In optical metrology, there is a need for modulators orchoppers of optical radiation emitted by multimodefibers. Micro-optical solutions offer a high dynamicrange of a modulation combined with a compact me-chanical setup. In Fig. 1 we present a simple beammodulation concept with microlens arrays in a con-focal arrangement.1 An array of diaphragms, whichcan be translated by a piezoelectric actuator, is placedin the common focal plane of the microlens arrays.Other micro-optical elements, for example, color fil-ters, polarization filters, phase plates, and so on, canalso be placed in that plane to modulate other param-eters of the beam. This concept is similar to systemsthat employ microlens arrays to improve the perfor-mance of spatial light modulators for optical signalprocessing.2 If the size of the microlens arrays ischosen in a way that efficient beammodulation can beachieved by small displacements of the diaphragmarray, then the switching can be fast. Formaximumdisplacements of 10–100-mm switching times of theorder of 1 ms are possible if piezoelectric actuatorsare used. This meets the requirements of manymetrological applications.It is well known that an array arrangement of
lenses also acts as a diffraction grating if it is illumi-nated with coherent radiation.3,4 However, there
The authors are with the Fraunhofer Institute for Applied Opticsand Precision Engineering Jena, Schillerstrasse 1, D-07745 Jena,Germany.Received 28 May 1996; revised manuscript received 12 August
are also a number of applications for incoherent illu-mination, for example, imaging devices or multimodefiber switches and modulators. Until now no theo-retical description for illumination with incoherentsources has been given. Therefore we must provewhether an efficient throughput from one to anothermultimode fiber with an arrangement like that inFig. 1 is possible.In Section 2 we investigate the system for incoher-
ent illumination with a remaining divergence in frontof the arrays. A system with multiple-switching ca-pabilities is analyzed. The results are quantified.We present configurations with high throughput,choosing appropriate lens arrays, focusing and cou-pling optics. In Section 3 we summarize and discussfuture work in this area.
2. Theoretical Approach
A. Analysis of the Point-Spread Function
The radiation emitted by a multimode fiber ~Fig. 1!can be expected to be spatially incoherent. We as-sume that the collimating lens transforms everyspherical wave emitted from a point within the core ofthe multimode fiber into a plane wave incident uponthe lens arrays. Because of the lateral extent of thesource, plane waves, which are incoherent to eachother, illuminate the lens arrays. We search for thepoint-spread function of incoming waves with an an-gle u1 against the optical axis in the focal plane of thefocusing lens ~Fig. 2!. The rays behind a lenslet ofthe second array have a tilt angle u2 against theoptical axis. The difference in u1 and u2 can becaused by a difference in the focal lengths of the len-slets of the microlens arrays. For simplicity we dealhere with the one-dimensional problem, which can be
1 March 1997 y Vol. 36, No. 7 y APPLIED OPTICS 1467
easily extended to the two-dimensional case. Figure3 shows the wave fronts before and after the micro-lens arrays; d and D are the lens pitch and the illu-minated length of the arrays, respectively. Wefurther assume D .. d. Introducing an optical fillfactor F ~0 , F , 1!, we assume that the lenses in thelast array are not completely filled with light. In theideal case the wave front after each single aperture ofthe last lens array is a plane wave. If the system isnot free of aberrations, then the aberrations deformthe wave front after each lens of the array in the sameway. Figure 4 shows the phase profile after the mi-crolens arrays. The phase can be described by a sumof three terms: a linear phase distribution over theentire aperture given by the phase distribution of theincoming wave front, a linear phase distribution overa single lens aperture with the periodicity of the lensspacings, and a periodic wavefront aberration. Inthe Appendix some definitions of mathematical func-
Fig. 1. Principle of a micro-optical intensity modulator for mul-timode fiber radiation.
Fig. 2. Search for the point-spread function of incoming waveswith tilt against the optical axis.
Fig. 3. Rays and wave fronts before and after the lens arrays.
1468 APPLIED OPTICS y Vol. 36, No. 7 y 1 March 1997
tions and relations, which we need for the followingcalculations, are given. The transmission functiont~x! of a single aperture of the lens array assembly isgiven by
t~x! 5 rectS xdFDexpF2i 2p
l~sin u1 1 sin u2!xGexp~ifab!.
(1)
Because the incoming wave front is tilted, the singleparts of the wave front after the lens arrays have aphase shift relative to one another. We can give thefield distribution u~x! after the lens arrays by the arraytheorem.5 The symbol R stands for convolution.
u~x! 51
ÎDF expSi 2p
lsin u1xDrectSxDD
3 H1d combSxdD ^ t~x!J. (2)
For convenience, the intensity is normalized~*2`
` uu~x!u2 dx 5 1!. The field distribution in the fo-cal plane of the following lens can be determined byFourier transformation ^ of u~x! with 8~ fx! 5^@u~x!# and 7~ fx! 5 ^@t~x!#:
8~ fx! 51
ÎDF dS fx 2sin u1
l D^ D sinc~Dfx! ^ @comb~dfx!7~ fx!#,
8~ fx! 51dÎDF H (
n52`
`
sincFDS fx 2sin u1
l2
ndDG7SndDJ,
(3)
7SndD 5 *2`
`
rectS xdFDexpF2i2p
l~sin u1 1 sin u2!xG
3 exp~ifab!expS2i2pndxD dx,
7SndD 5 *2dFy2
dFy2
exp~ifab!expF2i2p
3 Snd 1sin u1 1 sin u2
l DxGdx. (4)
Fig. 4. Phase distribution after the arrays: ~a! the phase can bedescribed as a sum of f1 and f2, ~b! a periodic wave front aberra-tion fab is introduced. Fill factor F is taken into consideration; cis a constant.
