the Rayleigh range of the

beam. The beam waist �10 at the plane z � d1 � d2,after passing through the Tx lens, will be17

�10 ��0

��1 � d1�f �2 � �ZR�f �21�2 , (4)

Fig. 1. Schematic of �a� free-space optical interconnect and �b�Gaussian diffraction model.

3470 APPLIED OPTICS � Vol. 41, No. 17 � 10 June 2002

where f is the primary focal length of the Tx lens givenby Eq. �2�. The transmission distance d2 is

d2 � �d1�n�1 � d1�f � � ZR

2�f�1 � d1�f �2 � �ZR�f �2 . (5)

The spot size at the plane z � d1 � d2 � d3 is givenby

�2 � �10�1 � �d3�Z R�21�2, (6)

where Z R � �102�� is the Rayleigh range. The

beam waist �20 after passing through the Rx lens isgiven by

�20 ��10

��1 � nd3�f �2 � �nZ R�f �21�2 , (7)

and the transmission distance d4 is

d4 � �n�d3�1 � nd3�f � � nZ R

2�f�1 � nd3�f �2 � �nZ R�f �2 . (8)

Therefore the total interconnect distance is L � d1 �d2 � d3 � d4, where d2 � d3 is its major fraction.

The above Eqs. �3� to �8� are equally applicable forthe fundamental Gaussian mode and the higher-order Gaussian modes because the higher-orderGaussian modes are characterized by the same set ofparameters as the fundamental mode itself, namelyby the beam waist and the radius of curvature.However, one needs to bear in mind that despite thefact that fundamental mode and higher-order trans-verse modes share the same waist �0, the transverseextents of the beams are very different. If the char-acteristic spot radius of the beam is calculated basedon the second moment of the beam intensity, thetransverse extent of the beam will be 1 �0 for thefundamental TEM00 mode, �2�0 for the first-orderLaguerre–Gaussian transverse mode TEM01, and�3�0 for the second-order Laguerre–Gaussian trans-verse mode TEM10, to mention only the first few.Similar results would be obtained with the other def-initions for the characteristic spot radius and thecorresponding M2 factors of the beam.19 As VCSELstypically operate simultaneously in fundamentalmode and higher-order transverse modes, this has to

be taken into account when estimating the diffractioneffects on the transmission of multimode beams inmicrochannel systems.

In general, the optical channel cross talk of an FSOIresults from the laser beam diffraction and misalign-ment. The diffraction cross talk can be associatedwith the Tx and�or Rx lenses. However, as long as thespot size �1 � D��3�2�, which gives a more than99.9% power transmission through the Tx lens,16 thecross talk arising from the diffraction at Tx lenses canbe neglected. In this case, also, the size and positionof the intermediate waist �10 will not be affected by thepassage through the transmitter lens aperture.20 Inpractical applications, this condition can be easily sat-isfied because d1 and R1 are usually small. The pho-todetector is typically placed at the position of thebeam waist �20 that allows for the maximum detectionof laser power. In addition, the beam waist �20 isnormally small compared with the diameter of a pho-todetector. The major fraction of the total intercon-nect distance is d2 � d3, and distances d1 and d3 can beadjusted according to the requirements of system de-sign, and in particular, the SNR. Therefore in thisstudy, we will only consider the optical cross talk at theplane z � d1 � d2 � d3. In accordance with this wedefine the useful optical signal S as a fraction of theoptical power emitted from the VCSEL labeled L0 andtransmitted through the Tx lens to the Rx lens labeled�0. The optical channel cross talk N is the laserpower emitted from the VCSELs labeled L1, L2, . . . , L8and transmitted through corresponding Tx lenses tothe Rx lens labeled �0 �see Fig. 2�. In addition, weassume that all VCSELs, microlenses, and photodetec-tors are identical, respectively.

When analyzing the optical cross talk and SNR forthe multi-transverse mode beam which is being re-layed through the multichannel imaging system de-scribed above, the interconnect designer can choosebetween the following two approaches:

�a� When the exact modal content of the beam isknown the beam can be modelled by a superpositionof several Laguerre–Gaussian modes and the signalS and the noise N at the detector plane can be calcu-lated directly. This approach has been adopted inthis study, and it is shown that for a small diameter

Fig. 2. Structure of �a� VCSEL array and �b� Rx lens array.


