70-nm-bandwidth achromatic waveguide coupler

Sergio B. Mendes, Lifeng Li, James J. Burke, John E. Lee, and S. Scott Saavedra

We report a general approach to the design of broadband waveguide couplers. A double-parallel gratingassembly is used to cancel the first chromatic order, and a proper choice of prism glass and base angle ismade to compensate for the second chromatic order. The technique was applied to a Corning glass7059 waveguide, and a spectral bandwidth of 70 nm was measured by the use of two complementaryprocedures.Key words: Achromatic coupler, grating coupler, prism coupler, planar waveguide, integrated optics,

surface spectroscopy.

1. Introduction

Multimode cylindrical waveguides have been widelyused for evanescent-wave spectroscopy of interfacialfilms and optically based chemical sensing.1–3 Analternative geometry that has seen increasing use isthe planar integrated optical waveguide 1IOW2.Planar IOW’s have several advantages over multi-mode fibers, including a much higher density of totalreflection 1up to several thousand reflections percentimeter of beam propagation2, which yields a con-comitant increase in path length. The sensitivityenhancement makes it possible to measure attenu-ated total reflection 1ATR2 from interfacial samplesthat would otherwise be problematic. Examples in-clude dissolved and adsorbed dyes,4–7 adsorbed pro-tein films,8,9 and Langmuir–Blodgett films.10The major disadvantage of measuring ATR with

planar waveguides is spectral bandwidth. In a givenconfiguration 1i.e., launch angle2, planar waveguideincouplers are usually efficient for only a narrowspectral range. Thus, to our knowledge, all planarIOW–ATR measurements reported to date have usedeither a monochromatic source or a narrow spectralband isolated from a polychromatic source.Different configurations to increase the spectral

bandwidth of planar waveguide couplers have beenproposed. Spaulding and Morris11,12 suggested two

S. B. Mendes, L. Li, and J. J. Burke are with the Optical SciencesCenter, University of Arizona, Tucson, Arizona 85721. J. E. LeeandS. S. Saavedra arewith theDepartment ofChemistry,University ofArizona, Tucson, Arizona 85721. E-mail may be addressed tosermende@ccit.arizona.edu.Received 30 November 1994; revised manuscript received 6

March 1995.0003-6935@95@276180-07$06.00@0.

r 1995 Optical Society of America.

6180 APPLIED OPTICS @ Vol. 34, No. 27 @ 20 September 1995

different approaches to the problem: 1a2 a surface-relief transmission grating on top of a prism couplerand 1b2 a grating coupler combined with a transmis-sion grating tilted at a specific angle. The reportedexperimental full-widths at half-maximum 1FWHM’s2were 41 nm and 13 nm for prism–grating and double-grating couplers, respectively. By using a transmis-sion-volume hologram tilted at an angle with respectto a grating coupler, Hetherington et al.13 were able toget a 5-nm bandwidth; Strasser and Gupta14 achieveda 17-nm bandwidth by combining a grating couplerwith a tilted reflecting grating. Recently, Li andBrazas15 obtained a 45-nm bandwidth by using aprism and a pair of parallel gratings, which greatlysimplifies alignment between elements. All experi-mental demonstrations above rely on canceling thefirst derivative of the input-coupling angle with re-spect to wavelength.In this paper we introduce a different approach to

the design of achromatic couplers. Instead of work-ingwith coupling angles,11–14 we describe the achroma-tization problem as an effective-index-matching pro-cess between waveguide and coupler, which is directlyrelated to coupling efficiency.16 The effective-indexmismatch function is defined and expanded into aTaylor series in which the coefficients are defined aschromatic orders. The achromatization process takesplace through correction of increasingly higher orders.The method is general and can be applied to anyspecific configuration with the appropriate descrip-tion of the effective index for each coupler.We follow the general configuration of Li and

Brazas,15 but we search for a design that corrects notonly the first but also the second chromatic order bymaking use of available parameters that have notbeen previously exploited in the configuration. Thestructure shown in Fig. 1, which combines two grat-

ings and a prism, has simplicity and sufficient degreesof freedom. The two gratings are parallel to thewaveguide, avoiding the critical alignment requiredin previous designs,11–14 and can be fabricated on asingle slide. As we show below, the gratings aredesigned with slightly different spatial frequencies tofunction as an equivalent long-period grating, andtheir linear wavelength dependence is used to correctthe first chromatic order. The prism offers the de-grees of freedom 1choices of glass and base angle2needed to correct the second chromatic order. Froman improved design we were able to extend theinput-coupling spectral bandwidth much beyond thatof previously reported experiments.12–15

