Optimized kinoform structures for highly efficientfan-out elements
D. Prongu, H. P. Herzig, R. Ddndliker, and M. T. Gale
We discuss the realization of highly efficient fan-out elements. Laser-beam writing lithography isavailable now for fabricating smooth surface relief microstructures. We develop several methods foroptimizing microstructure profiles. Only a small number of parameters in the object plane are necessaryfor determining the kinoform. This simplifies the calculation of M x N arrays also for large M and N.Experimental results for a 9-beam fan-out element are presented.
Phase elements that generate arrays of light spots arekey components in many optical processing systems.Space-invariant fan-out elements split a laser beaminto N quasi-plane waves, which are focused by aFourier lens, as shown in Fig. 1.
Different types of fan-out elements have beeninvestigated in the past. Binary phase holograms,such as Dammann gratings, are well known. Theirfabrication involves microlithographic techniques thatare well mastered and widely available. However,these elements are usually limited by their efficiency.In order to increase the diffraction efficiency, recentefforts have concentrated on multilevel phase struc-tures, such as quaternary phase elements1 or off-axismultilevel phase elements.2 Their fabrication alsoinvolves microlithography, but with the drawbackthat the number of masks increases with the numberof phase levels. Thus precise alignment must beperformed during each process.
In our approach we optimize a nonquantized phasetransfer function for an on-axis fan-out element andwe implement this phase function as a smooth sur-face relief grating. This process became possible withthe laser-beam writing system developed at the PaulScherrer Institute, Zrich. 34 This method has theadvantage of being able to generate smooth surfaces,
M. T. Gale is with the Paul Scherrer Institute, Badenerstrasse569, CH-8048 Zurich, Switzerland. The other authors are with theInstitute of Microtechnology, University of Neuchfltel, CH-2000Neuchdtel, Switzerland.
which can have maximum diffraction efficiency, andto draw the structure in one single step, which avoidserrors that are due to successive alignments. On theother hand the control of the modulation depth can bedifficult, depending on the knowledge and stability ofthe photoresist and development parameters.
We have developed a two-step method to optimizethe transfer function of a fan-out element.56 Theoptimization criteria are the efficiency and the unifor-mity of the outgoing beams. The optimization param-eters are the amplitudes and phases of an array ofrecorded virtual light sources. These sources arecalled virtual because they are used only to describethe construction of the fan-out element, but they arenot physically used to record the fan-out element.The small number of free parameters and the non-quantized phase function in the hologram plane leadsto fast optimization procedures.
We applied the optimization method to a nine-beamfan-out and we recorded its transfer function intophotoresist. The resulting element is an on-axis sur-face relief phase hologram.
II. Kinoform Optimization
The goal of our fan-out element is to focus a laserbeam onto a regular array of equally intense lightspots. The desired field distribution in the objectplane (Fig. 1) is then given by
NU(x, y) = E Am exp(i(m )8(x - Xm, y),
m=1(1)
where Am is the amplitude, (m . is the phase, and xm isthe position of the mth spot of a one-dimensionalarray. The phases (,, are free parameters.
tions of the object irradiance in the hologram planeare minimized, i.e.,
Because we are interested only in the intensity distri-bution of the object the phases (km are free parame-ters. For determining the optimum phases, the irradi-ance I(u, v) given by Eq. (4) is introduced intoexpression (8). We get
fan-out phase grating Fourier lens spots
Fig. 1. Readout of the fan-out element.
The field distribution U(u, v) in the hologram planeis related to the field U(x, y) by a Fourier transform:
U(u, v) = f_ U(x, y)exp[2,ri(xu + yv)]dxdy
(2)
The field U(u, v) written in terms of magnitude andphase is
(3)
where qk(u, v) = arg{ U). For the irradiance distribu-tion I(u, v) in the hologram plane we obtain
N NI(u, v) = I U(u, v) 12 = F Am + 2 > AmAn cos(flmn), (4)
m=1 m<n
where the arguments Q~mn stand for
(5)
For a constant distance s between the spots thearguments flmn in Eq. (5) become
flmn = 2u(m - n)s + 'm -)n- (6)
The first term on the right side of Eq. (4) is constantand is equal to the mean object irradiance,
(7)
The second term describes the irradiance variations,or intermodulations, in the hologram plane.
In order to perfectly reproduce the desired objectU(x, y), the hologram must have a transfer functionthat is proportional to U(u, v), which means anintensity transfer function proportional to I(u, v) anda phase transfer function equal to exp[i*(u, v)]. Witha single hologram, the intensity transfer function canbe made by absorption only. This will inevitablyreduce the efficiency.
