Weighted optical diffraction gratings

$Page 1: Weighted optical diffraction gratings$
Weighted optical diffraction gratings

Meirion F. Lewis and Christopher L. West

This paper describes the behavior of optical diffraction gratings which have been weighted in various waysanalogous to those used in surface acoustic wave devices. It is shown experimentally and theoretically that awide variety of 2-D responses can be achieved by these means. In this study we have chosen to concentrate onquasirectangular responses, and we describe the factors that limit the performance achievable in practice.Some possible applications of these techniques in the spatial, temporal, and spectral domains are mentioned.

1. Introduction

The optical properties of a diffraction grating com-prising a large number of slits within a rectangularaperture are well known,1'2 the narrow sinc-functionresponse of such a device having long been exploited inmonochromators and spectrometers. Few variationson this arrangement have been described in the litera-ture, the only one of any importance being deliberateor accidental apodization through the use of an inputbeam with a Gaussian spatial profile, when the re-sponses of the individual orders also tend toward aGaussian form.1 An acoustic component closely anal-ogous to the optical diffraction grating is the interdigi-tal transducer (IDT) which in its basic form comprisesa large number of parallel linear equispaced sources ofacoustic waves, usually surface acoustic waves(SAWs). By analogy with the optical diffraction grat-ing, the acoustic frequency response of such a trans-ducer is of sinc-function form. However, in contrast tothe optical diffraction grating, the IDT has been modi-fied in numerous ways in recent years to achieve manydesirable frequency responses for use in electronics, 3 ' 4

the basic sinc-function response itself rarely being ofinterest. For example, weighting procedures havebeen devised to provide >60-dB frequency-domainsidelobe rejection in bandpass filters, while chirpedIDTs (transducers with monotonically varying spatialperiod) are regularly employed in spectrum analysis,and in pulse compression radar for waveform genera-tion and matched filtering. In this paper we describethe results of an investigation into the use of such IDT

The authors are with Royal Signals & Radar Establishment, Mal-vern, Worcestershire WR14 3PS, U.K.

Received 18 September 1987.0003-6935/88/112357-06$02.00/0.

weighting techniques in optical diffraction gratings.Many phenomena familiar to the SAW community arereproduced in the optical devices, but several new ef-fects also appear. For example, weighted optical dif-fraction gratings display an apparently anomalous be-havior in their zeroth and even orders, which effectsare not encountered in SAW devices since they do notrespond at their zeroth or even harmonics. Further, asshown later, the two dimensionality of the output fieldcan hardly be ignored in weighted optical diffractiongratings, whereas almost all SAW device designersstrive to avoid this aspect of their devices. Ironically itis only recently that the benefits of exploiting thesecond dimension in SAW devices have been recog-nized, and then often by analogy with optics.5 Thedescription below represents a succinct summary ofour investigations. A fuller account is available onrequest and includes details of the experimental tech-niques used to provide a logarithmic response and toextend the instantaneous optical dynamic range to>40 dB.6

II. Weighted Diffraction Gratings

The response of a classical optical diffraction gratingis invariably derived from phasor summation proce-dures' but for the present purposes it is more conve-nient to employ the language of Fourier optics.7 Thisdiscipline revolves around the Fourier transform (FT)relationship between the (complex) fields in the input(x,y) and output (u,v) focal planes of a convex lens,namely,

F(u,v) = U dxdyf(x,y) exp[ -j2ir (ux + vy)]ff IXfo II (1)

where fo is the focal length of the lens and X is theoptical wavelength.7 Since this geometry is commonlyused in diffraction grating measurements, the gra-ting's response can be calculated from FT theory.8Thus, the transmittance, f(x,y), of a grating with slits

