Design of optical-waveguide homogeneous refracting lenses
Giancarlo C. Righini and Giuseppe Molesini
A class of multielement homogeneous refracting lenses for optical waveguides is presented, and their designprocedure is described. These lenses appear to be particularly suitable for use in integrated opticalprocessors, as they can be fabricated by fully planar photolithography and reliable diffusion processes eitherin glass or in lithium niobate; their optical characteristics can be very good. For example, the design of anf/3.3 waveguide optical system, constituted by five elements, with no field curvature and very well correctedup to a 20° total field of view, is presented.
1. Introduction
Waveguide lenses are a key component of manyintegrated optical devices, from multiplexers to signalprocessors. The functions accomplished by wave-guide lenses range from the simple collimation of adiverging laser beam, as one coupled from a solid statelaser, to the Fourier transforming of a spatially modu-lated signal. Most applications require nearly diffrac-tion-limited operation over a certain field angle to-gether with a short focal length to keep the device sizesmall. In any case fabrication techniques which aresuitable for batch processing, e.g., making use of fullyplanar photolithographic techniques, are greatly pre-ferred.
Geodesicl- 5 and Luneburg 6-9 lenses have demon-strated very good overall focusing properties. Be-cause of their circular symmetry, they also have intrin-sically superior off-axis performance. However, thereare tooling and optical alignment difficulties whichseriously limit the possibility of mass production andkeep their cost high. Both these problems are over-come by planar structures such as chirp-grating,Bragg, and Fresnel lenses10- 4; on the other hand, theperformance of these lenses is limited either by thefield angle, which is quite small for Bragg and chirp-grating structures, or by the errors in the photolitho-graphic reproduction of the ideal grating pattern.
Giancarlo Righini is with CNR Instituto di Ricerca sulle OndeElettromagnetiche, Via Panciatichi 64, Florence, 50127, Italy; G.Molesini is with National Institute of Optics, Largo E. Fermi 5,50125 Florence, Italy.
Some recent papers15-' 9 have demonstrated the ad-vantage of using homogeneous refracting lenses, whichare not affected by any of the previous limitations. Asa matter of fact, very good overall performance is of-fered by a new class of corrected homogeneous acircu-lar refractive lens systems (CHARLES) that we havesuggested very recently.20 Here we present a detailedanalysis of the design procedure of these lenses basedon the extension to integrated optical circuits of theconventional optical design techniques. For example,a five-element f/3.3 lens is described, and its perfor-mance is analyzed.
11. Features of Optical Design in Thin-Film Waveguides
Homogeneous integrated optical lenses consist ofareas of planar waveguide of high effective refractiveindex ne2 immersed in the lower-effective-index ne1waveguide, where the optical waves propagate. Such astructure can be easily implemented thanks to recentdevelopments in two-stage waveguide formation pro-cesses, which have provided simple and reliable tech-niques to produce two waveguides, with different re-fractive index, on a common substrate. For example,the use of Ti-indiffusion and proton-exchange in lithi-um niobate19,2' and the use of a double ion-exchange inglass22 appear quite advantageous for producing re-fracting optical elements.
Since the beginning of integrated optics it has beenproved that Snell's law applies also to planar-wave-guide optical boundaries, provided that one uses theeffective refractive index of the propagation mode(which depends on the thickness and refractive indexof the guiding layer, and of the order of the mode)instead of the refractive index of the material.2324
Therefore, it has been quite obvious to apply the prin-ciples of classical lens design also in the design ofwaveguide lenses, even if focusing with refracting ele-ments inside a planar optical waveguide implies anumber of problems unusual to bulk optics.
First, the refractive index n, defined as the ratio ne2/
nel, is very close to unity, calling either for a highnumber of refractions or for strong curvatures to endup with some focusing power. Compactness is also ofparticular concern in integrated optics, so that specialcare has to be given to the problem of reducing thenumber of refracting boundaries and the thicknessoccupancy to a minimum, still keeping the opticalperformance at the desired level.
A major peculiarity of optical design in planar wave-guides is given by the linear aperture there of concernand by field angles that extend only in the tangentialplane. Referring to standard design methods in bulkoptics, both features can be taken care of by consider-ing tangential rays. An interesting consequence of theplanar geometry is that the spot diagram reduces to asegment bar produced by a comb of rays at the pupil,linearly accounting for the height ray contributions tothe focus energy distribution. Because of the manu-facturing processes used in making planar waveguidelenses and their narrow operational wavelength band,optical design can conveniently use only a single re-fracting material and a single value of the refractiveindex at a time. Also, since light propagates and isprocessed inside the same waveguide where the detec-tor itself can be immersed, the number of refractingboundaries can be made uneven.