The losses that are due to absorption by theintensity transfer function are minimized if the varia-
which means that the intermodulation terms must beminimized. The intermodulation terms can be rewrit-ten by collecting terms of the same spatial frequency,namely
N N-1 N-kA,,A cos(fQmn) = X E AmAm+k cos(2rrksu + m4m+k)
m<n k=i m=i
where k = m - n. The coefficients Bk are obtainedfrom
N-k2
Bk2= AmAm+k cos((m - m+k)
Lm=l I
+ N-k- 6k)2+ AmAm+k sin(¢m - m+k)] . (11)
m=s
Expression (9) then becomes
ff [ ' Bk cos(27rksu + Dk )]2dudv min.
Since the intermodulation terms Bk cos(2rksu + Pk)in expression (12) have different spatial frequencies,they are orthogonal and therefore one gets, fromexpression (12),
Expression (13), together with Eq. (11), supplies thecriterion for determining the optimum phases (km fora fan-out element with maximum efficiency.
The criterion in expression (13) reduces the inter-modulation terms of Eq. (4) to a minimum but thereis still a residual intermodulation. However, to avoidany absorption in the hologram plane, and also forfabrication reasons, we opt for a pure phase element.The phase is implemented as a smooth surface reliefhologram without quantization. Thus the intensitytransfer function of the fan-out element is clipped toI(u, v) = 1. The phase transfer function is equal toexp[ii+(u, v)], as obtained for the optimum phases (kmfrom Eqs. (2) and (3).
Clipping the residual intermodulation terms hardlyalters the high efficiency, but it does reduce theuniformity of the fan-out. The amplitudes Am' of thereconstructed spots are slightly different from thedesired amplitudes Am that are assumed for therecorded virtual light sources (Fig. 2). Furthermore,some weak side spots appear on each side of thedesired N beams. There are different solutions forsolving the uniformity problem. One is to add addi-tional weak light spots to the virtual recording lightsources. Thus the new parameter set contains N'(N' > N) light sources, where the intensity of Nobject beams only is of interest. Now the minimum ofexpression (8) can be reduced significantly. The effectof clipping becomes negligible. The readout of thisfan-out produces N' light spots with perfect unifor-mity of the amplitudes Am' for m = 1 to N. On theother hand, the N' - N additional light spots (Ai, i)increase the number of optimization parameters. As aresult, the computing time rises strongly. Thereforefor improving the uniformity of the fan-out, wepropose another optimization process. By changingthe amplitudes of the recorded virtual light sourcesslightly to Am the resulting amplitudes Am' (n) in theoutput can be balanced; the efficiency does not de-crease much.
Below we demonstrate the optimization step bystep.
Ill. Application of the Optimization
First Optimization
For weighted fan-outs, any distribution of the Am canbe assumed and optimized by following the proceduredescribed in Section II. However, here we treat onlythe particular case of uniform fan-outs (Am = 1). Forsymmetry considerations, we impose an even distribu-tion of the phases (()m = (N+l-m)- In this case, Eq.(11) reduces to
N-kBk = O cos(4m - ()m+k). (14)
m=1
hologram plane
FT-1Iing I(u,v) --* cos
I
clipping: (u,v) -+ const
object plane
1mpt|input:optimized object(virtual recording)
From expression (13) and Eq. (14) we get the opti-mum phases (m. The minimum in expression (13) isfound by numerical optimization with the Downhillsimplex method. Figure 3 shows the optimizationprocedure, and Table I lists the optimum phases (km.
From the optimized set of phases )m, the field U(u, v)in the hologram plane and the corresponding inten-sity I(u, v) and phase Jk(u, v) are obtained throughEqs. (2) and (3). After clipping the intensity transferfunction to I(u, v) = 1, we get a phase-only fan-outelement with phase tp(u, v), which is shown in Fig. 4.We calculate the reproduced field in the object plane,which now consists of N spots with amplitudes Am' •1, as listed in Table I. The uniformity error is due toclipping of the residual intermodulation terms in theintensity transfer function I(u, v) of the hologram(Fig. 2).
For a nine-spot fan-out element, this first optimiza-tion leads to 99.38% of the incident light beingdiffracted into the nine beams with ±5.35% unifor-mity error.
We have imposed a symmetrical distribution ofphases (m = m+i). Optimization runs without thisrestriction have also been performed. For odd fan-outnumbers, we never found higher efficiencies withnonsymmetrical phases. However, we did find higherefficiencies with nonsymmetrical phases for evenfan-out numbers. Therefore the restriction of symmet-rical phase has been dropped for the even fan-outnumbers.