2. Theory

The problem of broadband coupling into a planarwaveguide can be simply formulated in terms of thematching of the effective index 1or b, the projection ofwave vector along the direction of mode propagation2of the waveguide with that of the coupler 1prism,grating, or combination2 for a broad spectral range.For this purpose we define an effective-index mis-match function F as the difference between wave-guide and coupler effective indices:

F1l2 ; Nw1l2 2 Np1l2 2 Ng1l2, 112

where each term on the right-hand side is defined,referring to Fig. 1, as follows:

1a2 Nw is the waveguide effective index obtainedfrom the well-known characteristic equation

2p

ltw1nw2 2 Nw

221@2

5 mp 1 tan2131nwnc 22r

1Nw2 2 nc2

nw2 2 Nw221@2

41 tan2131nwns 2

2r

1Nw2 2 ns2

nw2 2 Nw221@2

4 , 122

where r 5 0 applies to TE mmodes and r 5 1 applies toTMm modes. We note that Eq. 122 has an explicitwavelength dependence and an implicit dependencedue to the dispersion of materials involved.

Fig. 1. Schematic representation of overall structure and symbolsused for each element.

1b2 The effective index Ng for the two gratings, AandB, which are aligned parallel with the waveguide,is described by

Ng 5 ma

l

La1 mb

l

Lb;

l

L*, 132

where L* is an equivalent grating period. We re-strict ourselves below to the first diffraction orders byassuming thatma 5 21 andmb 5 11. Note that thegrating term gives only a linear wavelength contribu-tion.

1c2 From simple geometry and Snell’s law, theeffective indexNp of the prism is given by

Np 5 ni sin ui cos w 1 1np2 2 ni2 sin2 ui21@2sin w. 142

To search for a solution over a broad spectral range,the mismatch function F is expanded in a Taylorseries centered at the wavelength of interest 1l02:

F1l2>F1l021dF

dl 0l0

1l 2 l021d2F

dl2 0l0

1l 2 l022

21O1Dl32.

152

Setting the zero-, first-, and second-order coeffi-cients to zero yields

F1l02 5 Nw 2 ni sin ui cos w 2 1np2 2 ni2 sin2 ui21@2

3 sin w 2l0

L*5 0, 162

dF

dl 0l0 5dNw

dl2

np sin w

1np2 2 ni2 sin2 ui21@2

dnpdl

21

L*5 0,

172

d2F

dl2 0l0

5d2Nw

dl22

np sin w

1np2 2 ni2 sin2 ui21@2

d2npdl2

1ni2 sin2 ui sin w

1np2 2 ni2 sin2 ui23@2 1

dnpdl 2

2

5 0. 182

Equation 162 is the usual coupling condition for aparticular wavelength. Making the first chromaticorder zero 3Eq. 1724 gives a stable solution around thedesign wavelength, and setting the second-chromatic-order coefficient 3Eq. 1824 to zero enlarges the couplingregion.Thus the problem of designing an achromatic cou-

pler for a specific waveguide consists of searching forvalues of the angle of incidence ui, equivalent gratingperiod L*, index of refraction np of the prism, andbase angle w that simultaneously cancel the firstthree coefficients in the Taylor series. We used asimple approach for solving the system of coupledequations by initially selecting a specific glass andneglecting the grating term 1Ng 5 0 or L* 5 `2. Theincident coupling angle ui and the prism base angle wwere then calculated by simultaneously solving Eqs.

20 September 1995 @ Vol. 34, No. 27 @ APPLIED OPTICS 6181

162 and 182. The grating parameter L* was thenobtained from the first-order Eq. 172. Of course whena grating term 1Ng fi 02 is introduced we must recalcu-late. However, as the grating contribution is gener-ally small, usually one or two additional iterations aresufficient for the variables to converge to the achiev-able experimental precision. Several glass materi-als were tested, and only certain choices with theappropriate index of refraction and Abbe numberyielded a solution for our particular waveguide.On correction up to the second chromatic order, the

