The behavior of afocal microlens-array �MLA� tele-scopes for agile beam steering of coherent light hasbeen discussed by several authors.1–8 It was shownthat for coherent light the periodic arrangement ofthe microlenses in a MLA telescope results in a typ-ical blazed-grating-like behavior with respect to in-tensity distribution in the far field or in the focalplane of a focusing optics. The blaze angle therebyvaries with the degree of decentering of the MLAtelescopes.
In the research reported in Refs. 9 and 10, MLAtelescopes were examined for spatially incoherent il-lumination by an ensemble of tilted plane waves rep-resenting light emitted by a multimode fiber aftercollimation. However, to the authors’ knowledge,until now it was not possible to describe the actualintensity distribution and transfer efficiencies of abeam steering system based on MLA telescopes sit-uated between collimating and focusing optics, in thecase of an extended source, and decentering of the
J. Duparre �[email protected]� is with theFraunhofer Institut fur Angewandte Optik und Feinmechanik,Winzerlaer Strasse 10, D-07745 Jena, Germany. R. Goring�[email protected]� is with Pyramid Optics GmbH, Lin-denstrasse, D-07589 Lederhose, Germany.
Received 30 October 2002; revised manuscript received 17 April2003.
MLA telescopes. Also, in the analytical models thepropagation between the two MLAs was always ne-glected to preserve the possibility of analytical treat-ment. It can be expected that this failure to includepropagation will cause some substantial errors in thedescription of Keplerian MLA telescopes.
For the development of scanners, field-of-view mul-tiplexers, and also switches or modulators for multi-mode fiber transmission systems with highthroughput efficiencies, a rigorous treatment is de-sirable to explain how transfer efficiency depends onthe various system parameters. Only in this waycan the system be optimized for a given application.A typical setup that we describe below is shown inFig. 1.
New computer power allows MLA telescope sys-tems to be treated in a numerical wave optical way.Here, all important physical effects are inherentlytaken into account. The treatment is based on geo-metrical transmission through optical elements andwave optical free-space propagation between the in-dividual components.
Thus we try for the first time to produce an almostcomplete description of a MLA-telescope-based sys-tem, taking into consideration the grating behavior ofthe MLAs, the remaining divergence of the incominglight after collimation, decentering of the MLAs, thetelescope magnification, and all possible sources ofspurious light.
However, before starting this detailed analysis weslightly modify the previously used analytical model
such that we can explain qualitatively the behaviorobserved later �limited to one dimension�.
The final results of previous publications are typesof grating equations that determine intensity distri-bution in the image plane as a function of MLA pa-rameters and their lateral decentering2–5 or tilt withrespect to the optical axis of the incoming and outgo-ing plane waves.9 In a similar derivation includingthe influence of an extended source, lateral decenter-ing of the MLAs, and angular magnification of thetelescope in the transmission function of the MLAtelescope, one can describe the distribution of theintensity between the diffraction orders of an incom-ing plane wave with tilt angle � to the optical axis andwavelength �, as follows:
p� r0
� f2�� �1 � M�sin �� � m� �m � Z�. (1)
Here M � �f1�f2 is the angular magnification of thetelescope, p is the pitch of the MLAs, and f1 and f2 arethe focal lengths of the lenslets in the focusing andrecollimating MLAs, respectively. r0 is the amountof lateral decentering of the MLAs, and m is thenumber of the diffraction order.
Equation �1� represents a strong nonisoplanatic be-havior and can now be considered for the two differ-ent types of telescope.
The angular magnification of the Galilean setup isM � 0. For �M� � 1 �small axial spacing between theMLAs; almost equal focal lengths of focusing andrecollimating MLAs�, Eq. �1� reduces to
pr0
� f2�� m� �m � Z�. (2)
The optical system becomes isoplanatic, and the im-aging of an extended source can be understood as aconvolution of the source distribution with the im-pulse response of the optical system. The image willlook similar to the source.
