Synthesis of analogapodizers with binary angular sectors

Adolf W. Lohmann, Jorge Ojeda-Castaneda, and Alfonso Serrano-Heredia

We describe a new procedure for generating complex amplitude distributions along the optical axis of anoptical processor by the use of binary masks in the form of binary angular sectors. This type of binaryspatial filter acts as a gray-level apodizer with rotational symmetry. Experimental verifications areincluded.Key words: Axial signal synthesis, computer-generated holography, Strehl ratio versus defocus,

angular spatial filters.

1. Introduction

The impulse response of an optical system can beconveniently tailored for performing special tasks,such as processing of large amounts of data whencomplex mathematical operations are implementedor for satisfying image-quality requirements. Forthese purposes, sometimes it is necessary to modifythe transfer function of optical systemswith the use ofspatial filters, or apodizers, whose transmittanceprofile follows a continuous gray curve. In otherwords, the apodizer is an analog filter. However,binary masks are easy to fabricate and to replicate.Consequently one always attempts to substitute ana-log filters by binary filters whenever possible.Jaquinot and Roizen-Dossier have proposed trad-

ing, or sacrificing, one dimension in an imagingsystem for implementing one-dimensional 11-D2 ana-log apodizers by use of two-dimensional 12-D2 binaryscreens.1 In a different context we have reportedtwo techniques for synthesizing 1-D complex ampli-tudes by use of an optical processor and 2-D binarypatterns in rectangular coordinates.2,3 These laterresults can be thought of as trading one dimension ina 2-D optical processor for producing 1-D complex

A. W. Lohmann is with the Angewandte Optik, PhysikalischesInstitut der Universitat Erlangen, Nurnberg, Staudstrasse 7,Erlangen 91058, Germany. J. Ojeda-Castaneda and A. Serrano-Heredia are with the National Institute for Astrophysics, Optics,and Electronics, Apartado Postal 216, Puebla, Puebla 72000,Mexico.Received 4April 1994; revised manuscript received July 1994.0003-6935@95@020317-06$06.00@0.

r 1995 Optical Society of America.

amplitude variations. Of course, the whole proce-dure is similar to the binary computer-generatedholograms first described by Brown and Lohmann.4Our aim here is to present the concept of trading

dimensionality for generating 1-D complex ampli-tudes along the optical axis of an optical processor bythe use of a 2-D binary mask in the form of angularsectors. We show that, from the viewpoint of thelongitudinal irradiance, angularly periodic, binaryscreens are equivalent to angulary symmetric, non-negative, gray screens. Our results may be consid-ered as applications of McCutchen’s theorem.5To achieve our aim, in Section 2 we present the

concept of trading dimensionality for binary objects inpolar coordinates. In Section 3 we apply the previ-ous result for experimentally synthesizing slow-varying functions along the optical axis of an opticalprocessor. In this manner we increase the axialirradiance depth of an optical system by the use ofnew binary spatial filters, which are equivalent tocertain gray-level apodizers.

2. Trading Dimensionality

The impulse response of an optical processor P1x, y2, asshown in Fig. 11a2, is related to the coherent transferfunction P1n, µ2 by a 2-D Fourier transform:

P1x, y2 5 e2`

` e P1n, µ2exp3i2p1xn 1 yµ24dndµ. 112

Now, according to Ref. 1, if one sacrifices one dimen-sion by setting y 5 0 in Eq. 112 and if one employs

10 January 1995 @ Vol. 34, No. 2 @ APPLIED OPTICS 317

binary filters of the form

P1n, µ2 5 rect3µ@W1n24rect3n@2V4, 122

as is shown in Fig. 11b2, then the impulse response inEq. 112 becomes

P1x, 02 5 e2V

V e2W1n2@2

W1n2@2

exp1i2pxn2dndµ

5 e2V

V

W1n2exp1i2pxn2dn. 132

In Eq. 122 we denote the edges of the pupil along the naxis as 6V and the variable width of the pupil alongthe µ axis asW1n2. It is apparent fromEq. 132 that oneis able to generate 1-D impulse responses of gray-levelor analog filters,W1n2, by using a 4-f optical processorand a binary spatial filter.Because W1n2 is a real function, P1x, 02 is a Hermi-

tian function, P1x, 02 5 P*12x, 02; see Ref. 6. Thismeans that this technique is able to synthesize sig-nals only with certain phase variation. In a generalsense it is a computer-generated hologram for asubset of signals. SeeAppendixA.Next we show that a similar situation applies in

polar coordinates. For this purpose we consider the

Fig. 1. Schematic diagrams: 1a2 optical setup, 1b2 binarymask forgenerating 1-D analog filters along the n axis.