VCSEL lasing simultaneously in only few transversemodes, this method is accurate and efficient.

�b� When the VCSEL beam comprises the un-known mixture of modes, or if the number of coexist-ing modes is too large, rendering the approach �a� toocumbersome, the beam can be characterized by theexperimentally determined M2 factor, and the funda-mental mode equations can be used as scaled by theM2. This approach will, however, work fine only forcylindrically symmetrical beams as we know fromother applications of the M2 formalism.21

Based on the Gaussian-beam diffraction model formulti-transverse mode operation, and neglecting thecross talk at the transmitter lens array, the intensitydistribution for a multi-transverse VCSEL beam atthe plane z � d1 � d2 � d3 is given by

I�r, �, d3� � �p,m

CpmIpm�r, �, d3�

� �p,m

Cpm

4p!�2

2� p � m�!�1 � �0m�� 2r

�2�2m

� �Lpm�2r2

�22��2

exp��2r2

�22� cos2�m��,

(9)

where �0m � 1 if m � 0, otherwise �0m � 0. Lpm�2r2��2

2� is a Laguerre polynomial of order �p, m�.Therefore the useful optical signal is given by

S � ��0

I�r, �, d3�rdrd�. (10)

In our model, only the optical channel cross talk thatresults from the eight neighboring VCSELs, L1,L2, . . . , L8, is considered �see Fig. 2�. To determinethe channel cross talk we need to calculate the laserpower falling on the lens aperture �0 from the neigh-boring lasers L1, L2, . . . , L8. However, the samemathematical result can be obtained by calculatingthe laser power falling from laser L0 on lens aper-tures �1, �2, . . . , �8 by use of the variable transfor-mation. Consequently, the optical channel crosstalk can be written as

N � 2 ��1��2��3��4

I�r, �, d3�rdrd�. (11)

The above integration is performed only over the lensapertures labeled �1, �2, �3, and �4 because of thesymmetry of interconnect architecture and the laserintensity distribution. The numerical solution toEq. �11� can be obtained by applying variable trans-formation.15 As we have mentioned before, this pro-cedure can be used to calculate the signal-to-noiseratio of the FSOI, provided that the modal content ofthe VCSEL beam is known.

3. Modal Content of the Vertical-CavitySurface-Emitting Lasers Used in Free-SpaceOptical Interconnect

To explain the effect of the VCSELs modal contenton the interconnect performance and to investigatethe interplay between several feature parameters ofthe microchannel architecture, we will analyze theinterconnect based on the commercial 2 � 8 VCSELarray �Lase-Array22�. The Lase-Array VCSELsare AlGaAs planar, top-surface emitting devicesfabricated with the ion implantation isolation.Lase-Array VCSELs have 10 �m circular aperturesin the top mirror contact, fundamental mode beamradius �0 � 2.25 �m, and � � 250 �m pitch. Thisparticular VCSEL array was chosen as an exampleof a transmitter that can operate in both singlemode and multimode regime, depending on the la-ser �driving� current.

Figure 3 shows measured lasing spectra for cur-rents between the lasing threshold Ith � 2.6 mA andI � 2.7 � Ith. The spectra were measured with the0.275 m monochromator with a resolution of 0.1 nm.From the spectra one can see that VCSEL startslasing in a single transverse mode �TEM00� and thatwith the increase of laser current, higher-ordermodes start to appear. Figure 3 also shows fournear-field images of the VCSEL beam taken at I �2.7 � Ith. The image obtained by the CCD camera�labeled �a� is a superposition of all three coexistingtransverse mode families present at higher currents.Spectrally resolved modal patterns obtained by thenear-field scanning beam profiler are shown in Fig. 3�labeled �b�, �c�, and �d�.23 From these images onecan see that three families of transverse modes arelasing simultaneously. From spectral measure-ments we calculated the light-current curves for eachtransverse mode, shown in Fig. 4. The fundamentalmode can be seen to saturate its power where the firsttransverse mode reaches the threshold �4.6 mA�.Similarly, the first transverse mode saturates itspower on the threshold of the second transverse mode�6.0 mA�. The modes were found to be separatedspectrally by 0.4 nm, which is consistent with theresonator diameter of �10 �m.24

This set of measurements provide insight into thegeneral behavior of the VCSEL. Information re-garding the groups of transverse modes that are las-ing simultaneously �with modal beam profilesindicating the order of individual modes� is the infor-mation we need to design the optical interconnectusing this VCSEL array.