For the Keplerian arrangement the situation iscompletely different. Because of the upright imag-ing property of the lenslet telescopes in the Kepleriansetup we have M 0. For �M� � 1, Eq. �1� becomes
p� r0
� f2�� 2 sin �� � m� �m � Z�. (3)
The single source points of the extended source areimaged �without deflection� in higher diffraction or-
ders with increasing �. The imaging of the extendedsource by a Keplerian setup is strongly noniso-planatic and thus cannot be explained by a simpleconvolution.
This major difference in the imaging properties ofthe two telescope types will be found again in thesimulated and measured intensity distributions inthe images of multimode fiber end faces, as shownbelow.
2. Numerical Wave Optical Model
In the numerical wave model the extended lightsource is divided into a number of virtual sourcepoints. Each source point is associated with a tiltedplane wave whose angle with respect to the opticalaxis is a function of the focal length of the collimatingoptics FCOL and the lateral distance of the sourcepoint from the optical axis. The extension of thefield depends on the numerical aperture of the fiberand FCOL. The spectrum of tilted plane waves thatcorresponds to the extended light source is used asinput for the wave optical simulation. Figure 2 is aschematic diagram of the various simulation stepsand approaches. Each plane wave is separatelypropagated electromagnetically through the wholeoptical system. Free-space propagations are carriedout in the angular spectrum of plane waves �SPW�.A transmission through the MLAs can be simulatedin two different approximations: By use of the thin-element approximation �TEA� a physical height pro-file is directly converted into a phase map,11 whichcan be multiplied to the electromagnetical field atthis axial position. If the local plane waveapproximation–local plane interface approximation12
�LPWA–LPIA� is used, the electromagnetic field infront of the MLA is converted into a field of rays thatare traced through the MLAs by local application ofSnell’s and Fresnel’s laws. After its transmissionthrough the element the field of rays is convertedback into an electromagnetical field. The LPWA–LPIA is a more accurate model, especially if MLAswith larger numerical apertures of the lenslets orstronger surface deformations have to be simulated.
In both models, one can generate ideal surface pro-files from parameters to examine ideal conditions �in-cluding fill factor and surface deformation�, ormeasured surface profiles of experimentally testedMLAs can be used.
Fig. 1. Schematic diagram of a micro-optical scanner that couldbe used as a multimode fiber switch.
Fig. 2. Schematic diagram of the numerical wave optical simu-lation.
The number of source points that represent theextended light source in the simulation should bechosen such that this distribution is not resolvableinside the image. For two-dimensional �2D� simula-tions such a choice is not possible owing to limitedcomputer power. For one-dimensional �1D� simula-tions it is achieved when the number of source pointsis equal to or larger than the diameter of the extendedsource divided by the FWHM of the point-spreadfunction that corresponds to a source point.
We start the calculations with the plane wavesafter the collimation lens. These waves are propa-gated to the focusing MLA to simulate the exact lat-eral beam extension that illuminates the MLAs as aresult of the remaining divergence of the extendedsource after collimation. After transmission of thewaves through the first MLA, propagation betweenthe two MLAs, and transmission through the recol-limating MLA we obtain a tilted segmented planewave that has an additional tilt with respect to theoptical axis as a function of the relative lateral de-centering of the MLAs and the focal length of thelenslets in the second MLA. Finally, this wave hasto be propagated to the focusing optics, where thefocusing is simulated by a fast Fourier transforma-tion �FFT� and a coordinate transformation.
In the image plane of the focusing optics the inten-sity distributions that correspond to all the singlesource points are calculated, and all intensity distri-butions are superimposed to yield the incoherent im-age of the fiber end face. A virtual pinhole with thediameter of the source is placed over the image forcalculation of the overall system transfer efficiency.
3. Intensity Distributions after Transmission through aKeplerian or a Galilean Microlens Array Telescope
We measured and calculated the images and transferefficiencies of 1D Keplerian and Galilean MLA tele-scope arrangements �cylindrical microlens arrays�.Figure 3 shows our experimental setup for measure-ment of transfer efficiency. We observed the imagesof the fiber end faces with a CCD camera after theirmagnification by a microscope objective.