318 APPLIED OPTICS @ Vol. 34, No. 2 @ 10 January 1995

three-dimensional impulse response P1x, y, z2 of theoptical processor in Fig. 11a2:

P1x, y, z2 5 ee2`

`

P1n, µ2exp32iplz1n2 1 µ224

3 exp3i2p1xn 1 yµ24dndµ, 142

which can be written in polar coordinates as

P1r, f, z2 5 e0

2p e0

`

P1r, u2exp12iplzr22

3 exp3i2prr cos1f 2 u24rdrdu. 152

Now, we employ as the spatial filter a binary sector, asshown in Fig. 2, which can be conveniently repre-sented as

P1r, u2 5 rect3u

UW1r2@224 . 162

In Eq. 162, U denotes the maximum angular width ofthe sector. The fraction of the maximum value U asa functionof r is described by W1r2@22, such that 0 #W1r2@22 # 1. The limits of the radial spatial fre-quency r are from r 5 0 to r 5 V.

Fig. 2. Binary mask for generating radial analog filters with onesector: 1a2 display in Cartesian coordinates 1r, u2, 1b2 display inpolar coordinates.

By substituting Eq. 162 into Eq. 152 and by setting r50 and f 5 0, we have

P10, 0, z2 5 U e0

V

W1r2@22exp12iplzr22rdr

5 2U e0

V2@2

W1r2@22

3 exp32i2plz1r2@224d1r2@22. 172

This result may be interpreted conveniently as aFourier transform by use of a suitable change ofvariables, z 5 1r@V22 2 0.5 and Q1z2 5 W1r2@22, as inRef. 7. That is,

P10, 0, z2 5 UV2 exp12iplzV2@22

3 e20.5

0.5

Q1z2exp32i2p1lzV2@22z4dz. 182

Consequently it is apparent from Eq. 182 that theimpulse response along the z axis, P10, 0, z2, is propor-tional to the Fourier transform of the analog filter

Fig. 3. Same as Fig. 2 but for a symmetric version with 2M 5 16sectors.

Q1z2, which is implemented with a binarymask. Thisuseful result can be thought of as a new application ofMcCutchen’s theorem.5Our previous proposal generates highly asymmet-

ric diffraction patterns outside the optical axis whenr fi 0. If this feature is undesirable in some applica-tions, then one can reduce the asymmetric behaviorsubstantially by using several 1say, 2M2 sectors in theform of a pie or a daisy. Each sector contains thebinary pattern in Eq. 162 but is now centered at u 512m 2 12p@2M, for m 5 1, . . . , M, as shown in Fig. 3.In this case, Eq. 162 becomes

P1r, u2 5 om51

2M

rect3u 2 12m 2 12p@2M

1p@M2W1r2@22 4 , 192

and consequently Eq. 172 becomes

P10, 0, z2 5 om51

2M

12p@M2 e0

`

W1r2@22

3 exp32i2plz1r2@224d1r2@22

5 2p e0

`

W1r2@22exp32i2plz1r2@224d1r2@22

5 pV2 exp32iplz1V2@224 e20.5

0.5

Q1z2

3 exp32i2p1lzV2@22z4dz. 1102

Again as in Eq. 182, the amplitude distribution alongthe optical axis is generated as the Fourier transform

Fig. 4. Amplitude transmittance of the gray-level apodizers forincreasing focal depth.

10 January 1995 @ Vol. 34, No. 2 @ APPLIED OPTICS 319

of a gray spatial filter, Q1z2, which is implementedwith a binary screen. But now the diffraction pat-tern is symmetric.If the asymmetry of the lateral impulse response is

of no consequence, then a single sector with U 5 2phas the highest light-gathering power. Of course, ifU , 2p, then M sectors with U 5 2p@M and M . 1have higher light-gathering power than a single sector.In what follows we illustrate our results with a simpleexample of 16 sectors.

3. Nonzero Axial Irradiance: Extended Range

Next we apply Eqs. 192 and 1102 to shape the amplitudeimpulse response along the axis of the optical

Fig. 5. Binary masks with 16 sectors that encode the apodizers inFig. 4: 1a2 n 5 1@2, 1b2 n 5 1, 1c2 n 5 3@2.

320 APPLIED OPTICS @ Vol. 34, No. 2 @ 10 January 1995

processor shown in Fig. 11a2 for increasing the nonzeroaxial irradiance. According to Ref. 7, the amplitudedistribution along the optical axis can be shaped toachieve large axial irradiance if one uses the analogspatial filter

W1r2@22 5 512r@V22n2131 2 1r@V224n21@2 if r # V

0 otherwise,

1112

with n 5 1@2, 1, 3@2, 2, 5@2, . . . , where, as before, Vdenotes the edge of the pupil, which is the maximum

Fig. 6. In-focus irradiance distributions generated with the bi-nary masks in Fig. 5.

value of r. The amplitude transmittance of this typeof apodizer 3Eq. 11124 is shown in Fig. 4 for severalvalues of n.In fact, in Ref. 7 it is mentioned that this type of

spatial filters generates axial amplitude distributionsproportional to the Bessel functions of the first kindand order n. Here we note from Eqs. 192 and 1102 thatthis type of apodizer can be implemented by use of 2Mbinary sectors if the angular width of each sectorfollows the functionW1r2@22 in Eq. 1112.For the experimental verifications we generate

some binary sectors, shown in Fig. 5, that implementthe apodizers in Fig. 4, for n 5 1@2, n 5 1, and n 5

Fig. 7. Same as Fig. 6 but for out-of-focus irradiances. The clearcircular aperture generates zero axial irradiance.