1 June 1988 / Vol. 27, No. 11 / APPLIED OPTICS 2357

in the y direction can be regarded as the product of its2-D rectangular aperture and a picket fence or shah-function 8 in the x direction, all convolved with a 1-Drectangular function in the x direction derived fromthe finite slit width. The response, F(u,v), in the zeroorder is a product of sinc functions in the u and vdirections. The higher-order responses arise by con-volution of this with a shah-function in the u direction,and their intensities are determined by multiplicationby a wide sinc function, this being the FT of the profileof a single slit. Evidently the response in any individ-ual order derives predominantly from the aperture ofthe grating, and it is this function that we have modi-fied in this paper to provide responses of nominallyrectangular form along the u axis of the output plane.This particular form was chosen as it could be of valuein the spectral or spatial domains, while the generaliza-tion of our procedures to other functions is entirelystraightforward, being based on Fourier transformtheory. It is clear from the preceding discussion thatto obtain a rectangular response requires an aperturefunction in the x direction equal to the (inverse) FT ofthe rectangular function, i.e., of sinc-function form.Unfortunately, however, this cannot be realized exact-ly for two reasons. First, the sinc function is of infiniteextent, and so must be truncated in practice; as shownlater this results in some degree of in-band ripple, andthe appearance of out-of-band sidelobes (e.g., Figs. 2and 3 later). The extent of these deviations dependson the manner of truncation. Second, it should benoted that the sinc function changes sign in alternatesidelobes and this precludes implementing a rectangu-lar response in the zeroth grating order unless oneresorts to sophisticated techniques such as evaporat-ing a dielectric layer of appropriate thickness over thenegative slits. However if we initially confine ourattention to the d 1 grating orders we can resort to theSAW technique of shifting the slit locations by half aperiod to implement the required change in sign.3

This procedure is equivalent to the optical detour-phase technique,9 and in this section we describe itsuse in conjunction with apodization and withdrawalweighting.

A. Apodized Gratings

In this technique the weighting is implemented byvarying the lengths of the slits. A series of sevenpatterns employing apodization has been designed andtested. These patterns all include the main lobe of thesinc function, together with a symmetrical sidelobepattern containing on each side m = 0, 1/2, 1, 11/2, 2, 21/2,

and 3 sidelobes, respectively. One such slit patternwith m = 11/2 is illustrated in Fig. 1(a). These devicesemployed a period p = 16 gtm, the slit and gap widthseach being nominally 8 gim. This is sufficiently largeto avoid any polarization-dependent phenomenawhich arise when the slit width is of the order of theoptical wavelength, X.2 The main lobe contained 200slits giving each quasirectangular response a fractionalfull width of 1%. Figure 2 shows the calculated re-sponses and those measured along the symmetry axis

y

.x.

(a)

(b)Fig. 1. Schematic diffraction gratings incorporating truncatedsinc-function weighting. (a) An apodized grating with m = 1/2

sidelobes on either side of the main lobe. The inset illustrates thehalf-period step used to implement the sign reversals between side-lobes. (b) A withdrawal-weighted grating with m = 3 sidelobes on

either side of the main lobe.

of the output plane (v = 0). These measurements areon a linear optical power scale and show excellentagreement between the measured and calculated levelsof the in-band ripple. Figure 2(a) shows that, as moresidelobes are retained in the grating, more ripples ap-pear across the passband. For integer values of m thetotal number of ripples is 1 + m. It is in this sense thatthe overall response approaches a rectangular functionwith increasing m, but strong ripples always remain atthe band edges if the grating patterns are truncatedsharply, as here. Perhaps more surprising at firstsight is the fact that the passband is generally flatterfor patterns truncated halfway through the outer side-lobes than for those truncated at a null between side-lobes. The explanation of this is discussed below,where it is shown that this improved in-band responseis accompanied by a higher spurious sidelobe level out-of-band. Experimental confirmation of this latterfeature appears in Fig. 3 where we have used the elec-trical network analyzer technique described in Ref. 6to present results on a decibel scale. The reason forthis behavior can be deduced by noting that the (sev-en) responses of Fig. 2 are each the ideal rectangularresponse function convolved with a sinc function, thelatter being the Fourier transform of the appropriaterectangular truncating function. The specific proper-ties of the convolving sinc functions responsible for thein-band and out-of-band structures of Figs. 2 and 3discussed above are (a) the reversal in sign of alternatesidelobes, and (b) the fact that the main lobe is twice aswide as each sidelobe. This is made clear in an extend-ed discussion in Ref. 6. Although this trade-off be-tween the in-band response and the out-of-band re-sponse is also an inherent feature of SAW filters, theauthors are unaware of any prior recognition of it; inpractice SAW device designers use computer proce-