The cylindrical MLAs for the experiments wereproduced by a special replication technique in a poly-mer film upon a glass substrate at the FraunhoferInstitute Angewandte Optik und Feinmechanik,Jena, Germany. They were characterized by a sur-face profiling instrument and a shearing interferom-eter. In Table 1 the parameters of the MLAs aregiven. The multimode fiber source that we used hada core diameter of 200 m and a numerical apertureof 0.22. The focal length and the free aperture of thebulk collimating and focusing optics were 20 and 12mm, respectively.
To concentrate on the major physical effects of im-age formation by the two MLA telescope systems inthis section we set the fill factor of the MLAs to 1 andneglected the lenslet aberrations in our calculations.The experiments were performed with a laser diodesource at 635 � 5 nm. In the calculation we as-sumed that the light was monochromatic because a
variation of spectral width of several nanometerswould not change the results.
Figure 4 shows the experimentally observed fiberend images and Fig. 5, the corresponding calculatedimages. The window width is always 540 m � 540m. It can be seen that measured and simulatedimage figures coincide quite well in terms of typicalintensity distributions. The rough pixel structure inFig. 5 results from insufficient computer power for a2D calculation with high resolution. For the samereason, to obtain quantitative information we reliedon 1D simulations with the full aperture and largenumber of source points �110 source points were suf-ficient for the given case�. The 1-D simulation alsocorresponds to the experimental use of one-dimensional �cylindrical� microlens arrays.
Figures 6–9 show the results of one-dimensionalsimulations of the image intensity distributions andthe cross sections of the corresponding experimen-tally observed images. For comparison, Fig. 10shows the image cross section without the MLA tele-scope between the collimating and focusing optics.In the simulation the collimating and focusing opticsare assumed to be paraxial lenses.
For the Galilean arrangement we can easily under-stand the resultant measured and simulated inten-sity distributions in Figs. 4�a�, 4�b�, 5�a�, 5�b�, and thecorresponding cross sections in Figs. 6 and 7 for f1�� f2�
Fig. 3. Experimental setup for determination of the transfer ef-ficiencies of the MLA telescope as a function of lateral decentering�here coupling into a pinhole in front of a photodetector�.
Table 1. Parameters of Cylindrical MLAs Used in Experiment andSimulation
� 1. They are the result of convolution of the coher-ent diffraction pattern with the source distribution Eq. �2��.
From Fig. 6 it can be expected that a deflectionexactly into the direction of a diffraction order willgive a higher efficiency of a 1:1 imaging system thana deflection between two diffraction orders. In thelatter case the image extension in the direction of thetransformation in the cylindrical MLA telescope be-comes much larger than the object extension. It alsobecomes clear that this behavior will result in oscil-lation of the system’s efficiency with beam deflection.
A comparison of Fig. 7�a� with Fig. 10 also demon-strates experimentally the rather good 1:1 image ca-pability of the Galilean setup when deflection goes inthe direction of a grating order. We show quite dif-ferent behavior of intensity distribution in the imagefor the Keplerian arrangement Figs. 4�c�, 4�d�, 5�c�,5�d�, and also Figs. 8 and 9�.
The characteristic rearrangement of intensity dis-tributions in the image has already been explainedqualitatively by Eq. �3�. Here, we illustrate it by asimple figure �Fig. 11�. For simplification we as-sume that M � �1 and r0 � 0.
Tilt angle � of the incoming plane wave fulfills thegrating equation for the Keplerian setup, Eq. �3�,when the distance of the source point from the opticalaxis is exactly a multiple of the half grating orderdistance in image space. These source points �pointsa and d in Fig. 11� are imaged into one major diffrac-tion order. The number of the corresponding diffrac-tion order increases with increasing distance of asource point from the optical axis �zeroth order for
points a and first order for points d�. One can alsoexpect that arbitrarily located source points will bedistributed into various diffraction orders such thatthe center of intensity represents the coordinate ofthe source point. However, the maximum illumi-nated diffraction orders are situated at positions com-pletely different from the expected 1:1 imaging of asource point �points b, c, and e in Fig. 11�.