The final result is
8~ fx! 51dÎDFH (
n52`
`
sincFDS fx 2sin u1
l2ndDG
3 *2dFy2
dFy2
exp~ifab!expF2i2p
3 Snd 1sin u1 1 sin u2
l DxGdxJ. (5)
The grating behavior of the system is obvious, with nbeing the number of the diffraction order. With theenergy conservation *2`
` uu~x!u2dx 5 *2`` u8~ fx!u
2d fx 51 and the general relation
limD3`HÎD sincFDS fx 2sin u
l2ndDGJ
2
5 dS fx 2sin u
l2ndD,
the energy distribution to the single orders can begiven by
hn 51Fd2U*
2dFy2
dFy2
exp~ifab!expF2i2p
3 Snd 1sin u1 1 sin u2
l DxGdxU2 . (6)
In the case of aberration-free operation ~fab 5 0! ofthe lens arrays hn is given by
hn 5 F sinc2FSnd 1sin u1 1 sin u2
l DdFG . (7)
Light is diffracted to only one diffraction order if F 51 and u1, u2 solve the grating equation; d~sin u1 1 sinu2! 5 ml, with m being an integer number. If aber-rations are present, the diffraction efficiency hideal forangles u1, u2 solving the grating equation is given by
hideal 5
U*2dFy2
dFy2
exp@ifab~x!# dxU2d2F 2 F 5 VF . (8)
V is the Strehl ratio of a single optical train of the lensarray assembly.6 As expected, the performance ofthe lens arrays can be described as the performanceof a single optical train combined with the array the-orem. Therefore testing of single elements of lensarrays and testing of the homogenity of the lens ar-rays give complete information about the system.With fx as the spatial frequency and x as the spatial
coordinate in the image plane of Fig. 2, the field dis-tribution u is given by8~ fx! 5 8 ~ xylf ! 5 u~ x, sin u1!.The system is not isoplanatic because of the depen-dence of the incident angle u1. The input and outputangles of the rays before and after the lens arrays are
related to each other by u2 5 Mu1, with M being theinternal angular magnification of the lens array as-sembly. With the assumptions that the incident an-gles are small ~sin u1 ' u1! and that the wave-frontaberrations are the same for different incident anglesin a certain field of view, which can be guaranteed bya proper optical design, the image shows a periodicityup:
u~ x, u1! 5 u~ x 1 Mupf, u1 2 up! with
up 5 lyd~1 1 M!. (9)
To describe the imaging capabilities of the setup, onlyone period of incident angles has to be investigated.Because of the nonisoplanatic behavior, the opticalsystem cannot be described by a modulation transferfunction. However, if the smallest object details arelarge, then the radiation incident upon the arrays hasa broad propagation angle spectrum ui . up andtherefore an averaging process occurs. This averag-ing process results in a nearly ideal transmission oflow spatial frequencies of the object.
B. Investigation of a System with Multiple SwitchingCapabilities
To ensure switching with small displacements of amicro optical element, the light concentration in thecommon focal plane of the microlens arrays must beefficient. The maximal spot area 2h is given by thefocal length f1 and the maximum incident angle umax:2h 5 2f1 tan umax ~see Fig. 5!.With f1y# as the f-number of a lenslet of the first
array, k gives the number of individual micro opticalelements, for example, filters, that can be placedwithin one period d. Therefore k gives the numberof different discrete switching states possible withthat configuration. Further, Fig. 5 shows the ar-rangement in the focal plane of filters of differentkinds. The diaphragm ~intensity filter! and theother filters, for example, color filters, can be movedseparately to allow a combined continuous-intensitymodulation and a discrete switching operation:
k 5d2h
51
2f1y# tan umax. (10)
One must pay attention to the fact that k is boundedby the diffraction of the wave at the first lens.Therefore Eq. ~10! holds only for umax . 1.22lyd.
Fig. 5. Light concentration in the common focal plane of thelenslets of the arrays.
1 March 1997 y Vol. 36, No. 7 y APPLIED OPTICS 1469
For example, with l 5 670 nm, d 5 200 mm: kmax .120yf1y#.To avoid spurious light losses, which can be seen in
Fig. 5, the focal length f2 of the second array must bechosen to be slightly smaller than f1:
f2 5 f1k 2 1k 1 1
. (11)
The optical fill factor F is then given by F 5 f2yf1 andis identical with the inverse of the internal angularmagnification M of the lens-array assembly:
M 51F
5k 1 1k 2 1
. (12)
All relations hold only for ideal lenses, which impliesthat aspheric surfaces are required if low f-numbersare required to achieve efficient light concentration.Figure 6 shows k and the fill factor F for differentmaximum incident angles umax.