2358 APPLIED OPTICS / Vol. 27, No. 11 / 1 June 1988

��I . I .1 I �� U

(a) EXPERIMENT dB THEORY

(a) / \-20 (a

-40

dB0

(b)

-20

-40

dB0

(c)

-20

-40u (LINEAR SCALE)

o2

m=O/

EXPERIMENT

Fig. 3. Calculated (a)-(c) and measured (d)-(f) responses of threeapodized gratings of Fig. 2 plotted on a decibel scale. The particularresults shown derive from gratings with low values of m (namely, 0,1/2, 1) because these display the greatest trade-off between the in-

band and out-of-band responses.

1 5

LOCATION.. FIRST ORDER

Fig. 2. Measured and calculated responses of apodized diffractiongratings along the symmetry axis, v = 0, of the output plane (a) forinteger numbers of sidelobes on either side of the main lobe; (b) forhalf-integer numbers of sidelobes. As discussed in the text, thecorresponding results for withdrawal-weighted gratings are indistin-

guishable from these.

dures which optimize the overall performance numeri-cally, at the expense of physical insight.

There are two other aspects of the responses of apo-dized diffraction gratings that are worthy of comment,namely, the two dimensionality of the optical field in

the output plane, and the behavior of the grating in thezeroth and even orders. These are now discussed inturn. In the case of the apodized gratings illustratedin Fig. 1(a), f(x,y) is no longer separable in x and y, sothe same is true of F(u,v). This feature is evident fromthe complex sidelobe structure of Fig. 4(a), which mea-surements are in close agreement with our calculationsinvolving a numerical 2-D Fourier transformation.The behavior of apodized gratings in the even gratingorders is also of some interest. The measurementsbelow have been confined to the zeroth order becausefor an ideal 1:1 mark/space ratio the responses in thesecond, fourth, .. . orders all vanish.1 Figure 5 com-pares a typical zero-order pattern with the correspond-ing first-order pattern; this employed an apodizedgrating containing m = 3 sidelobes on each side of themain lobe. The dramatic difference is readily seen toarise from the breakdown of the sign-reversal proce-dure described earlier, for in the zeroth order any slitsanywhere in the input plane are in phase at ( = 0, v =0) for any optical wavelength. A similar response tothe zeroth order obtains in the second, fourth, ...orders, if present, because the effect of the half-periodstep is to introduce additional path lengths of preciselyX, 2X, etc. Conversely in the third, fifth, ... orders theadditional path lengths generating the detour phasebecome 3X/2, 5X/2, . . , etc., restoring the behavior inthe first order.

Some potential applications of weighted diffractiongratings involve the separation of different optical wa-velengths in spectroscopy, wavelength division multi-plexing, etc. In this connection we illustrate in Fig. 4


THEORY

(d) i

I \~~~~~~~~~~m=O

1

2

e)

LOCATION.

THEORY

FIRST ORDER

(b)

B t

IA

(a)

0

(b)

476.5 488.0 496.5 501.7 514.5I I I -~ I

I I I470 4B0 490 500 510 520

WAVELENGTH (nm)

Fig. 4. Responses of an apodized grating with m = 2 when illumi-nated by an argon-ion laser operating in its all-lines mode: (a) the 2-D output plane in perspective illustrating the off-axis sidelobe struc-ture and (b) the response along the symmetry axis, v = 0. Fivediscrete lines with wavelengths ranging from 476.5 to 514.5 nm are

clearly visible. The ordinates are linear in optical power.

the response of an apodized grating when illuminatedby an argon-ion laser operating in its all-lines mode.The responses of the five laser lines all appear clean,but to achieve this it was necessary to employ achro-matic optics to ensure that all responses focused in acommon plane.