A decentering of the MLA telescopes in the Keple-rian arrangement corresponding to a deflection of onegrating order results in intensity distributions thatare similar to those with no decentering in Fig. 5�c� Fig. 4�c�� and the corresponding cross sections in Fig.8�a� 9�a��. For decentering corresponding to a de-flection of half a grating order we obtain distributionsas in Fig. 5�d� 4�d��, and the corresponding crosssections in Fig. 8�b� 9�b��, which have a slightly dif-ferent form.
It can be also seen that the spot variations duringdeflection are not so remarkable as for the Galileansystem; thus we also expect a smaller variation insystem efficiency. The ratio of the source extensionto the half grating order distance thus determines thenumber of segments into which the source will beimaged by a Keplerian setup without deflection. Inour example this number is seven, as can be foundexactly, for instance, in Figs. 4�c� and 5�c�.
Figure 12 gives an example of the use of the modelpresented here for simulating the influence of severalparameter sets that were not achieved experimen-tally. The image figures become increasingly simi-
Fig. 4. Experimentally recorded 2D spots: �a� Galilean arrange-ment, centered; �b� Galilean arrangement, laterally decentered fordeflection of the amount of half a grating order; �c� Keplerianarrangement, centered; �d� Keplerian arrangement, laterally de-centered for deflection of the amount of half a grating order.
Fig. 5. Numerical wave optically simulated 2D spots: �a� Ga-lilean arrangement, centered; �b� Galilean arrangement, laterallydecentered for deflection of the amount of half a grating order; �c�Keplerian arrangement, centered; �d� Keplerian arrangement, lat-erally decentered for deflection of the amount of half a gratingorder.
Fig. 6. Simulated 1D intensity distribution for a Galilean arrangement: �a� centered �93.4% efficiency�, �b� laterally decentered fordeflection of the amount of half a grating order �80.4% efficiency�.
Fig. 7. Cross section of the experimentally recorded image of the end face of a multimode fiber with 200-m core diameter imaged withthe use of a MLA telescope in the Galilean arrangement: �a� centered �87.2% efficiency�, �b� laterally decentered for deflection of theamount of half a grating order �73.2% efficiency�.
Fig. 8. Simulated 1D intensity distribution of the Keplerian arrangement: �a� centered �81.4% efficiency�, �b� laterally decentered fordeflection of the amount of half a grating order �85.1% efficiency�.
lar to the source when the source extensioncorresponds to a larger number of diffraction orders.
4. Numerically and Experimentally Obtained TransferEfficiencies
In Figs. 13–15 the results of several approximationsin the simulations are compared with the experimen-tal results. The parameter of interest is the achiev-able transfer efficiency for an extended source withvariable deflection. A Keplerian, a reversed Keple-rian �the substrates of the MLAs are oriented towardeach other in contrast to the Keplerian setup in whichthe lens structures face to each other, which causeslarger aberrations�, and a Galilean arrangement areexamined. The parameters used are the same asthose described in Section 3.
The system’s transfer efficiency is determined byuse of a 1D simulation in the direction of transfor-mation of the cylindrical MLA telescopes. The grat-
ing behavior and vignetting �spurious light� arefunctions of the distance of a source point from theoptical axis in this direction. However, the source istwo-dimensionally extended and of circular shape.We assume equidistant sampling of the source. Thenumber of sampling points on a secant perpendicularto the direction of transformation is proportional tothe secant’s length. For a circular source this secant
Fig. 9. Cross section of the experimentally recorded image of the end face of a multimode fiber with 200-m core diameter imaged withthe use of a MLA telescope in the Keplerian arrangement: �a� centered �72.5% efficiency�, �b� laterally decentered for deflection of theamount of half a grating order �73.5% efficiency�.