If uniformity is more important than efficiency, wecontinue with the second optimization step.
Second Optimization
Uniformity errors can be minimized by a secondoptimization procedure, which is shown in Fig. 5. Wenow maintain the optimum phases (km from the firstoptimization and change the amplitudes Am, untilperfect uniformity in the reproduced image (Am' = 1)is obtained after clipping I(u, v) in the hologramplane. The amplitudes Am(n) of the virtual sources arechanged individually at each iteration loop of theoptimization process to correct for the nonuniformamplitudes Am'(n) of the diffracted spots. The virtualsource amplitudes A.(n+l) for the next iteration are
v~uv) |111 FO,|lltl 'M |
output:non-uniform object
Fig. 2. First optimization process for the phases Etm to gain highefficiency. The reconstructed object after clipping is nonuniform
(Am' d 1).
output M(n)
Fig. 3. Numerical optimization of phases for maximum efficiency.
where (Am'(n)) is the average over all the diffractedamplitudes. The ratio Am'W/)I(Am'(n) ) represents theweight of the mth diffracted amplitude and serves asthe correction factor for the next iteration (n + 1).The new optimized set of amplitudes Am and phases(km of the virtual sources for the 9-beam fan-outelement is shown in Table II.
This parameter set produces the intensity I(u, v)and the phase qk(u, v) in the hologram plane. Afterclipping I(u, v) = 1] the reproduced spot amplitudesare perfectly uniform (see Table II). The fan-outphase distribution qk(u, v)(n) is similar to that shownin Fig. 4. This second optimization has reduced theefficiency only slightly, from 99.38% to 99.28%.
IV. Discussion: Results of Numerical Optimization
We have applied our optimization methods to fan-outelements with different numbers N of beams. Thecomputed efficiencies and uniformities resulting fromthe first and second optimization are shown in TableIII.
For the 9-beam element, the computing time on aMicro VAX 3400 computer was 250 ms for theefficiency optimization and 900 ms for the unifor-mity optimization. The computing time for large
6.0 -
5.0-
~4.0-3-_3.0
1.0
0.0--5 -4 -3 -2 -1 0 1 2 3 4 5
u [mm-1 ]
Fig. 4. Phase transfer function c(u, v) of a high-efficiency 9-beamfan-out element.
output : A' m(n)0lm(n)
Fig. 5. Second optimization step (uniformity): 4M(O) comes fromthe first optimization step and (n) counts the number of iterations.
fan-outs can be reduced considerably by choosing agood starting parameter set. Good starting sets can beobtained by cascading optimum solutions of smallerfan-outs, as shown in Fig. 6. For example, an efficient117-beam fan-out is obtained by cascading the twooptimized solutions for N = 9 and N = 13.
Adding or subtracting a few points in the solutionfor a large optimized fan-out also gives a good startingparameter set. For example, the 100-beam fan-outhas been obtained just by taking away one virtualsource in the optimum solution for 101 beams.
Although the results of the optimization have beenimplemented only for a one-dimensional on-axis ele-ment, they can also be used for two-dimensionalelements and for off-axis elements. The two-dimen-sional M x N parameter set is defined by Aij = AiAjand (ij = (i + (Aw, where i E [1 ... M] and j E [1 . . .N]. For example, an optimized 9 x 9 fan-out elementleads to 98.6% efficiency with perfect uniformity.
Reflection fan-out elements can be obtained bydeposition of a reflective coating on a specially de-signed kinoform relief. Another solution is to replaythe fan-out element under total internal reflection
Table II. Perfectly Uniform 9-Beam Fan-Out Element with Amplitudeand Phase of the Virtual Recording Light Sources and the Resulting
conditions. Such elements do not need special reflec-tive coatings and are of great interest for integratedplanar optics. Two-dimensional and total internalreflection fan-out elements are actually under investi-gation.
Another interesting type of element combines fan-out and focusing properties. Focusing can be added bya Fresnel lens superposed to the fan-out phase struc-ture, the whole structure being written in one rasterscan.
The photoresist relief structure of all elementsdiscussed above can be converted into a metal masterrelief by electroplating techniques for mass produc-tion of low-cost replicas.
V. Comparison with Other Optimization Procedures
The first and second optimization steps describedSection III are quick. They lead to fan-out elementswith good efficiency and perfect uniformity. Someother optimization procedures have also been testedfor comparison.
