Design of thin-film Luneburg lenses for maximum focallength control
Ettore Colombini
The application of thin-film Luneburg lenses to integrated optical circuits will require accurate control oftheir focal length to permit the necessary alignment between the various circuit elements. Of particular in-terest is the design of lenses for application to a silicon-based integrated optical rf spectrum analyzer. Thisstudy analyzes the sensitivity of the focal length of Luneburg lenses to thickness variation at the lens centerresulting from fabrication process tolerances. It is shown that this sensitivity can be minimized by properlyselecting the refractive index of the waveguide material, using a larger focal length and employing a longeroptical wavelength.
1. Introduction
Lenses used in guided-wave optical circuits like theintegrated optical rf spectrum analyzer (IOSA) willrequire both diffraction-limited performance and ac-curate control on focal length. The latter requirementpermits the necessary alignment and focusing betweenthe various elements in the IOSA (particularly thetransform lens and detector array).
The thin-film, Luneburg lens has been proposed' foruse in a silicon-based IOSA and has been demonstrated 2
to yield near diffraction-limited performance. Thefocal length of this lens, however, is particularly sensi-tive to slight variations in the waveguide thickness atthe lens center and places very severe demand on fab-rication process tolerances if the necessary opticalalignment is to, be realized.3 From computed thicknessprofile curves4 for Ta2O5 lenses on a Corning 7059waveguide, it can readily be perceived that a o-Athickness variation can cause a focal length change of0.25 mm for a 1-cm diam lens. When one considers thatthe depth of focus of such a lens may only be 10 gm, itis clear that severe defocusing could result, compro-mising the analyzer's frequency resolution.
Other planar lenses have been proposed which couldpermit the desired focal length control, but these haveother drawbacks. The chirp-grating Bragg diffractionlens made by photolithographic techniques has a highly
The author is with Bell-Northern Research, Ltd., P.O. Box 3511,Station C, Ottawa K1Y 4H7.
reproducible focal length, dependent only on gratinggeometry (the chirp rate) and is insensitive to fabrica-tion process variations. Diffraction-limited perfor-mance with high throughput has been demonstrated,5
although the sidelobe level due to scattering was inferiorto that obtained with the Luneburg lens. Unlike theLuneburg lens, the chirp-grating lens is not circularlysymmetric and can only accept rays incident at theBragg angle. As a result, angular positioning of thegrating would be necessary. If used as the transformlens of an IOSA, there may be further complicationssince the input beam, after being steered by the acousticwave, would arrive at slightly different incident anglesover the device bandwidth. The geodesic lens has alsoproven to yield sufficient control on focal length butrequires extremely high precision diamond machiningtechniques6 to control the depth of the aspherical de-pression to within tolerances of 1 .Lm.
This work investigates the possibility of optimizingthe Luneburg lens design so as to minimize the focallength sensitivity on lens thickness and relax fabricationtolerances to acceptable levels. By examining the de-pendence of this sensitivity on such lens parameters asthe waveguide material index, focal length, and opticalwavelength, a set of optimum design specifications hasresulted.
11. Luneburg Lens Structure
The thin-film Luneburg lens is a dome-shaped cir-cular waveguide overlay which can be made as a sput-tered deposition through an edge-shaped mechanicalmask.2 Although high-index materials like Ta 2O5 havebeen used3 to define lens overlays on Corning 7059 op-tical waveguides, there is no reason why the same ma-terial cannot be used for both guide and lens. Ahigher-index material for the lens may only be advan-
Fig. 1. Cross section of the thin-film Luneburg lens and layeredwaveguid& structure.
tageous in making thinner lenses requiring shortersputtering runs but at the expense of more stringentthickness tolerance requirements.
We propose and have analyzed a lens structure,shbwn in Fig. 1, where both the lens and guide aresputtered with the same material so that no interfaceexists between the two. The superstrate is air, while thesubstrate in silica (SiO2) formed as ail optical isolationlayer on a silicon wafer.