Fig. 10. Cross section of the experimentally recorded image of theend face of a multimode fiber with 200-m core diameter imagedwithout the use of a MLA telescope �imaged only with collimatingand focusing optics�.
Fig. 11. Schematic diagram of the distribution of intensities ofseveral source points onto the diffraction orders when they areimaged by the Keplerian telescope setup. Left, five source points�a –�e of the laterally extended object. Their normalized intensityis divided roughly into three equal parts. Right, intensity distri-bution of several source points between various diffraction ordersas a function of the distance of the source point from the opticalaxis. The oblique thinner lines that connect object and imagespace represent the directions of propagation of the zeroth diffrac-tion order for each source point. The oblique thicker lines repre-sent the direction of propagation of the center of intensity of theilluminated diffraction order for each source point for 1:1 imagingby the Keplerian setup. In accordance with Eq. �3� one can findthe diffraction order with maximum intensity for a given sourcepoint by determining the direction of the zeroth order for this pointand counting the diffraction order numbers until the order that isclosest to the direction of the 1:1 image by the Keplerian setup isfound.
becomes shorter as it is separated from the center ofthe source. Thus a weighting of the source points inthe 1D simulation as a function of their distance from
the optical axis is applied for correct representation ofthe 2D circular source.
A comparison of ideal structures �fill factor 1, per-
Fig. 12. Simulated 1D intensity distribution for the centered Keplerian arrangement for four values of the pitch of the MLAs: �a� 50-mpitch, �b� 100-m pitch, �c� 200-m pitch, �d� 400-m pitch �the numerical aperture is fixed to 0.1�.
Fig. 13. Transfer efficiencies of the Keplerian arrangement �forsystem parameters see Table 1 and Section 3�.
Fig. 14. Transfer efficiencies of the reversed Keplerian arrange-ment �for system parameters see Table 1 and Section 3�.
fect spherical surface profile, no focal-length varia-tion� in simulation with the real structures used inexperiment and simulation is of major importance fordistinguishing between the physical limits of the ar-rangements and the effects of imperfections of the
fabricated MLAs on the achievable transfer efficien-cies.
The ideal lens structures are represented in a TEAand in a LPWA–LPIA to permit us to analyze theeffects of the approximation in the model. Real lensstructures can be modeled in a TEA with the knownfill factor and the optical path difference �obtained bya shearing interferometry� of the lenslets or in aLPWA–LPIA just with an experimentally obtainedsurface profile of the MLAs. For comparison, in Fig.14 the results of a simple commercial ray-tracingprogram �ZEMAX� are presented.
From Figs. 13–15 the very good agreement of thesimulations made with the parameters of the realstructures and those of the experiments can be ob-served. However, simulations with the ideal struc-tures show the physical limits of the various setupsand thus give an idea of what would be achievablewith optimized lens structures. The continuous de-crease in efficiency with deflection has its main originin the generation of spurious light with growing lat-eral displacement of the MLAs, which suggests theuse of field lens arrays in the reversed Keplerianarrangement, as proposed in Ref. 2.
The observed oscillations of the transfer efficiencieswith displacement result from the grating effect andwere explained in Section 3 as a function of variationin overall image beam dimensions. Even if the di-verging lens arrays used here show large surface er-rors we obtain almost ideal efficiency values for smalldeflections, partly as a result of aberration compen-sation in the Galilean setup �Fig. 15�.
The Galilean setup that we used experimentallyhas a magnification of 1.06. Because the collimatingand focusing optics have the same focal lengths, theimage of the fiber end face has a 1.06 times largerextension in the direction of transformation than theoriginal. This is why, even for an ideal Galilean
Fig. 15. Transfer efficiencies of the Galilean arrangement �forsystem parameters see Table 1 and Section 3�.
Fig. 16. Influence of lens pitch on transfer efficiency for a given fiber core diameter of 200 m: �a� reversed Keplerian arrangement, �b�Galilean arrangement �the numerical aperture is fixed at 0.1�.