Amplitude Optimization
In the described first optimization step we minimizethe variance of the object irradiance in the hologramplane (expression 8). Wyrowski defines an upper limitfor the diffraction efficiency of phase-only holograms[Eq. (19b) of Ref. 8]. This upper limit is reached whenthe variations of the object field amplitude UI aroundits average value (IUI) are minimum. This leads to a
TU)
Cenc
different optimization criterion, namely,
ff (I I - (I J))2dudv , min. (16)
We have optimized a few fan-outs by following thiscriterion. The optimization process was much slower.It yields the same set of optimum phases.
Phase Spectrum Optimization
In this optimization we work with a set of parametersthat are defined in the hologram plane. The gratingphase function is constructed from a limited sum ofharmonics:
U(u, v) = exp[it,(u, v)],
P
t(u, v) = I A, cos(27rpu).p=1
(17)
(18)
The amplitudes Ap of these harmonics are the optimi-zation parameters, while the efficiency and the unifor-mity in the object plane are the criteria for theoptimization. Again, this optimization process is muchslower. For a 5-beam fan-out element, built with the10 first odd harmonics, 3 min of computing time on aMicroVAX 3400 was necessary to find the sameoptimum phase function with 92.1% efficiency and<0.1% uniformity error, as was already shown inTable III. However, this method is more general andcan easily take into account the limitations for theresolution of the hologram writing system by limitingthe number of harmonics.
VI. Kinoform Fabrication
The above results can be used for the fabrication ofeither an on-axis or an off-axis hologram (recordedwith a reference beam). For our application we haveopted for an on-axis hologram in which the phasefunction is transferred into photoresist as a surfacerelief element. By the time of fabrication of thekinoform, only the first optimization step had beendeveloped. Therefore only the phase *(u, v), whichresults from the first optimization, has been imple-mented.
The element was fabricated at the Paul ScherrerInstitute, Zurich, where a high precision laser-beamwriting system is available.4 A photoresist layer isdeposited on a flat substrate. It is exposed with a
1.0 -
>, 0.8-Cna) 0.6 -
- 0.4-
0.2 -
0.970.95
-4
0.94 1.00 0.98 0.910.86 0.88 0.92
1 I I I 1 l
*2 0 oDiffraction order
4
Fig. 7. Profile thickness of the fabricated fan-out.
Fig. 8. Relative intensities of the fabricated fan-out.
scanning laser beam of controllable intensity. Then itis etched in developer. The etching rate depends onthe received exposure dose. The fidelity of the reliefphase element relies on the knowledge and reproduc-ibility of the photoresist response. Finally the phaseelement is baked and measured with a stylus profilo-meter. One typical measured profile of photoresistthickness is shown in Fig. 7.
The element has been tested by analyzing theFourier image that is produced when illuminatedwith a plane-wave laser beam (Fig. 1). The diffractedbeams are focused onto a CCD camera with a lens of100-mm focal length. The measured relative values ofthe spot intensities are shown in Fig. 8. The unifor-mity is within 7%. The absorption of the substrateand the photoresist has been measured to be 2%. Themeasured efficiency, i.e., the fraction of the transmit-ted light that was focused into the nine spots, was92%.
VII. Conclusions
A two-step method of optimizing the transfer func-tion of fan-out elements has been developed. The firststep optimizes the efficiency. The result is an elementwith minimized variations of the intensity transferfunction. To obtain a pure phase element, the resid-ual intermodulations in the intensity transfer func-
tion are clipped. This introduces uniformity errors.The second step optimizes the uniformity of the purephase element. The efficiency is only slightly reduced.The theoretical efficiency for a perfectly uniformnine-beam fan-out is better than 99%.
Both optimization steps are fast compared withother methods. The computing time for a 9-beamelement is 250 ms for the first step and 900 ms forthe second step on a MicroVAX 3400. Our optimiza-tion parameters are defined in the object plane.Generally, optimization in the hologram plane re-quires more parameters and, consequently, morecomputing power.
The optimization method is easily extendable tolarge and to two-dimensional fan-outs. Good startingparameters and, therefore, a short computing timefor large fan-outs are obtained by cascading optimumsolutions of smaller fan-outs. The optimum solutionfor a 117-beam fan-out with a 97.6% efficiency and aless than 0.1% uniformity error has been obtainedfrom cascading the two optimized solutions for anine-beam and a 13-beam fan-out.
We have recorded an optimized 9-beam fan-out as asurface relief kinoform with a laser-beam writingsystem. The implemented phase is a result of the firstoptimization step and has a theoretical efficiency of99.38% and a uniformity of +5.35%. This elementhas been investigated experimentally. It shows a highefficiency (92%) and a moderate uniformity (± 7%),which is due mainly to fabrication tolerances.
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