To examine the performance of the FSOI for beamswith different modal content we used the beams ob-tained at three representative bias currents. ForI1 � 4.6 mA, only the fundamental mode is lasing:This mode corresponds to Laguerre–Gaussian LG0

0

mode. For I2 � 6.6 mA, besides the fundamentalLG0

0 mode, the first transverse mode is also lasing.As we have discussed elsewhere,25 this VCSEL modeis a superposition of up to four modes with slightlydifferent frequencies and polarizations. However,


in terms of the beam-propagation characteristicsthrough the microchannel system, they can be mod-elled as an LG0

�1� mode, a combination of LG0�1 and

LG0�1, usually termed a doughnut mode. For I3 �

7.6 mA, three mode families simultaneously lase:fundamental LG0

0, first-order transverse LG0�1�, and

up to six modes of the second order. As Fig. 3�b�shows, the second-order mode is a four-hole patternusually termed optical leopard. Optical leopard is asuperposition of three modes of the order j � 2p � �l�� 2: LG0

�2, LG0�2, and LG1

0. The first two modesare doughnut shaped, while the third mode is a brightring separated by the dark �nodal� ring from thehigh-intensity center. To avoid the quandaries oforienting the leopard mode with respect to the pho-

todetector array when numerically calculating theSNR, we represent this mode family only with theLG1

0 mode that has the same transverse character-istics as a complete leopard mode.

We use these three representative beams contain-ing, one, two, and three different modes, respec-tively, in order to examine the influence of the beammodal content �or conversely, the VCSEL bias cur-rent� on the interconnect performance. To charac-terize the receiver we assumed NEP � 0.1 nW��Hz, B � 500 MHz. The value of the total opticalpower P0 of a VCSEL depends on the injection cur-rent. The relative contribution of the differentmodes changes with the injection current, I, isshown in Fig. 5.

Fig. 3. Lasing spectra for various injection currents to a VCSEL.

Fig. 4. Optical power for different modes of a multi-transversemode beam versus the injection current.

Fig. 5. Contribution of different modes to the total output powerof the VCSEL.


4. Simulation Results and Analysis of theSignal-to-Noise Ratio

In this section we introduce the simple design meth-odology for the interconnect, assuming that we startthe design from the existing VCSEL array with al-ready defined beam waist �0 and the array pitch �.Second, we examine the performance of the intercon-nects for three different multimode regimes of oper-ation of the VCSEL.

The design space of the FSOIs has seven basickinematic parameters: the array pitch �, lens fillfactor � � D��, the beam-clipping ratio k, VCSEL-to-microlens distance, d1, and microlens focal lengthf �or equivalently, the radius of curvature R�, andfinally the required interconnect distance �or the dis-tance between the transmitter and the receiver lens�the VCSEL beam waist �0. One needs to specify atleast eight additional parameters defining the dy-namics of a VCSEL, photodetector, and an intercon-nect, namely, the required aggregate bandwidth andthe interconnect capacity, the acceptable bit-error-rate that is related to the signal-to-noise ratio SNR,individual VCSEL modulation bandwidth BT and op-tical power P0, detector bandwidth BR, and the noiseequivalent power NEP. Also we show that themodal structure of the VCSEL beam will also signif-icantly affect the performance of the interconnect.

Which of these parameters will be defined as freedesign parameters and which will be set as designrequirements �e.g., interconnect distance, intercon-nect capacity, etc.� will depend on the particular de-sign. In general a multivariable and multiobjectiveoptimisation procedure would be needed to performthe optimal design taking into account constraintsand requirements of the design task.

In the Section 5 we describe a simplified designprocedure that starts from the given parameters ofthe available VCSEL and photodetector arrays ��0, �,B, P0, NEP� and modifies the parameters of the op-tical system to achieve the design goals, such as therequired SNR or interconnect distance L.