Standard optimization procedures are further sim-plified by relaxed constraints on central and edgethicknesses; no minimum edge thickness needs to beimposed, although some minimum axial thickness isrequired to be able to actually draw the mask necessaryfor fabrication.
Ill. System Compactness: First-Order Analysis
The packing of the refracting boundaries within aminimum thickness occupancy can be preliminarlystudied in the mainframe of first-order optics. Theproblem is set by assigning a fixed overall thickness Tand solving for the resulting optical power due to full-arch circular boundaries as the number of such bound-aries is varied. The linear aperture extent D is alsospecified (Fig. 1).
To properly insert a single refracting circular bound-ary within the area D*T, the condition D 2T has tobe satisfied. Thickness partitions T = Tk are thenmade (k integer), thus defining a series of k equalrefracting boundaries within T. The radius Rk of eachcircular boundary may be obtained from the formula
Rk = (T)2 + (D )22Tk
(1)
It is useful to define a geometrical shape factor mk asmk = D/(2Tk) with the constraint mk 2 1. The previ-ous formula may be written accordingly:
Rk= (1 + 2). (2)
For a single refracting boundary of the kth partitionthe associated power (1/fsingle)k is given by
1 -(n-1) = (n -) -(single)k = - T h1+
(3)
After combination of k refracting boundaries, ne-glecting thickness effects, the resulting power (1/ftotal)kis given by
( 1 ' (n- 1)2k(htotal )k (1 + M ) Tk
(4)
The corresponding relative aperture f/No. = fD is
(V/N°-)k 4k-I + mk(f~.h 4k(n - )Mk (5)
The asymptotic value (f/No.)kH is worked out as
(f/No.)k~, - 8T(n - 1)
leading to the result
(f/No.)k I 1
(f/N-)k-X m2
(6)
(7)
Independently of the refractive index, the above ex-pression states, for example, that a geometrical shapefactor mk = 5 already allows the asymptotic relativeaperture to be achieved within 4%.
Although obtained with some simplifications, theabove expression accounts satisfactorily for real lensesand paraxial ray tracing computations, as it appearsfrom Fig. 2 where continuous values for the geometri-cal shape factor have been used in abscissa.
One sees that increasing the number of refractingboundaries within the assigned area does improve theresulting relative aperture. Such improvement, how-ever, may not be worthwhile beyond a certain value ofthe geometrical shape factor, as it would imply anexcessive number of boundaries. This leads to theconcept of an optimum factor mo, resulting from theabove trade-off and singling out the most effectiverefracting boundary configuration with respect to op-tical power and thickness occupancy.
Assuming that a value mo of the optimum shapefactor is selected, one can investigate the relationshipbetween thickness occupancy and expected relativeaperture. For this purpose a series of h optimumboundaries (h integer) at refractive index n is cascad-ed, and the resulting f/No. is evaluated in the first-order approximation. Still neglecting thickness ef-fects, in analogy with Eqs. (4) and (5), one can write
Fig. 2. Percent relative aperture (f/No.)k/(f/No.)k -t vs the shapefactor m. The dashed area corresponds to a forbidden region ac-cording to the condition mk 2 1. The full line represents thebehavior after the first-order approximation; the dots indicate the
numerical results from a few design procedures.
( 1 2h(n-1) (8)ftotal h T0(1 + mO)
with To = D/(2m0 ). The final expression results:
1 + mO
(f/No.)h 4hm(n - 1)
Such an expression is reasonably well verified numeri-cally, at least for a number of boundaries not too large,as it appears from the real examples reported in TableI assuming mo = 5. Data suggest, for example, thatwith n = 1.055 a compact /13 system can in principle beobtained with only eight boundaries. The boundariesof course should be faced two by two to give four planarlenses in the usual sense.
In practice, however, such upper-limit performancehas to be relaxed to some extent. So far, results havebeen obtained referring to on-axis operation with first-order optics neglecting any analysis of optical aberra-tions. Going to more realistic systems, which should
Table 1. Relative Aperture Obtainable with Specified Number h ofOptimum Boundaries (m = 5) at Different Refractive Indices
a Results obtained from complete lens designs.b Results calculated for comparison from Eq. (9).
cover an angular field of view of practical interest andshow reduced aberration, a departure from optimumcompactness has to be accepted. Such a departure canbe quite large. In fact, using circular boundaries only,an optical system, to be corrected over a total fieldangle of the order of 150, needs about thirty lenses.17"18
However, properly acting on the available degrees offreedom, one can get a nearly diffraction-limited f/3.3optical system by using ten acircular boundaries: thesystem, which exhibits a useful field of view of 20°(±100), is described in the following.