B. Withdrawal-Weighted Gratings

In this technique all the grating slits retain the samelength in the y direction, and weighting is achieved byomitting slits at appropriate x locations in the grating[see Fig. l(b)]. As in the case of apodized gratings, aneffective phase reversal in the +1 (and all odd) gratingorders can be implemented by displacing the appropri-ate slits by p/2. A set of seven such patterns hasbeen made for direct comparison with the apodizedgratings of the previous section. These were designedwith the same fractional bandwidth as the apodizedgratings, so that the main lobe could accommodate upto 200 slits, and each sidelobe up to 100 slits leading toa 1% full fractional bandwidth. The patterns weregenerated by integrating the area under the sinc func-tion numerically, working out from the center. A slitwas called for each time this area increased by unity.In the negative sidelobe regions, the slits were dis-placed by zp/2 symmetrically with respect to the cen-ter of the grating. The largest grating contained 118slits in the main lobe, and 14, 8, and 6 in each successivesidelobe. These even numbers of slits made itstraightforward to implement the half-integer side-lobe patterns. This weighting procedure proved mostsuccessful, as can be judged from the fact that themeasured and calculated grating responses are indis-tinguishable from those of the corresponding apodizedgratings in Fig. 2 (and are therefore not reproducedhere).

There are, however, two significant differences inthe detailed behavior. First, the aperture functionf(x,y) has again become separable in x and y. Thus theresponse in the v direction of the output plane has asinc-function variation as in the case of the conven-tional diffraction grating discussed earlier. The otherprincipal difference in behavior of the apodized andwithdrawal-weighted diffraction gratings concernstheir behavior between grating orders. In the case ofthe apodized gratings the response on the symmetryaxis v = 0 declines dramatically between orders in amanner reminiscent of a conventional grating with asinc-function response [see Fig. 6(c)]. This is con-

tLINEAR POWER

LOCATION u (a) (t) (C)

Fig. 5. (a) Response of an apodized grating (with m = 3) in the +1 grating order, see Fig. 2(a). (b) The corresponding response in the zeroth

order is much narrower. This arises because all slits are in phase, effectively extending the width of the main lobe to the full grating aperture.

Although this response is close to calculation it is slightly distorted by optical breakthrough due to the finite optical density (OD = 3.5) of the

opaque area of the mask. (c) The much more severe distortion of the zeroth-order response of a mask with background OD = 2.5.


firmed experimentally to the limit of our measurementaccuracy, as shown in Fig. 6(d). By contrast, in thecase of withdrawal-weighted gratings the sidelobe levelincreases greatly between orders as shown in the calcu-lated and experimental curves of Figs. 6(a) and (b),respectively. This behavior is well known in the disci-pline of the SAW bandpass filter design, and here wegive only a physical description of its origin. We sawearlier that the technique described for implementingsign reversals in the + 1 grating orders is ineffective inthe even orders because it does not introduce a true1800 phase shift. Instead it introduces path lengthchanges equivalent to ±X/2 at the center frequency(wavelength) of the ±l grating orders. However,apart from this effect, an (untruncated) apodized grat-ing is a uniformly (over)sampled version of the sinc-function envelope, and by the sampling theorem willtherefore reproduce the required bandlimited re-sponse exactly.8 Now if we compare the physical con-struction of apodized and withdrawal-weighted dif-fraction gratings (Fig. 1) we see that a withdrawal-weighted pattern could be generated from thecorresponding apodized grating by moving severalweak slits by integer numbers of periods p to produce asmaller number of full-length slits. Such movementsare of no significance at the center frequencies of thei1 grating orders as they introduce path lengthchanges of an integer number of wavelengths. Howev-er at other frequencies they do affect the response. Inthe particular patterns described here these pathlength changes amount to several wavelengths, and soaffect the response at frequencies removed from thecenter of the grating order by as little as -10%.