As an example, let us start from the already de-scribed VCSEL array with the beam waist �0 � 2.25�m and the pitch � � 250 �m. Based on these waistand pitch values, we adopt the lens fill factor � �D�� 1.0, and the clipping factor of k � Dlens�2�lens � 2.12, where Dlens is the lens diameter and�lens is the beam spot size on the lens. This deter-mines the size of the VCSEL lens, D � 250 �m, thespot size of the beam at the VCSEL lens �1 � 59 �m,and finally the distance from VCSEL to transmitterlens d1 � 1.70 mm. Therefore the only free param-eter that we can still change to optimize the perfor-mance of the interconnect is f, the focal length of thetransmitter lens. Clearly, as we want our intercon-nect distance to be as long as possible, we need tochoose the focal length f such that we obtain thewell-collimated beam that can be focused again onthe receiver side. The geometrical optics would sug-gest the solution f � d1, or in other words, placing theVCSEL in the focal point of the transmitter lens.

According to the Gaussian beam theory the beam isconsidered collimated �a� when the divergence of thebeam is minimum, or �b� when the distance to thenext beam waist is maximum. It can be shown thatthe condition �a� is satisfied when the input beamwaist is at d1 � f, thus producing the intermediatewaist at the distance f from the lens. The condition�b� is satisfied when d1 � f � ZR, and the position ofthe intermediate waist is

d2 � �n � 1� f�2n � f 2�2nZR. (12)

In case of a symmetrical system this approximatelydetermines the maximum interconnect distance L �2d2. As for the VCSEL in our example the Rayleighrange is ZR � 65 �m, the achievable interconnectdistance with such an architecture is L � d1 � d2 �d3 � d4 � 18 mm. We will base the preliminarydesign on the condition �b� and show that there is aposition between these two collimation positions�d1 � f and d1 � f � ZR� for which the interconnectdistance will be maximum �for a given SNR� or, con-versely, position for which SNR is maximum �for agiven interconnect distance�.

Let us first examine the sensitivity of the SNR onthe interconnect distance L, the pitch of the laserarray �, and the lens fill factor � � D��.

Figure 6 shows the dependence of the SNR on theinterconnect distance. It can be seen that for theshort interconnect distances �L � 15 mm� the SNRdoes not depend on L. The explanation is simple.For L � 15 mm the full optical power of the VCSELbeam �regardless of its modal content� falls on thedetector. As the cross talk is negligible, the P0Nterm in Eq. �1� tends toward zero and the SNR isdetermined only as the ratio of the signal power P0Sand the detectors’ NEP �B product. Consequently,for larger injection currents �and higher optical pow-ers� the SNR is higher. Therefore for very shortdistances the modal content of the beam �or quality ofthe beam� is not important: The SNR is determinedsolely by the optical power of the beam. The overlapin the curves for 6.6 and 7.6 mA results from the

Fig. 6. Dependence of SNR on interconnect distance L. �0 �2.25 �m, � � 0.25 mm, D � �, d1 � 1.70 mm.


shape of the VCSELs light-current characteristics.As the 7.6 mA operating point is located after thethermal roll-over point, the complete optical powerfor both pump currents �6.6 and 7.6 mA� is the same,therefore providing the same SNR in the short dis-tance range. In the intermediate range between the15 and 20 mm the SNR for beams with higher-ordermodes steeply decreases, while the SNR for the beamcontaining only the fundamental mode �4.6 mA� re-mains almost unchanged. Owing to the presence ofthe LG1

0 mode at 7.6 mA, the SNR for this currentdrops quickly below the other two curves. Finally,in the long-distance region between 20 and 30 mm,the SNR is predominantly determined by the modalcontent of the beam. The beams containing trans-verse modes show consistently lower SNR than thebeam containing only the fundamental mode in spiteof its lower total power. In conclusion, if the FSOIhas to be used at maximum achievable distances, thespectral quality of the beam is essential. VCSELshould be running at lower-output powers, just beforethe threshold for the higher-order modes.