IV. Acircular Refracting Boundaries
In planar optical waveguides no advantage is in prin-ciple attributed to circular refracting boundaries bythe manufacturing process. Thus one can assume ageneral boundary profile z(y) mathematically ex-pressed by the equation
z(y) = Cy211 + [1- (1 + K)C2 y 2]1/21-1
+ Aly 4 + A2y 6 + A 3y8 + A 4 y'0 , (10)
where C is the paraxial curvature, K is the conic coeffi-cient, and the Ai coefficient are the ith-order deforma-tion terms.25 So far, compactness first-order analysisonly involved the paraxial curvature C. According toEq. (10), now each boundary contributes five moreshape parameters. Clearly, due to the increase in thedegrees of freedom, the optimization problem rapidlygrows with the number of boundaries. In practice,owing to the improved performance of acircular pro-files, the number of refracting boundaries can be keptreasonably small without loosening the compactness ofthe system or its quality. Available computationpackages are thus suitable to deal reliably with systemsof actual interest.
With the help of one such package26 various focusingsystems have been designed and optimized to diffrac-tion-limited performance covering wide ranges of rela-tive aperture and field angle. The number of refract-ing boundaries has been varied from 6 to 18,corresponding to a number of ordinary lenses from 3 to9. In Fig. 3 symbols represent successful attempts todesign diffraction-limited lenses at given f/Nos., semi-field angle, and specified number of boundaries, therefractive index being set at n = 1.055. The parameterin abscissa (f/No.)-1 represents the maximum theoret-ical spatial frequency in wave units, the referencewavelength being the one in the waveguide, i.e., XO/nei.For a fixed number of refracting boundaries, Fig. 3singles out existence areas (on the left of each brokenline considered) where diffraction-limited perfor-mance can be achieved. The frontier extends furtherin the semifield angle against the f/No.-' domain as thenumber of refracting boundaries is increased. Ac-cording to such areas, for example, to successfully de-sign a diffraction-limited f/4 lens covering ±100 offield of view, at least eight refracting boundaries haveto be included in the optical system.
As the refractive index varies, the above areas andfrontiers evolve, as shown in Fig. 4. Thus an /4 sys-tem requiring eight boundaries for n = 1.055 would
Fig. 3. Acircular optical systems optimized to diffraction limitoperation; each symbol marks a successful design. The numbers inthe plot indicate the numbers of boundaries required to design thediffraction-limited system having given a semifield angle and f/No.Equivalent symbols correspond to systems with the same number of
boundaries.
1.09
n 6 a 10 12
1.07-
1.05
1.03
M O1 f/S Iji. f12.5 I/ 2
(f, )1Fig. 4. Acircular optical systems with a given field angle of 80optimized to diffraction-limited operation as a function of the avail-able effective index ratio n. The number of refracting boundariesfor each optical system is indicated in the figure close to the corre-
sponding symbol.
require ten boundaries if n is reduced to 1.04 or just sixboundaries if a value n = 1.07 is available.
The above figures on the number of required bound-aries are a little in excess of the expected ones after Eq.(9) due to the actual nonzero field angle of interest.As a first approximation, after Fig. 3 an empiricaldependence of the achievable relative aperture on thefield angle 0 at a fixed number of boundaries can bewritten as
(f/No.J 1) 6 n (f/No.'))0.0(1 - aO). (11)
Still after Fig. 3, with expressed in degrees, a fairvalue for the parameter a is a = 0.03 for any number ofboundaries. The value of (f/No.)o=0 can be obtainedfrom Eq. (9) for each specified number h of boundaries.Combining Eqs. (9) and (11) one can derive the generalexpression
1+ m2
4mo(n - 1)(1 - aO)(f/No.) (12)
giving the approximate value of the number h (con-tinuing) of boundaries required to obtain the chosenrelative aperture over the semifield angle 0 when therefractive index n is available. Once m and h arechosen, on the basis of Eq. (12), one can also study the
'4.64
1.07
20 1.03
Fig.5. Three-dimensional plot of the achievable f/No. as a functionof n and 0 according to Eq. (12) for a = 0.03 with m = 5 and h = 10.The refractive index n ranges from 1.03 to 1.07 and the semifield
angle 0 from 0 to 200.
obtainable f/No.(n,O). Figure 5 shows a 3-D plot ofsuch a function with m = 5 and h = 10 within aspecified range of n and 0 values.