effective-index mismatch function exhibits a cubicwavelength dependence. We then define a meritfunction to be the sum of squares of the mismatchfunction over the spectral region of interest andminimize the function by refining the variables.Basically the final procedure compensates the cubiccontribution by reintroducing a small amount of thefirst-order termwith the opposite sign. This is analo-gous to the usual procedure in lens design that is usedto initially correct third-order aberrations 1Seidelaberrations2; when fifth-order terms subsequentlybecome dominant, small amounts of the third orderwith the opposite sign are reintroduced for balancing.The procedure was implemented into a computer

code for an automatic search of solutions. The disper-sion of all the materials involved has been taken intoaccount as well as the explicit wavelength depen-dence of the grating and waveguide effective indices.As usual in the waveguide case,Nw and its derivativesare calculated by the use of a numerical routine.In Fig. 2 examples of the effective-index mismatch

function for four different couplers are given: asingle-grating coupler, a prism coupler, the achro-matic design specified in Li and Brazas,15 and ourachromatic design, which is corrected up to the

Fig. 2. Comparison of effective-index mismatch function fordifferent incoupler designs: 1a2 grating coupler L 5 313.4 nm; 1b2prism coupler with LaSF3 glass and base angle 52.206°; 1c2 Li andBrazas15 double grating La 5 309.5 nm and Lb 5 308.8 nm, prismSF1 glass, base angle 60°; 1d2 present design, with double gratingLa 5 305.4 nm and Lb 5 313.4 nm, prism LaSF3 glass, base angle52.206°. For all designs, the waveguide material is Corning glass7059 with a thickness of 400 nm, the cover is air, and TEpolarization is assumed. Designs 1a2, 1b2, and 1d2 are centered atl0 5 550 nm, and the substrate is fused silica; for design 1c2 l0 5

685 nm, and the substrate is pyrex.

6182 APPLIED OPTICS @ Vol. 34, No. 27 @ 20 September 1995

second chromatic order. For the grating and prismcouplers where no achromatization has been at-tempted, a strong linear dependence rapidly in-creases the mismatch function for small detunings ofthe design wavelength. In the design of Li andBrazas, where the first order has been canceled, aquadratic behavior is observed. The mismatch func-tion of our design shows a weak cubic dependence andis kept to small values over a much broader region.Waveguide thickness is another parameter that can

be included in the design process. It should be notedthat we were able to correct the first chromatic orderof a conventional prism coupler merely by adjustingwaveguide thickness. Though its mismatch functionis inferior to the achromatic design, it nonethelessrepresents an important improvement over the con-ventional prism or grating couplers and can be usedfor certain applications.Our condition for canceling the first derivative of

the mismatch function with respect to wavelength isequivalent to the condition given by Spaulding andMorris,11 which imposes a zero derivative of theincident coupling angle with respect to wavelength1dui@dl 5 02. This can be verified by taking our mis-match function as an implicit function of wavelengthand coupling angle and setting it equal to zero,

F1l, ui2 5 0, 192

so we can write the following relation:

dui

dl5 2

≠F@≠l

≠F@≠ui

. 1102

We conclude that, if the angular variation againstwavelength is zero, the first derivative of the effective-index mismatch function with respect to wavelengthis also zero.From the effective-index mismatch function F we

calculate the normalized coupling efficiency, followingBrazas and Li.16 The functional form is simple whenone of the length parameters involved 1grating lengthLg, beam size v0, or Lc ; 1@a, where a is the wave-guide leakage rate2 is much smaller than the others.In Table 1 the expressions for each limiting case aregiven, where we have used Db1l2 ; 12p@l2F1l2. Unlikeprevious studies11–14 that used the incident couplingangle, here there exists a straightforward connectionbetween the design function F1l2 and the variable h1l2that is experimentally determined.

3. Fabrication and Experiment

A prism satisfying our conditions was manufacturedfromLaSF3 Schott glass 1nd 5 1.80801 and nd 5 40.612with a base angle of 52.206°.

Table 1. Normalized Coupling Efficiency for Each Limiting Case

Condition Normalized Coupling Efficiency 1h2

Lg 9 Lc, Lg 9 v0 sinc21DbLg@22Lc 9 Lg, Lc 9 v0 a2@1a2 1 Db22v0 9 Lg, v0 9 Lc exp3211@221Dbv0224

Fig. 3. Measured index of refraction versus wavelength for several Corning glass 7059 films fabricated by the use of sputtering deposition.