Ill. Conclusion and Discussion

In this paper we have described the use of apodiza-tion and withdrawal-weighting techniques to general-ize the sinc-function response of a classical opticaldiffraction grating. However, other techniques de-rived from the field of SAW devices are equally appli-cable, for example, the use of linearly chirped gratingswhich also produce a quasirectangular response, albeitwith a nonlinear (dispersive) phase response. Similar-ly one might employ variable mark/space ratios orrotated slits. The former could be useful as it wouldcombine two beneficial features of the individual tech-niques of apodization and withdrawal weighting,namely, low interorder sidelobe levels, and a sinc-func-tion response in the v direction of the output plane.The use of rotated slits could be interesting in that theresponse may have a natural tendency to be rectangu-lar.'0

The Fourier transform relationship between the re-sponse and the aperture function is important as itensures that, in principle at least, any 2-D response canbe obtained, and also defines the 2-D structure neededto achieve it. In reality the performance is limited byvarious practical constraints, and these have been de-scribed in the case of nominally rectangular responsesimplemented with amplitude-only gratings. It hasbeen shown that for narrowband responses the detour-

111i0111a

-45

(c) f-l

I

I.I

()

(d)

Fig. 6. Comparison of the behavior of apodized and withdrawal-weighted gratings between orders: (a) and (b) show, respectively,the calculated and measured responses of a withdrawal-weightedgrating (with m = 3) over a range covering the zeroth and first gratingorders; (c) and (d) show the corresponding responses for apodizedgratings. The fine structure in (c) is of no consequence; it arisesfrom the sampling process in the numerical computation procedure.

phase technique can be employed to implement thenecessary sign reversals as far as the i 1 grating ordersare concerned, but that this technique is ineffective inthe zeroth and even orders. We have shown that forsharply truncated aperture functions there is a naturaltrade-off between the in-band and out-of-band re-sponses, and described a curious phenomenon where-by the in-band response is smoothest when the (sinc-function) aperture is truncated halfway through theouter sidelobes. It should be emphasized that awealth of literature exists on superior truncation pro-cedures, many developed for digital electronics, andthat the results presented here should not be regardedas optimized in any sense. In particular, the paper byRabiner et al.'1 describes such optimization proce-dures for 1-D FIR (finite impulse response) filters, andadditionally comments on asymmetric weighting,which results in more compact gratings at the expenseof a nonlinear phase response. This paper also dis-cusses the design of 2-D filters and is therefore relevantto an extension of the present investigations in whichthe response in the second dimension is exploited rath-er than studied as a feature of the 1-D response.

Any reader interested in the technique of apodiza-tion can perform an extremely simple demonstrationof the principles by overlaying a conventional diffrac-tion grating with a square aperture whose edges areinclined at 450 to the slits; the response on the symme-try axis (v = 0) is then of sinc-squared form, displaying-26-dB close-in sidelobes, rather than the -13-dBsidelobes of a conventional sinc-function response. Asdiscussed in Sec. II, such an apodized grating alsoshows a strong off-axis sidelobe structure.

Concerning applications, responses of the type de-scribed here may be of use in certain forms of spectros-copy (e.g., Raman) to exploit the very low sidelobe


(a)

4- ZERO ORDER FIRST OR

o

of l

L-4D ._AS: _

11

RDER_"

levels realizable (-60 dB?) and/or the inherent notch-filter capability of the technique. Another such appli-cation is in wavelength division multiplexing (WDM),especially if point detectors are mandatory, e.g., tohandle large electrical bandwidths. Nominally rect-angular responses themselves could be of value in com-munications systems employing WDM in which the(laser) frequencies are not precisely known before-hand, or which drift with time, temperature, etc. Fol-lowing the successful exploitation of SAW devices aswaveform generators,3 4 we speculate that weightedgratings could also be used to generate sophisticatedoptical waveforms, e.g., when impulsed with subpico-second optical pulses. They could also be used asmatched filters for such waveforms, or as group delayequalizers, etc.