Table 2. Summary of Determined Transfer Efficiencies
setup with those parameters, we can achieve �94%efficiency at maximum.
The upright imaging property of the lenslet tele-scopes together with the grating effect in the Keple-rian setup results in a rearranged image intensitydistribution, as was explained in Section 3. Becausethe pinhole should have the same extension as thesource, it will be considerably overfilled, which willcause larger losses than in the Galilean setup.
A comparison of Keplerian and reversed Kepleriansetups shows slight differences because there are dif-ferent aberrations. Of course, in the ideal model inthe TEA there is no physical difference between thesetwo setups.
Because of wave-front distortions, which enter intothe calculations more strongly as the model becomesmore realistic, the effective focal lengths of the lensletsin the simulations change. We found that in waveoptical simulations it is necessary to optimize the axialdistance of the MLAs for a certain setup as in a realexperiment to obtain the optimum recollimation.
Table 2 gives a summary of the theoretically andexperimentally achievable transfer efficiencies forthe systems that we examined. For example, thecapabilities of the simulation tool described above fortesting various parameter sets are demonstrated inFig. 16 �variable lens pitch; fixed source and pinholeextension�. We also tested variable source extensionfor fixed lens pitch, fill factor, spherical aberration,numerical aperture, magnification of the MLA tele-scopes, field lenses, statistical variation of focallengths, and tolerances.
Figure 16 illustrates quantitatively our conclu-sions from the discussion in Section 4:
• For increasing distance of diffraction orders inimage space the transfer efficiencies decrease, espe-cially for the Keplerian setup, because an increasingpart of light is imaged outside the dimensions of theoriginal source size.
• The transfer efficiency shows an oscillating be-havior with MLA displacement, which is much moreobvious for the Galilean setup.
• With decreasing influence of diffraction, thebeam deflection function becomes more nearly con-tinuous.
5. Conclusion
The numerical wave optical model presented in thispaper gives new possibilities for examining microlens-array telescope systems for beam deflection applica-tions of spatially incoherent illumination. Because ofits nature, this model automatically takes into accountall relevant physical effects within one closed simula-tion for a certain parameter set and thus permits pa-rameter optimization and a prediction of thetolerances. It can be seen as a numerical experiment.
With the help of this model it was possible, for thefirst time to our knowledge, to calculate the imagedfiber end faces of multimode fibers for several tele-scope types. The results could be verified by corre-sponding experiments and explained by a modified
analytical model. Major differences between the im-aging behavior of Galilean- and Keplerian-type tele-scopes were proved. The segmentation of the imageof the Keplerian-type telescope was explained andfound to be a main reason for transfer efficiencieslower than those of a Galilean-type telescope withsimilar parameters.
The most important advantages of the numericalwave optical model presented here are the following:
• Transfer efficiencies of systems with ideal geo-metric parameters can be calculated for arbitraryamounts of beam deflection. For a specific configu-ration the theoretical limits of the achievable transferefficiencies can be found in this way.
• The parameters of experimentally achieved sys-tems can be implemented in the simulation. So theeffects of imperfections of the MLAs and other exper-imental components on the maximum achievabletransfer efficiencies can be distinguished from thelosses caused by the principal physical effects thatapply for a given system.
We have demonstrated quasi-continuous deflectioneven for inputs that are highly divergent owing todiffraction effects.
A 1 � 4 fiber switch for multimode fibers with600-m core diameter based on the principle pre-sented here, in which one MLA is laterally displacedby piezoelectric actuators, is currently under devel-opment.
This research was performed while the authorswere with Piezosystem Jena GmbH, Jena, Germany.We are grateful to Peter Dannberg �Institut Ange-wandte Optik und Feinmechanik, Jena, Germany�for fabrication and characterization of the cylindricallens arrays, Frank Wyrowski �Lighttrans GmbH,Jena, Germany� for temporary free use of the waveoptical simulation software DIFFRACTICA, and AndreasTunnermann for many valuable discussions.
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