C. Efficiency Calculations
We again consider the setup of Fig. 3. Plane waveswith uniformly distributed energy within a cone ofangles 2umax # u1 # umax against the optical axis areincident upon the first array. Every wave is dif-
Fig. 6. k and fill factor F for different maximum incident anglesumax and f-numbers f1y#.
1470 APPLIED OPTICS y Vol. 36, No. 7 y 1 March 1997
fracted to different orders and in different directionsbehind the second lens array. Because the singleplane waves are spatially incoherent relative to oneanother, the intensity or energy of the output of thewaves after the second lens array is added. Wesearch for the part of the energy that is containedwithin a cone of output angles2Mumax # u2 #Mumax.This gives us the efficiency x of the transmission ofthe input beam ~Fig. 7!, which can be compared witha coupling efficiency into a multimode fiber. Thefocusing lens of Fig. 1 guarantees the overall magni-fication of unity.We use the following algorithm: with given lyd,
f1y#, and umax, we obtain k, f2y#, F, and M with Eqs.~10!–~12!; with the help of Eq. ~6! we numericallycalculate the overall efficiency x, which is shown fordifferent f-numbers in Figs. 8 and 9. To avoid spu-rious light losses, the f-number of the second array,f2y#, and the magnification M are a function of themaximum incident angle umax. According to the pe-riodicity in the point-spread function @Eq. ~9!# a mod-ulation is obvious; the same is true if the aberrationsare independent of the incident angle for a certainfield of view. The aberrations diminish only the ef-ficiency x and the modulation depth. There arethree cases for which the beam transmission is veryefficient:
1. umax 5 0 and small aberrations: coherent ra-diation.
Fig. 7. Definition of the efficiency x of the system: x is the ratioof the output energy in directions 2Mumax # u2 # Mumax and theinput energy in directions 2umax # u1 # umax.
Fig. 8. Efficiency x for f1y# 5 2.5, ly2d 5 1.67 mrad.
2. umax ' 0 and small aberrations: incoherentcase, highly collimated beam.3. umax .. ly2d: incoherent case, weakly colli-
mated beam.
There are different applications for case 1, for exam-ple, laser-beam modulation and laser-beam steeringwith decentered lens arrays. It requires an appro-priate optical design to achieve nearly aberration-free operation over a certain steer angle.7 Theincoherent case is interesting for light modulators formultimode fiber radiation, adaptive illumination sys-tems, andmultimode fiber switches. Radiation withdifferent wavelengths can be transmitted simulta-neously with high efficiencies in a setup like Fig. 1 forcases 2 and 3. Case 3 is less sensitive to aberrationsthan is case 2.
3. Summary
We showed that microlens arrays in a confocal ar-rangement with filter elements in the common focalplane are suited for efficient modulation of light.The performance of the setup depends on the remain-ing divergence of the incoherent light, which is inci-dent upon the arrays, in comparison with theperiodicity of the point-spread function, which isgiven in the angle domain by ly2d. For weakly col-limated beams the transmission of the systemreaches values above 90% and the diffraction gratingbehavior of the lens arrays can be neglected. Thetransmission of light in different wavelength bands ispossible. The introduction of color filters, polariza-tion filters, diaphragms, and phase plates in the focalplane offers a number of applications. The switch-ing function can be discrete ~individual filters! or acontinuous-intensity modulation ~the scanning of adiaphragm over the focal spot!. The beam diver-gence can be adjusted with an appropriate collimat-
Fig. 9. Efficiency x for f1y# 5 5, ly2d 5 1.67 mrad.
ing lens in front of the lens arrays. The approachoffers a high dynamic range of the modulation. Thenumber of the output states for discrete switching ormodulation can be increased if two-dimensional ar-rangements of lenses and filters are used. For ex-ample a system with an f-number f1y# 5 5, a lenspitch of d 5 200 mm, and square filter dimensions of20 mm offers 100 output states. In the future, thefollowing problems have to be addressed: the fabri-cation of microlens arrays with good aberration per-formance, suited focal length, fill factor and arraylength; the fabrication of miniaturized filters; and theinvestigation of the actuator concept.
Appendix A: Definitions
We use the following definitions and Fourier trans-form relations:8
rect~x! 5 H10 ::
uxu # 1y2uxu . 1y2 , (A1)
sinc~x! 5sin ~px!
px, (A2)
comb~x! 5 (n52`
`
d~x 2 n!, (A3)
^@rect~x!# 5 sinc~ fx!, (A4)
^@comb~x!# 5 comb~ fx!. (A5)
This research was supported by the Federal Min-istry of Education, Science, and Technology ~BMBF!,Federal Republic of Germany, within the nationaljoint project ~Piezoelektrisch gesteverte mikroop-tische Systeme!.
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