The grating fabrication follows the procedure re-ported in Li et al.17 We used the 441.65-nm line of aKimmon He–Cd laser that was expanded, spatiallyfiltered, and collimated. The intensity for the ex-panded beam was 0.38 mW@cm2, and the photoresistwas exposed for 45 s. A real-time control of diffrac-tion efficiency gave an optimum time of 90 s for thedevelopment process. The photoresist grating pat-tern was transferred to the fused-silica substrate bythe use of a reactive-ion-milling process that usesfreon gas.On the fused-silica slide that was to be coated with

the waveguide, two gratings separated by 12.5 mmwere fabricated; one grating was part of the inputachromatic coupler 1see Fig. 12, and the other was usedas a simple output coupler for power measurements.We determined the grating period of the input couplerto be Lb 5 313.4 nm by measuring the diffractionangle in Littrow configuration.The waveguide was Corning glass 7059 deposited

on fused-silica substrates 150.8 cm 3 2.54 cm2 with aPerkin-Elmer 2400 RF diode-sputtering system.The bias voltage on the material target was set to1150 V, and a power of 250 W was established for apressure of 3.9 3 1023 Torr in the chamber. Thepredeposition vacuum was 2 3 1026 Torr, and areactive atmosphere of Ar 170%2 and O2 130%2 wasdynamically controlled during deposition by the use ofanMKS two-channel flowmeter. Wemeasured a 4006 2 nm thickness for the waveguide used in ourexperiment and estimated an average deposition rateof 49.5 Å@min. Many samples have been preparedunder the same condition for a precise characteriza-tion of the material dispersion as required for designpurposes. In Fig. 3 we present the dispersion resultsfor several 7059 waveguides obtained from prism-coupler measurements, from which a standard devia-tion of less than 60.001 in the characterization of theindex of refraction was determined. Typical loss was2 dB@cm at l 5 633 nm.

On a second fused-silica slide another grating witha spatial periodLa 5 306.2 nm, which closely approxi-mates the design value 1La 5 305.4 nm, as indicatedin Fig. 22, was fabricated.The two slides were brought into contact from the

unworked sides by an index-matching liquid of refrac-tive index 1.46. After the grooves were opticallyaligned the two slides were glued together, creating alaminate structure, as shown in Fig. 1, with overallthickness of 2.0 mm. The equivalent grating periodL* obtained from the combined pair was 13,328 nm.A liquid with a high index of refraction 11.802was usedto bring the prism and gratingA into contact.Figure 4 illustrates the experimental setup used in

the coupling-efficiency measurements. To cover abroad spectral range, a Coherent Innova 70 argon-ionlaser, a Coherent CR-599 dye 1Couramin C62 laserscanning from 525 to 580 nm, and a Melles-GriotHe–Ne laser were used as sources. Special attentionwas paid to guarantee that beams from differentsources reached the input coupler at the same posi-

Fig. 4. Experimental setup for coupling-efficiencymeasurements.

20 September 1995 @ Vol. 34, No. 27 @ APPLIED OPTICS 6183

tion and direction by sending them through two irisesseparated by 1.05 m. Transverse electric 1TE2 polar-ization was selected with a Glan–Thompson polarizer.The waveguide and the coupler were mounted on arotary stage driven by a stepper motor, and angularmeasurements were digitally controlled. After beingcoupled in and propagated through the waveguide,the guided mode was coupled out by a single-gratingoutcoupler and projected on a screen. A referencesignal was collected from the beam with a spectrallyneutral beam splitter and directed to the same screen.Both spots were simultaneously photographed with aCCD camera 1Photometrics CH210 with TektronixTK512CB@AR chip2.

4. Results and Discussion

We initially measured the dependence of couplingefficiency on angle of incidence for several discretewavelengths. Two different incouplers were investi-gated: the achromatic coupler designed and fabri-cated as described above and a conventional gratingcoupler with spatial frequency L 5 313.4 nm. Fig-ure 5A shows the results for the achromatic coupler;the angular dependence of coupling efficiency wasmeasured at 525, 550, and 580 nm. Strong overlapbetween all three curves is observed, with the 525-and the 580-nm curves intersecting at 87% of thenormalized coupling efficiency. An otherwise equiva-lent measurement was made for the single-grating

Fig. 5. Dependence of coupling efficiency on angle of incidence forA, our achromatic waveguide coupler at three particular wave-lengths, l 5 525, 550, and 580 nm, and B, a conventional gratingcoupler at three particular wavelengths, l 5 545, 550, and 555 nm.