Quite obviously, the validity of Eq. (12) is approxi-mate since it is obtained by making use of a number ofsimplifying assumptions, and it is intended to give justa rule of thumb for selecting the minimal configurationto start with in the optimization process. However, byselecting an optimum shape factor mo of the order of 5,the agreement with Eq. (12) of the source data for Figs.3 and 4, corresponding to the results of independentoptimization processes, is extremely good.
V. Numerical Example
In practice, the optimization process produces ap-parent departures in the single boundaries from theclaimed optimum shape factor. The overall compact-ness evaluation, however, averaging over the singlegeometrical characteristics, is still in agreement withthat optimum. This is confirmed by the validity ofEq. (12) with m = 5 as reported above.
To show what a refractive lens with acircular bound-aries looks like and to give a numerical example, com-putation details of an /13.3 lens are given here. Such alens, working from infinity to focus, is made of tenboundaries and is designed to be diffraction limitedover a field angle of ±100 with zero field curvature.The refractive index is assumed n = 1.049, a value thatcorresponds to the ratio of an effective extraordinaryrefractive index 2.326 of the TIPE (lens) region to the2.217 index of the TI region in a lithium niobate wave-
guide. The stop aperture is set at the first refractingboundary.
After the optimization process, the layout of theresulting lens is shown in Fig. 6. The final configura-tion values are detailed in Table II. The stop linearaperture is set at 6 mm, causing an effective focallength of 19.85 mm and an overall thickness of 25.30mm from first boundary to image. Tangential rayaberration in microns at different field angles is plot-ted in Fig. 7 as a function of the ray height at theaperture. Scaling down the linear aperture, the focallength and geometrical aberrations are scaled as well.The outstanding first-order parameters of the lensdesigned are listed in Table III.
As far as tolerances on the refractive index are con-cerned, the numerical analysis reveals that freedomfrom aberration is kept over variations of n of the orderof +0.001, provided the best image plane is suitablyshifted. According to first-order analysis the focusshift Af due to a refractive index variation An is of theorder of
100704000
Fig. 6. Layout and ray tracing of a ten-boundary f/3.3 correctedhomogeneous acircular refractive lens system. The detailed charac-
teristics of the lens are reported in Table II.
Af = -fAn/(n - 1). (13)
Referring to the designed lens, whose data are givenin Tables II and III, Eq. (13) would indicate Af = 0.405mm for a An = 0.001. This value is only slightly larger(-10%) than the actual focal shift resulting from thelens design repeated by using for the refractive indexthe corrected value n + An.
An accurate control of the diffusion processes, andthus of the waveguides effective indices, is, therefore,necessary; otherwise an experimental check of the fo-cal length has to be done before assembling the othercomponents of the integrated optical circuit.
VI. Conclusions
The use of conventional optical design techniques inthe field of waveguide optical systems has proved to beeasy and very fruitful. First-order analysis has sug-gested a practical concept of optimum lens system,
Table . First-Order Characteristics of the Lens Specified in Table 11
Conjugation Infinity to focusRefractive index 1.049Reference wavelength 850 nmEffective focal length 19.8518 mmFront focal length -14.1319 mmBack focal length and image distance 14.7845 mmThickness occupancy 10.5177 mmLength from first boundary to image 25.3022 mmf/No. 3.3086Effective pupil extent 6 mmStop location At first refracting boundaryExit pupil extent 8.4285 mmExit pupil location -13.1022 mmSemifield angle 10°Paraxial image height 3.5004 mm
Table 11. Lens Geometrical Data
Boundary number Acircular profile Thickness or spacing Aperture extent
Note: Linear dimensions are in millimeters. Thickness or spacing is the axial distance to nextrefracting boundary. Profile A(h) of hth boundary is specified by the values of the parametersaccording to symbols in Eq. (10).
Insertion losses, which can be caused mainly by re-flection and modal mismatch at each refracting bound-ary, are expected to be low: a complete analysis will bereported elsewhere. Masks of CHARLES lenses havealready been generated by computer drawing and pho-toreduction. Experimental devices are being fabricat-ed both in glass and lithium niobate.
One of the authors (GCR) wishes to thank VeraRusso and Peter J. R. Laybourn for stimulating discus-sions.
This work has been supported by CNR's SpecialProject on Materials and Devices for Solid-State Elec-tronics.
0.40 RELATIVEFIELD HEIGHT
( 4.00 )-0.0050
0.-0050
0.00 RELATIVEFIELD HEIGHT
C 0.00
-0 .0050Fig. 7. Residual ray aberration (mm) as a function of aperture forfour field heights (0,4,7, and 100) of the CHARLES lens depicted in
Fig. 6. The reference wavelength is 850 nm.
matching the two requirements of size compactnessand of a limited number of refracting boundaries. Arule of thumb has been derived numerically, whichallows one to select the minimal configuration to startfrom in the design optimization process.