In the spatial domain output spots of nominallyrect-function form with low-sidelobe levels could behighly desirable to minimize interchannel crosstalk invarious optical computing and interconnectionschemes.' 2 However the issue of efficiency discussedbelow appears to make the use of volume (phase) ho-lography more attractive for this purpose at present.A disadvantage of the free-space amplitude-onlystructures described in this paper is their inherentlylow efficiency. This would not be a problem, however,if our weighting techniques were employed in othergrating structures such as reflective gratings in planaroptics'3 and fiber optics.14 These may or may not beoperated in the FT mode. In such structures theincident light encounters many grating elements inseries and the absence or weakness of particular re-flecting elements does not incur a loss of optical power,but simply causes stronger illumination of succeedingelements. Of course, one must always be careful toensure that the spectrum of such gratings does notallow phase matching to unwanted modes in the medi-um itself, or in the surrounding free space, as this coulddeplete the required response and/or generate spuri-ous illumination.

It is a pleasure to acknowledge the assistance ofMike Hazell in the early stages of this work and theskill of George Gibbons in producing the gratings.References

1. E. Hecht and A. Zajac, Optics (Addison-Wesley, Reading, MA,1974).

2. R. Petit, Ed., Electromagnetic Theory of Gratings (Springer-Verlag, New York, 1980).

3. D. P. Morgan, Surface Wave Devices for Signal Processing(Elsevier, New York, 1985).

4. M. F. Lewis, C. L. West, J. M. Deacon, and R. F. Humphreys,"Recent Developments in SAW Devices," Proc. Inst. Electr.Eng. Part A 131, 186 (1984).

5. M. F. Lewis, "A Versatile SAW Filterbank Derived from theOptical Diffraction Grating," IEEE Trans. UFFC-33, 681(1986).

6. M. F. Lewis and C. L. West, "A Theoretical and ExperimentalInvestigation into the Use of SAW Weighting Techniques inOptical Diffraction Grating," (1988). Royal Signal & Radar Es-tablishment Memorandum #4128.

7. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill,New York, 1968).

8. R. Bracewell, The Fourier Transform and Its Applications(McGraw-Hill, New York, 1965j.

9. B. R. Brown and A. W. Lohmann, "Complex Spatial Filteringwith Binary Masks," Appl. Opt. 5, 967 (1966).

10. A. P. van den Heuvel, "Use of Rotated Electrodes for AmplitudeWeighting in Interdigital Surface Wave Transducers," Appl.Phys. Lett. 21, 280 (1972).

11. L. R. Rabiner, J. H. McClellan, and T. W. Parks, "FIR DigitalFilter Design Techniques Using Weighted Chebyshev Approxi-mation," Proc. IEEE 63, 595 (1975).

12. H. J. Caulfield, J. A. Neff, and W. T. Rhodes, "Optical Comput-ing: the Coming Revolution in Optical Signal Processing," LaserFocus/Electro-Optics (Nov. 1983), pp. 100-110.

13. A. C. Livanos, A. Katzir, A. Yariv, and C. S. Hong, "Chirped-Grating Demultiplexers in Dielectric Waveguides," Appl. Phys.Lett. 30, 519 (1977); see also Proceedings, IEEE InternationalWorkshop on Integrated Optical and Related Technologies forSignal Processing, Firenze, Italy (10-11 Sept. 1984).

14. I. Bennion, D. C. J. Reid, C. J. Row, and W. J. Stewart, "High-Reflectivity Monomode-Fibre Grating Filters," Electron. Lett.22, 341 (1986).

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Weighted optical diffraction gratings