6184 APPLIED OPTICS @ Vol. 34, No. 27 @ 20 September 1995

coupler at 545, 550, and 555 nm. In contrast, Fig. 5Billustrates that a spectral separation of only 5 nm cansplit apart the coupling-efficiency curves, and theoverlap is negligible.From the measured angular separation of these

lines 1dui@dl2 and the angular bandwidth 1Du50%2, thespectral bandwidth is estimated with

Dl50% 5 Du50%0 ≠F@≠ui

≠F@≠l 0 5Du50%

0dui@dl 0, 1112

yielding Dl50% 5 97 nm for the achromatic couplerand Dl50% 5 1.1 nm for the conventional gratingcoupler.A direct measurement of coupling efficiency over

the spectral range of interest was also performed.The procedure here was to initially adjust beamlocation and to tune the angle of incidence for amaximum of the coupled power into the waveguide atthe center wavelength 1550 nm2. We then performedall measurements by tuning the wavelength withoutany further change in the configuration. The mea-sured input-coupling angle ui was 11.25° 6 0.12°, inexcellent agreement with the expected theoreticalvalue 111.26°2. The estimated absolute input-cou-pling efficiency was 0.04%, because of the low diffrac-tion efficiency of grating A that was used in thepresent demonstration and the mismatch betweenthe grating-coupler leakage rate 1a2 and the spot sizeof the input beam.Figure 6 shows the measured normalized coupling

efficiency for the achromatic coupler as a function ofwavelength. Aspectral bandwidth 1FWHM2 of 70 nmwas obtained. Also plotted is the coupling efficiencypredicted from the effective-index mismatch functionF when the inverse of the waveguide leakage rate1Lc ; 1@a2 is estimated to be the shortest of thelength parameters 1see Table 12. Although the re-sults are satisfactory for the region close to thecentral wavelength, they clearly disagree at the side-bands. However, as pointed out by Strasser andGupta,14 lateral shift of the beam as a function ofwavelength 1illustrated in Fig. 72 plays an importantrole for large deviations from the central wavelength.

Fig. 6. Normalized coupling efficiency for an achromatic coupleras a function of wavelength. Experimental results 1U2 and theo-retical calculation 1solid curve2 are shown without consideration oflateral shift effects.

The expressions in Table 1, which are used for thetheoretical calculation of coupling efficiency, take intoaccount only the effect of phase mismatch and do notconsider lateral displacement of the beam with re-spect to the central wavelength. We calculate for ourachromatic coupler a lateral shift of 60.12 mm fordetuning of 625 nm from the central wavelength,mainly because of gratingAand a negligible contribu-tion from the prism. Shorter wavelengths are shiftedtoward a region without grating modulation, andlonger wavelengths are moved inside the grating andfar from the edge. In the former case the couplingefficiency is reduced because only part of the beam isdiffracted into the waveguide; in the latter case alonger propagation length inside the grating in-creases losses as a result of outcoupling. Both casestend to reduce coupling efficiency and become thedominant effect, rather than phase mismatch, atsubstantial deviations from the central wavelength.However, this lateral shift can be substantially re-duced by the use of a much thinner substrate andlonger spatial frequencies for the gratings 1with thesame equivalent grating period L*2.

5. Conclusion

We have described a simple and straightforwardapproach to the design of achromatic waveguidecouplers based on correction of chromatic ordersobtained from a Taylor series expansion of the effec-tive-index mismatch function. The formulation isapplied to a prism–double-grating achromatic cou-pler, and for the first time to our knowledge a designthat corrects up to the second chromatic order wasexperimentally implemented. A spectral bandwidthof 70 nmwasmeasured, which is substantially greaterthan previous reported values.12–15Based on programmatic considerations, we have

focused our attention on increasing the bandwidth ofthe waveguide coupler, leaving the absolute couplingefficiency aside. Such an approach is justified be-cause the absolute efficiency of the proposed couplingdevice can be improved without affecting its band-width. Additionally, for many applications in chemi-cal sensors, the efficiency that we report here isadequate. If necessary, a higher absolute efficiencycan be obtained by choosing appropriate groove depths

Fig. 7. Schematic representation of lateral shift for differentwavelengths.

of gratingsAand B for maximum diffraction efficiencyand optimum matching between beam size and leak-age rate, respectively.As currently practiced, planar waveguide ATR is a

highly sensitive spectrometric technique for character-ization of thin films and interfaces. The achromaticcoupler presented here shows considerable potentialfor extending planar waveguide IOW–ATR to thebroadband regime. Development of a technology toacquire broadband ATR spectra at the surface of aplanar IOW would find immediate application in atleast two important areas: 1a2 spectral characteriza-tion of interfacial organic and biological thin-filmassemblies, particularly at solid–liquid interfaces,and 1b2 simultaneous chemical sensing of multipleanalytes at spectrally resolved wavelengths, based onincreased information content available in the poly-chromatic domain.