Various optical systems of the CHARLES type canbe designed: their characteristics make them verysuitable for utilization in integrated optical circuitsand particularly in signal processing devices. In factthey are capable of near-to-diffraction-limit operationover large field angles and can be mass-fabricated bythe usual waveguide diffusion processes through easilyreproducible mask patterns with clear advantages withrespect to other kinds of waveguide lens. For exam-ple, the design of a five-element system with n = ne2/n, = 1.049 is presented: this f/3.3 lens is capable ofdiffraction-limited operation over a total field of 20°with zero field curvature.
References
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4. S. Sottini, V. Russo, and G. C. Righini, "General Solution of theProblem of Perfect Geodesic Lenses for Integrated Optics," J.Opt. Soc. Am. 69, 1248 (1979).
5. G. F. Doughty, R. M. De La Rue, J. Singh, J. F. Smith, and S.Wright, "Fabrication Techniques for Geodesic Lenses in Lithi-um Niobate," IEEE Trans. Components, Hybrids, and Manu-facturing Technology CHMT-5, 2, 205 (1982).
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13. C. Warren, S. Forouhar, S. K. Yao, and W. S. C. Chang, "DoubleIon Exchanged Chirp Grating Lens in Lithium Niobate Wave-guides," Appl. Phys. Lett. 43, 4245 (1983).
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19. D. Y. Zang and C. S. Tsai, "Titanium-Indiffused Proton-Ex-
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21. A. L. Dawar, S. M. Al-Shukri, R. M. De La Rue, A. C. G. Nutt,
and G. Stewart, "Fabrication and Characterization of Titanium-Indiffused Proton-Exchanged Optical Waveguides in Y-LiNbO 3," Appl. Opt. 25, 1495 (1986).
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NASA continued from page 4160
ga .
This work was done by Robert Miserentino of Langley ResearchCenter and William C. White of Wyle Laboratories. Refer to LAR-
13731.
19Channels Fig. 3. Low-angle light-Validation of sensitive system can track
Blnary-to-Blnary- Data CommandCoded4ecImal a the moving leading edge of a
Encoder model.Binay-CodedDecmal 8 Unes
Data |
Binary-CodedDecimal 8 Unes
Output forEutemal DigitalProcessor
Digital-to-Analog
Converter
AnalogSignalConditioner
Output for ExternalAnalog Control Clrcult
as small as ordinary office light. The model can be as much as 1.8 maway and moving, and it does not have to be moving only in the planeof the sensing device. The field of view of each sensing element is acone less than 8° wide. The output from each sensing element isinterpreted by electronic latches to keep track of the moving leadingedge of the model. The system can locate a model that has littlemass and is as small as a human finger moving at large amplitudes,and it does not intrude on the model being measured.
While motivated by requirements for large space structures, thisdevice could have other uses, including the opening and closing ofautomatic doors as far and as fast as needed. This type of low-anglelight-sensitive device can be designed as a self-contained unit tomeasure large amplitudes, even at low frequencies where accelerom-eters are not adequate.
Fast data acquisition for mass spectrometerThe data generated by a time-of-flight mass spectrometer are
captured by a fast data-acquisition system. The system relies on afast, compact waveform digitizer with 32K memory that can bereadily coupled to a personal computer. With this newly availabledigitizer, the system captures all the mass peaks on each 25- to 35-Jscycle of the spectrometer. The computer conveniently controls thecollection and management of data. It automates the data-acquisi-tion process, including mass calibration, logging of pertinent experi-mental parameters, and presentation of the recorded spectral data ina variety of ways.
The system samples vapor plumes produced by laser-pulse heat-
ing of various materials. After the 1-ms pulse of vapor that follows
each laser pulse, the system displays the following on video screens:* a 3-D view of the time-resolved spectra from the vapor with the
major mass peaks labeled by their ratios of mass to change and* a presentation of laser intensity vs time and total ion current vs
time. Simultaneously, the system prints out pertinent experimen-tal parameters, including the identity of the sample, data and time,laser-power-supply voltage, pressure in the vacuum chamber, andpotential of the ion deflector. The operator can request prints andplots of data (see Fig. 4). In addition, the system can be used tocapture spectra in the averaging mode over much longer periods, forexample, several tenths of a second or more. These longer measure-ments analyze vapors from continuous laser heating or calibrationgases. Except for the three-dimensional format, the same displaysand printouts are available.
..�I92
--
a2
12 24 2 36mia (Ratio of Mass to Charge)
Fig. 4. Pseudo-3-D plot gives a time-resolved view of the massspectra.