3@2. In Fig. 5 we set the number of sectors 2M equalto 16.In Fig. 6 we show the in-focus irradiance distribu-

tions that are obtained with an optical processorsimilar to that shown in Fig. 11a2. In our experimen-tal setup the lenses used have a focal length f5 75 cm,and we employ a helium–neon laser for l 5 632.8 nm.The radius of the pupil was r 5 1.5 cm; consequentlyV 5 316 cm21.In our first experimental verifications the irradi-

ance point-spread functions have central lobes withroughly the same radius as for the three cases shownin Fig. 5. In other words our apodizers do notsubstantially increase the lateral spread of the im-pulse response.Figure 7 shows the out-of-focus irradiance distribu-

tions profiles in the plane where the clear circularaperture shows a zero value at the axial point. Forthe other screens the irradiance at the axial points is

Fig. 8. Axial irradiance distributions of the out-of-focus imagesshown in Fig. 7.

10 January 1995 @ Vol. 34, No. 2 @ APPLIED OPTICS 321

nonzero. We also verified this result by scanning theirradiance distributions at this particular plane, asshown in Fig. 8.From the above results it is apparent that the

binary sectors in Fig. 5 are able to increase the axialirradiance depth without severely deteriorating thein-focus impulse response. To the best of our knowl-edge, the above results are the first experimentalverifications on the use of the apodizers, shown in Fig.4 and proposed in Ref. 7.

4. Conclusions

We have shown that, by trading one dimension inpolar coordinates, one is able to substitute analogspatial filters with binary spatial fiters for shapingthe axial impulse response of an optical processor.This concept of trading dimensionality was applied tothe proposed binary screens in the form of angularsectors for synthesizing gray-level apodizers. Wehave reported some experimental verifications show-ing that the synthesized apodizers can be used toincrease the axial irradiance depth without severelydistorting the in-focus impulse response.

Appendix A

It is perfectly valid to express in a unique manner anyreal function, say,W1n2 in Eq. 122, as the sum of an evenreal function E1n2 5 E12n2 plus an odd real function O˜ 1n2 5 2O12n2. That is,

W1n2 5 E1n2 1 O1n2 5 E12n2 2 O12n2. 1A12

Now, it is easy to show 1see Ref. 6, for example2 thatthe Fourier transform ofW1n2 is a Hermitian function,namely,

S1x2 5 e2`

`

W1n2exp1i2pxn2dn 5 S*12x2. 1A22

Equivalently, by using Eq. 1A12, we have

S1x2 5 e2`

`

E1n2exp1i2pxn2dn 1 e2`

`

O1n2exp1i2pxn2dn

5 E1x2 1 iO1x2

5 E12x2 2 iO12x2. 1A32

This means that the modulus of S1x2 is an even

322 APPLIED OPTICS @ Vol. 34, No. 2 @ 10 January 1995

function,

S1x2 5 3E21x2 1 O21x241@2 5 0S12x20, 1A42

and that the phase of S1x2 5 0S1x20exp3if1x24 is an oddfunction,

tan f1x2 5 O1x2@E1x2. 1A52

From these results it is apparent that, by a suitablechoice of the even and the odd components of W1n2, inprinciple it is possible to encode the phase of thesignal S1x2.In other words the phase of the synthesized signal

is the real function

tan f1x2 5

2i e2`

`

O1n2exp1i2pxn2dn

e2`

`

E1n2exp1i2pxn2dn

. 1A62

A symmetric width function, W1n2 5 W12n2, generatesa purely amplitude signal S1x2, while any asymmetricvariation generates phase changes in S1x2.

We are indebted to the reviewers for useful com-ments on our terminology. J. Ojeda-Castaneda isgrateful to the University of Las Americas for finan-cial support. We thank Vıctor Arrizon and GustavoRamırez for useful discussions on our experimentalverifications.

References1. P. Jaquinot and B. Roizen-Dossier, ‘‘Apodisation,’’ in Progress in

Optics, E. Wolf, ed. 1North-Holland, Amsterdam, 19642, Vol. 3,pp. 41–43.

2. A. W. Lohmann, J. Ojeda-Castaneda, and A. Serrano-Heredia,‘‘Synthesis of 1-D complex amplitudes using Young’s experi-ment,’’ Opt. Commun. 101, 17–20 119932.

3. A. W. Lohmann and J. Ojeda-Castaneda, ‘‘Computer-generatedholography: novel procedure,’’ Opt. Commun. 103, 181–184119932.

4. B. R. Brown and A. W. Lohmann, ‘‘Complex spatial filteringwith binary masks,’’Appl. Opt. 5, 967–969 119662.

5. C. W. McCutchen, ‘‘Generalized aperture and the three-dimensional diffraction image,’’ J. Opt. Soc. Am. 54, 240–244119642.

6. R. Bracewell, The Fourier Transform and Its Applications1McGraw-Hill, NewYork, 19652, pp. 14–16.

7. J. Ojeda-Castaneda, L. R. Berriel-Valdos, and E. Montes, ‘‘Bes-sel annular apodizers: imaging characteristics,’’ Appl. Opt.26, 2270–2772 119872.