Figure 7 shows the SNR as a function of the arraypitch �channel spacing�. As the channel spacing in-creases, the SNR is increasing. For the large valuesof � the SNR is determined by the total power of thebeam rather then by the modal contents of the beam.However, the interconnects with the large interchan-nel distances, �, are not of much interest, because theprimary goal of FSOIs is to increase interconnectcapacity, i.e., the channel density. For the channelspacing close to 250 �m �which is the channel spacingof our array� the improvement in the SNR for a singlemode beam in comparison with that of multimodebeams is dramatic. Therefore one can conclude thatif the high density of channels is important, a singlemode beam has to be used. If a somewhat largerchannel spacing of 350 �m is acceptable, then thehigh SNR can be achieved even for the beams con-taining higher-order modes.

Finally, let us examine the effect of the lens fillfactor on SNR. Figure 8 shows that the SNR firstincreases with the lens fill factor � regardless of the

modal content and then decreases for � values above0.8. This is something that could be expected, be-cause the interconnect was initially designed for thefill factor of 1.0, and the clipping factor of 2.12, there-fore implicitly assuming the negligible diffraction onthe microlens aperture by design. With the reduc-tion of the fill factor the diffraction effects on thetransmitter lens aperture are becoming more pro-nounced, but the model does not account for that.The reduction of the SNR with reduced � in thismodel is simply the consequence of the reduced powertransmitted through the receiving aperture. For fillfactors above 0.8, the cross talk at the receiver lensbecomes dominant. The observed better perfor-mance of the single-mode beam over the multimodebeams could also be expected: The basic design wasdone with the assumption of fundamental mode op-eration. The increased transverse extent of thehigher-order modes clearly contributes to the crosstalk on the receiver lens. These effects would beeven more marked provided that the diffraction onthe transmitter lens aperture had been included inthe model.

Now that we have discussed the performance of theFSOI as a function of the interconnect distance, in-terchannel spacing, and the lens fill factor, we canreturn to the problem of the optimal focal length, i.e.,of determining the optimal radius of curvature of thelens for a given distance d1.

Figure 9 shows the dependence of the fundamentalmode spot size �2 on d1�f, the VCSEL-to-Tx-lens dis-tance d1 normalized to focal length f. The smallestspot size is achieved in the proximity of the collima-tion conditions f � d1 and f � d1 � ZR. It is evidentthat for each interconnect distance there is a differentoptimal value of d1�f that gives the minimum spotsize �2.

Figure 10 shows the dependence of the SNR ond1�f. The optimal value for d1�f is close to 1.03 overa wide range of interconnect distances. For exam-ple, for L � 18.5 mm, the optimal focal length f can berelated to d1 as d1 � 1.03f. Clearly, the optimal

Fig. 7. Dependence of SNR on channel spacing. �0 � 2.25 �m,D � �.

Fig. 8. Dependence of SNR on fill factor �. �0 � 2.25 �m, � �0.25 mm, D � ��.


focal length is not f � d1 as most designs in theliterature use, nor f � d1 � ZR, but somewhere inbetween these two. The SNR decreases signifi-cantly owing to the presence of the higher-ordertransverse modes. At this stage of the design oneshould check if the obtained f and d1 values lie withinthe limits set by the available implementation tech-nology.

It is also useful to estimate the maximum intercon-nect distance for the desired SNR and other condi-tions. Figure 11 shows that the ratio d1�f hassignificant influence on the maximum interconnectdistance for a given value of the signal-to-noise ratio,SNR � 10 dB. In addition, it can be seen from Fig.11 that the maximum interconnect distance de-creases in the presence of the higher-order modes.

The fact that the ratio d1�f has a significant impacton the SNR and the maximum interconnect distanceshows that it is critical to choose the location of theVCSEL carefully and to ensure the proper lateral andaxial alignment of the VCSEL with respect to thetransmitter lens.

5. Algorithm for the Design of FSOI’s

The basic requirements for the design of FSOI’s are:�1� the required bandwidth B, �2� the required SNRvalue corresponding to a given bit-error rate �BER�,and �3� the required interconnect distance. Fig. 12shows the algorithm for the design of FSOI’s for thesegoals. The procedure includes the steps �a� to �d� asfollows:

�a� Setting the initial values for R, �, D, k, P0,NEP, and B.