This is essentially a three-layer problem with thecentral guiding layer consisting of both lens and guide.The index profile in the lens region has been previouslycomputed 4 7 to yield the Luneburg design. A family ofcurves appears in Fig. 2 for varying values of 9, the focallength normalized to the lens radius. By applicationof Maxwell's ecuations in layered media3 8 one can re-alize the desired index distribution by controlling thethickness profile in the lefis region of the guiding layer.For a specific waveguide material index n2 and substrateindex n3 the variation of effective index with waveguidethickness is given by the modal dispersion relations.8
The TErn modes for an asymmetric slab waveguide(three-layer case) are described by
curve indicates the single-mode region. Both guide andlens are designed to operate in this region. The guidethickness is chosen so that its operating point is at B onthe curve. For purposes of computation Ithe percent ofthe single-mode region (PSMR) over which the thin-film guide operates has been chosen at 10%. This allowsguide operation to be reasonably far away from cutoffwhile permitting a sufficiently large region from B to Afor the lens index variation to be realized.
For the Luneburg lens, the index (and thickness) ismaximum at the lens center and decreases radiallytoward the periphery where it equals that of the guide.The total index variation is dependent on the designfocal length, with shorter focal lengths requiring agreater variation (yielding greater refraction of light)and thus thicker lenses. For a specific material index
1.20
1.16X0C
; 1.12
- 1.08
E0z 1.04
1.00 L0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Normalized Lens Radius
0.8 0.9 1.0
Fig. 2. Normalized index profiles for generalized Luneburg lenses.
tan(K 2d + mr) = K2(7 + 3)
2- 7173where
m - 0,1,2,....
yj = ko(n2- n2)1/2
K2 = ko(n2 - n2)1/2,
73 = ko(ne - n3)112
ko = 27r/Xo,
X = optical wavelength in free space,
n = index of superstrate,
n2 = index of waveguide,
n3 = index of substrate,
ne = effective waveguide index,
d = waveguide thickness.
An analysis for TM modes yields similar results.Typical dispersion curves for the two lowest order TE
modes appear in Fig. 3, where the solid part of the TEo
(1)
Waveguide Material Index
TEO - /,
()
C0
wM 1
Guide
/
Index
Lend
Waveguide Thickness
Fig. 3. Effective index dependence on waveguide thickness forsingle-mode operation of thin-film guide and lens.
1.9 I I I root search algorithm subject to Eq. (2) and. so that theeffective index at point A permits the full index varia-
Wavelength Independent tion at a specified focal length. The resulting value of1.8 - P.S.M.R. (%) 10.0 n 2 is called the optimum material index and has been
plotted in Fig. 4 as a function of normalized focal length.The optimum waveguide material index is independent
1.7 - of the optical wavelength, and as the focal length in-creases, its value approaches that of the substrate (1.47for silica).
1.6- A family of TEO single-mode dispersion curves isdisplayed in Fig. 5 with normalized focal length as theparameter. Each curve was computed using the opti-
1.5.I. mum value of material index n2. An optical wavelength0 2 4 6 8 10 12 14 16 18 20 of 900 nm, the upper limit for best responsivity in sili-
Normalized Focal Length con, was used in the computation so that an actualthickness could be indicated. Notice that selecting the
Optimum material index for waveguide lens which minimizes material index for each focal length to be at the opti-focal length sensitivity to thickness variations. mum value causes the longer focal length lenses to be
thicker than the shorter ones. This situation is the1.80 | | | 1 |reverse of what would occur had the index not been al-
Varying Norm. Focal Length S withi Wavelength (nm) = 900.0 tered.2 To illustrate the advantages of optimally choosing the
1.70 _ material index, consider the situationi for a lens havingan s value of 10. An optimum design would yield a
3 relatively large thickness range B, in Fig. 5, over whichthe lens profile can be realized. Alternatively, if a
1.60 4 higher material index had been chosen,say that which- I 6 7 is optimum for an s of 2, the allowable thickness varia-
1--K / /- -0112 -tion would be restricted to the range A, with the slope1.50 18 at the upper index limit being much steeper than in the
f ac - -- -20 previous case. The result would be a much higher focalP. SMR. M_ 10B0 B length sensitivity to thickness variation.