This work has been supported by the NationalScience Foundation under grant CHE-9403896 andthe National Institutes of Health under grant R03RR08097. Sergio B. Mendes gratefully acknowl-edges support from Coordenacao de Aperfeicoamentode Pessoal de Nivel Superior 1Brazilian Agency Sup-porting Graduate Programs2.

References1. O. S. Wolfbeis, ‘‘Fiber-optic sensors in biomedical sciences,’’

PureAppl. Chem. 59, 663–672 119872.2. R. E. Dessey, ‘‘Waveguides as chemical sensors,’’ Anal. Chem.

61, 1079A–1094A 119892.3. M. A. Arnold, ‘‘Fiber-optic chemical sensors,’’ Anal. Chem. 64,

1015A–1025A 119922.4. J. N. Polky and J. H. Harris, ‘‘Absorption from thin-film

waveguides,’’ J. Opt. Soc. Am. 62, 1081–1087 119722.5. S. S. Saavedra and W. M. Reichert, ‘‘Integrated optical attenu-

ated total reflection spectrometry of aqueous superstratesusing prism-coupled polymer waveguides,’’ Anal. Chem. 62,2251–2256 119902.

6. M. D. DeGrandpre, L. W. Burgess, P. L. White, and D. S.Goldman, ‘‘Thin film planar waveguide sensor for liquid phaseabsorbancemeasurements,’’Anal. Chem. 62, 2012–2017 119902.

7. D. A. Stephens and P. W. Bohn, ‘‘Absorption spectrometry ofbound monolayers on integrated optical structures,’’ Anal.Chem. 61, 386–390 119892.

8. S. S. Saavedra and W. M. Reichert, ‘‘In situ quantitation ofprotein adsorption density by integrated optical waveguideattenuated total reflection spectrometry,’’ Langmuir 7, 995–999 119912.

9. D. S. Walker, M. D. Garrison, and W. M. Reichert, ‘‘Proteinadsorption to HEMA@EMA copolymers studied by integratedoptical techniques,’’ J. Colloid Interface Sci. 157, 41–49 119932.

10. T. E. Plowman, M. D. Garrison, D. S. Walker, and W. M.Reichert, ‘‘Surface sensitivity of SiON integrated optical wave-guides 1IOWs2 examined by IOW-attenuated total reflectionspectrometry and IOW-Raman spectroscopy,’’ Thin Solid Films243, 610–615 119942.

11. K. E. Spaulding and M. Morris, ‘‘Achromatic waveguideinput@output coupler design,’’Appl. Opt. 30, 1096–1112 119912.

12. K. E. Spaulding and M. Morris, ‘‘Achromatic waveguidecouplers,’’ J. Lightwave Technol. 10, 1513–1518 119922.

20 September 1995 @ Vol. 34, No. 27 @ APPLIED OPTICS 6185

13. D. L. Hetherington, R. K. Kostuk, and M. C. Gupta, ‘‘Disper-sion compensation for an integrated optic grating couplerutilizing a transmission volume hologram,’’ Appl. Opt. 32,303–308 119932.

14. T. A. Strasser and M. C. Gupta, ‘‘Grating coupler dispersioncompensation with a surface-relief reflection grating,’’ Appl.Opt. 33, 3220–3226 119942.

15. L. Li and J. C. Brazas, ‘‘A method for achromatically coupling

6186 APPLIED OPTICS @ Vol. 34, No. 27 @ 20 September 1995

a beam of light into a waveguide,’’ U.S. patent 5,420,947 130May 19952.

16. J. C. Brazas and L. Li, ‘‘Analysis of input-grating couplershaving finite lengths,’’Appl. Opt. 34, 3786–3792 119952.

17. L. Li, M. Xu, G. I. Stegeman, and C. T. Seaton, ‘‘Fabrication ofphotoresist masks for submicrometer surface relief gratings,’’in Integrated Optical Circuit Engineering V, M. A. Mentzer,ed., Proc. Soc. Photo-Opt. Instrum. Eng. 835, 72–82 119872.