�b� Calculating d1: Distance d1 has to be such asto ensure that the transmitter lens is not overfilled.This can be expressed by the condition �1 � D��3�2�.

Fig. 9. Dependence of beam spot �2 on ratio d1�f. �0 � 2.25 �m,� � 0.25 mm, D � �, d1 � 1.70 mm.

Fig. 10. Dependence of SNR on ratio d1�f. �0 � 2.25 �m, � �0.25 mm, D � �, d1 � 1.70 mm.

Fig. 11. Dependence of maximum interconnect distance on ratiod1�f. �0 � 2.25 �m, � � 0.25 mm, D � �, d1 � 1.70 mm.

Fig. 12. Algorithm for the design of FSOIs. For detailed descrip-tion of the procedure refer to Section 5 steps �a� to �d�.


Under this condition more than 99.9% of the power isbeing transmitted through the Tx lens and the effectof beam clipping resulting from the Tx lens is negli-gible.20 Based on Eq. �3�, to meet the �1 � D��3�2�condition, d1 should satisfy the following relation:

d1 � ZR� D2

18�02 � 1�1�2

. (13)

It can be seen from Eq. �13� that the optimal valueof d1 is determined by the VCSEL beam waist and thediameter of lens aperture.

�c� Calculating the optimal primary focal lengthof the Tx lens for the given interconnect distance:The optical channel cross talk is determined by �2,the spot size of the beam on the receiver lens. �2 inturn depends on the intermediate beam waist �10.Equation �4� shows that �10 is a function of f and d1.Therefore the channel cross talk will depend on thevalues of f and d1.

�d� Checking the value of the SNR: the SNRshould be greater than the required �set� value toensure the desired bit error rate. In our case, weassume8,10 SNR � 10 dB, which corresponds to the biterror rate better than 10�15. If this condition is sat-isfied, the design procedure is complete and does notneed to be reiterated. Otherwise, the feature pa-rameters of VCSEL and photodetector arrays and�orthe interconnect distance should be readjusted andsteps �b� to �d� repeated.

It can be seen from the above analysis that it isunnecessary to limit the design of FSOIs to the con-focal system. In fact, if the VCSEL is located ap-proximately at the primary focal length, which is theoptimal position of the VCSEL in this case, d2 istypically small and is not the Rayleigh range of thebeam between the Tx lens and the Rx lens.13 As aresult, if the confocal system has to be employed, theinterconnect distance is limited to 2 � d1. For thesame reason, if it is required to make d2 as the Ray-leigh range, the VCSEL can not be located at theoptimal position and the effect of beam clipping aris-ing from the Tx lens would increase. However, thisconflict can be solved by employing a nonconfocalsystem as we discussed above. The nonconfocal sys-tem has advantage over the confocal system in thesense of extending the interconnect distance and�orimproving the SNR.

6. Conclusion

We have developed a systematic design algorithmfor the optical design of FSOIs based on arrays ofVCSELs, microlenses, and photodetectors. TheGaussian-beam diffraction model is employed to an-alyze the transmission of characteristics of single andmultitransverse mode beams. The dependence ofthe SNR on the feature parameters of both VCSELand photodetector arrays has been investigated indetail. The results have shown that the SNR in-creases with the increase of channels pacing and the

fill factor of lenses, but decreases with the increase ofinterconnect distance.

We have also analyzed the influence of the pres-ence of higher-order modes on the SNR and the in-terconnect distance based on the results obtained fora practical VCSEL. The results have shown that thepresence of higher-order modes may deteriorate theinterconnect distance and the SNR. The resultshave also shown that a single-mode VCSEL has ad-vantage over multimode one for achieving a long in-terconnect distance and dense channel spacing.

The design procedure for microchannel FSOI’s isproposed and analyzed. Our analysis has shownthat the ratio d1�f has a significant impact on theSNR and the interconnect distance. We have alsoargued that non-confocal system has advantage overconfocal system in the design of FSOI’s in the sense oflonger interconnect distance and�or higher SNR.

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Design of Microchannel Free-Space Optical Interconnects Based on Vertical-Cavity Surface-Emitting Laser Arrays