.40 To further quantify the sensitivity, Pig. 6 shows the0.00 0.40 0.80 1.20 1.60 2.00 dependence of the normalized focal length on the
Waveguide Thickness (jum) thickness at the lens center for varying values of mate-rial index. The least sensitive situation occurs at the
Single TEo mode dispersion curves for varying normalized left most part of each curve, where the index is at itsfocal length. optimum value. It is clear that the sensitivity increases
drastically with increasing s on each curve, particularlyfor higher values of material index. At the optimumrnger focal length lenses will operate over ever de- operating point, occurring at the lowest s value on each
3ing portions of region AB in Fig. 3, with the higher curve, the slope is nearly constant from curve to curve.Al.1 - P I . . uvteslphsnalecntnrrmcuv ocre
index limit A, moving rom A toward point . In sodoing, the sensitivity of the focal length to thicknesschanges, effectively increasing due to the increasingslope at A'.
111. Focal Length Sensitivity
The key behind minimizing the sensitivity of the focallength to thickness changes lies in choosing the mini-mum value of material index n 2, which permits the at-tainment of the full lens index variation while operatingin a single mode. The value of hod at the cutoff of theTE1 mode is 11 1 1
kod = n [tan- - 1/2 + rRewriting the2 d r I ()
Rewriting the dispersion relation (1) as
2.00
E 1.60
c, 1.20
a- 0.80
.5 0.40-J
(2)
f( ) t K dK2('Y + 73)0 f (n 2) = tanK2d - =K0 (-K2 c 71m3
the value of material index n2 can be determined via a
0.000 2 4 6 8 10 12 14
Normalized Focal Length16 18 20
Fig. 6. Dependence of focal length on the overlay film thickness atthe center of the Luneburg lens for varying optimum waveguide
Fig. 7. Evaluation of depth of focus from lens geometry.
This indicates that for an optimum design, the changein normalized focal length As for an incremental changein thickness Ad is nearly independent of s.
The focal length sensitivity given by the slope As/lAdis found to be inversely proportional to wavelengthmaking a longer wavelength more desirable. This resultfollows from the fact that in Figs. 5 and 6 the wavelengthwas soley used to denormalize the thickness whichwould otherwise have yielded the sensitivity as
As As)X0
A(d/Xo) Ad
IV. Optimum Lens Design
The depth of focus of the lens can be used to establisha criterion for acceptable focal length variation. Inother words it could be deemed acceptable for the focallength to vary by an amount not exceeding the focusingdepth of the lens. This would guarantee the retentionof a diffraction-limited spot at the design focal length.Figure 7 illustrates the lens geometry, and the depth offocus is given by
flg \a
where F = focal length,a = lens aperture,
ng = effective guide index, anda = aperture illumination factor.
Since the depth of focus varies as the square of the focallength, clearly a longer focal length should relax toler-ance requirements on the lens thickness.
The variation As in normalized focal length due to aspecified incremental lens thickness change will resultin an amount of defocusing given by
AF = As(D/2), (6)
where D = lens diameter. The normalized defocusingcan then be defined3 by combining Eqs. (5) and (6) toyield
AsD2
ND= 2 1. (7)
ng tarEquation (7) stipulates that the normalized defocusing
should not exceed unity if retention of a diffraction-limited spot at the design focal length is to be achieved.The inequality can be rewritten as
As < F ctXo
s -a 2 ng(8)
The left side of Eq. (8) represents the fractional changein focal length for an incremental thickness change ata given wavelength, that can be tolerated as a result ofthe lens parameters specified on the right side. Theseparameters are subject to geometrical, material, anddevice performance constraints.
For the IOSA, the aperture a of the transform lens isspecified uniquely by the desired frequency resolu-tion 9 :
a = ()I(bf), (9)
where va = acoustic velocity in the acoustooptic regionand f = frequency resolution.
A smaller frequency resolution will require a largerlens aperture and thus from Eq. (8), tighter focal lengthcontrol as expected. The focal length F is constrainedby space limitations on the 10-cm silicon wafer on whichthe IOSA would be fabricated. The upper wavelengthlimit is subject to the spectral responsivity of silicondetectors.
The inequality (8) is graphically displayed in Fig. 8to permit evaluation of the optimum Luneburg lensparameters which minimize fabricational toleranceswhile satisfying design constraints. The solid curvesrepresent the left side of the inequality and show thepercent change in focal length resulting from a specifiedlens thickness tolerance at an optical wavelength of 900nm. These results follow immediately from Fig. 6 byevaluating the slope at the optimum operating point.Thus, for each s value in Fig. 8, the optimum materialindex is assumed.
The dashed horizontal lines in Fig. 8 depict the rightside of the inequality and represent an upper boundbeyond which defocusing will occur and degrade thespecified frequency resolution. The only parameter
_ ~~ Frequency Resolution (MHz)Thickness Tolerance (A)
- \ IWavelength = 900 nm\100 P.S.M.R. = 10%
I ~~~~~ 1 0 9~~~~I = 1---
0 2 4 6 8 10 12 14 16 18 20
Normalized Focal Length
Fig. 8. Percent change in focal length at specific lens thickness tol-erances assuming an optimum material index at each value of nor-malized focal length. Lens design constraints are indicated as hori-
zontal lines.
which is slightly s dependent is ng, the effective guideindex, since the material index changes with s and thePSMR is fixed at 10%. However, this effect is negligi-ble, and the right side of Eq. (8) can be consideredconstant in s. The optical aperture a was computedfrom Eq. (9) with an acoustic velocity of 3200 m/sec.This yields apertures of 3.2 and 2.1 mm for frequencyresolutions of 1.0 and 1.5 MHz, respectively. A focallength of 40 mm was assumed, and the effective guideindex was 1.475. For truncated Gaussian illuminationthe factor a was taken to be 1.41.
The results clearly show that to relax the fabricationtolerance on lens thickness, longer focal lengths shouldbe employed. On the other hand, the choice of materialindex is somewhat restricted by the availability of lowloss waveguide glasses. Sputtered Corning 7059waveguides 3 exhibit a refractive index of 1.565. FromFig. 4, this material index is optimum for an s value of8.
Once the focal length is specified, fixing the s valuewill essentially determine the lens diameter. Apartfrom the waveguide material constraint, the diametershould be chosen to be at least twice the optical apertureso that lateral aberration at the lens edge due to animperfect lens profile will not significantly degrade lensperformance. 3 In this case, an s value of 8 will yield adiameter of 10 mm so that only 32% of the lens needs tobe apertured making near-diffraction-limited perfor-mance readily achievable.
For such a lens and a frequency resolution of 1 MHz,a tolerance on the center thickness of about 63 A or±0.84% of the lens thickness is required to maintain adiffraction-limited spot at the 40-mm design focallength. This thickness control is possible in practicesince the sputtering deposition process is rather slow,yielding typically a few nanometers per minute.
V. Discussion
An investigation has been carried out to determinethe sensitivity of the focal length of Luneburg lenses tothickness variation at the lens center. It is shown thatat any particular focal length, there exists an optimum
waveguide material index which will minimize thissensitivity. In addition, the use of a larger focal lengthand longer optical wavelength will further relax thethickness tolerance requirements.
The focal length sensitivity has been quantified.This, along with the stipulation of the depth of focus asbeing the criterion for acceptable defocusing, have re-sulted in a design procedure which permits evaluationof the optimum Luneburg lens parameters. These arechosen to minimize the fabricational tolerances whilesatisfying geometrical, material, and device perfor-mance constraints.
A realistic example of a lens design for application toan IOSA as the transform lens has demonstrated thatit is possible to fabricate near-diffraction-limited lenseswith sufficient focal length control to yield a 1-MHzfrequency resolution. By employing an optimum de-sign, the required thickness control was relaxed to ±63A representing 0.84% of the lens thickness. Had anonoptimum material index been used, the requiredtolerance to maintain a diffraction-limited spot at thedesign focal length could readily approach impossiblevalues of 1 A and below.
This work was performed for Defence Research Es-tablishment Ottawa, Department of National Defence,Canada under contract 2SR79-00034.
References1. D. B. Anderson, IEEE Spectrum 15, 22 (1978).2. S. K. Yao, J. Appl. Phys. 50, 3390 (1979).3. S. K. Yao et al., Appl. Opt. 18,4067 (1979).4. W. H. Southwell, J. Opt. Soc. Am. 67,1010 (1977).5. S. K. Yao and D. E. Thompson, Appl. Phys. Lett. 33, 635
(1978).6. D. Mergerian et al., in Digest of Topical Meeting on Integrated
and Guided-Wave Optics (Optical Society of America, Washing-ton, D.C., 1980), paper ME4.
7. E. Colombini, "Index profile computation for the generalizedLuneburg lens," to be published in J. Opt. Soc. Am.
8. P. K. Tien, Appl. Opt. 10, 2395 (1971).9. D. B. Anderson et al., IEEE J. Quantum Electron. QE-13, 268
