achromatic fourier transforming properties of a separated diffractive lens doublet: theory and...

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Achromatic Fourier transforming properties ofa separated diffractive lens doublet:theory and experiment

Enrique Tajahuerce, Vicent Climent, Jesus Lancis, Mercedes Fernandez-Alonso, andedro Andres

The strong chromatic distortion associated with diffractive optical elements is fully exploited to achievean achromatic optical Fourier transformation under broadband point-source illumination by means of anair-spaced diffractive lens doublet. An analysis of the system is carried out by use of the Fresneldiffraction theory, and the residual secondary spectrum ~both axial and transversal! is evaluated. Werecognize that the proposed optical architecture allows us to tune the scale factor of the achromaticFraunhofer diffraction pattern of the input by simply moving the diffracting screen along the optical axisof the system. The performance of our proposed optical setup is verified by several laboratory results.© 1998 Optical Society of America

OCIS codes: 050.1970, 070.2590.

1. Introduction

Some problems associated with the limitation ofavailable input–output devices and speckle noise re-strict the promise of coherent optical systems to per-form rapid information-processing operations.Fortunately, the above drawbacks can be partiallyovercome by use of noncoherent optical systems.1 Inthis way the use of spatially coherent but temporallyincoherent illumination permits, in addition tocoherent-noise reduction, the employment of broad-band spectrum sources, such as gas-discharge lamps,light-emitting diodes, etc., and especially the oppor-tunity to deal with color input signals.2 Neverthe-less, the use of the wavelength as a new parameterintroduces an additional factor in temporally incoher-ent optical processors, i.e., chromatic dispersion ofthe diffraction patterns. In particular, the Fraun-hofer diffraction pattern of an input transparencyprovided by means of a nondispersive refractive ob-jective suffers from strong lateral chromatic distor-

E. Tajahuerce, V. Climent, J. Lancis, and M. Fernandez-Alonsoare with the Departament de Ciencies Experimentals, UniversitatJaume I, 12080 Castello, Spain. P. Andres is with the Departa-mento de Optica, Universitat de Valencia, 46100 Burjassot, Spain.

Received 20 November 1997; revised manuscript received 5 May998.0003-6935y98y266164-10$15.00y0© 1998 Optical Society of America

6164 APPLIED OPTICS y Vol. 37, No. 26 y 10 September 1998

tion and thus broadband Fourier-based opticalinformation-processing setups cannot be direct repli-cas of their coherent counterparts.

Different approaches can be adopted to tackle thismatter. In this respect achromatic Fourier trans-formers try to compensate for the chromatic disper-sion of the Fraunhofer diffraction pattern, as theymake it possible to achieve, in a first-order approxi-mation, a wavelength-independent optical Fouriertransformation.3 The cornerstone of the procedurelies in achieving the incoherent superposition ofmonochromatic versions of the Fraunhofer diffractionpattern of the input that are generated by the differ-ent spectral components of the incident light in asingle plane and with the same magnification.

Needless to say, the achromatization of the opticalFourier transform pattern requires a strongly disper-sive optical system. All-glass achromatic Fouriertransform lenses4,5 and combinations of holographicand strongly dispersive refractive elements in cas-cade6,7 have been employed to reach the above goal.However, optical setups that combine diffractive andachromatic refractive lenses8–11 are certainly moreeasily implemented, as they employ only commercialavailable components. Moreover, these types of op-tical setup have permitted the development of someapplications in optical matched filtering and otheroptical processing techniques with white light.

In this paper we theoretically and experimentallydemonstrate that an air-separated diffractive lens

atmAa

scptStnts

U

doublet working under broadband convergingspherical-wave illumination provides an achromaticrepresentation of the intensity of the optical Fouriertransform of any input object. Chromatic compen-sation is performed by means of taking advantage ofthe severe chromatic aberrations associated with dif-fractive lenses. However, in contrast to other opti-cal setups previously reported, the system comprisesonly two common diffractive blazed lenses ~DL’s!, andthe axial location of the input provides a new degreeof freedom because it permits the variation of themagnification of the achromatic Fraunhofer diffrac-tion pattern. In this way, by simply moving the in-put along the optical axis, we are able to record a widecontinuous range of achromatic optical Fourier trans-forms of different size.

A discussion of the system is carried out by use ofthe Fresnel diffraction theory. The achromatic re-quirement and the analytical expression for the fielddistribution in the achromatic Fraunhofer plane ofthe diffractive doublet are derived within this frame-work. Moreover, we also obtain an analytical for-mula for evaluating the secondary spectra ~both axialnd transversal! by use of the wave-front aberrationheory, and we conclude that these residual chro-atic errors are low over the entire visible spectrum.n elementary geometrical-optics description of thebove optical architecture was reported in Ref. 12.In Section 2 we discuss the key idea on which the

etup is based. In Section 3 the behavior of the opti-al system is analyzed within the framework of thearaxial Fresnel diffraction theory. We emphasizehe scale-tunable nature of our proposal. Then inection 4 an evaluation of the secondary spectrum inerms of physical-optics concepts is carried out. Fi-ally, in Section 5 both the achromatic and the scale-unable properties of our diffractive Fourier transformetup are verified experimentally.

2. Heuristic Approach

To present the basic features of our achromatic Fou-rier transformer, we first remember that a DL has animage focal length of Z 5 Z0sys0, which is propor-tional to the wave number s of the incident light.The constant Z0 is simply the value of the focal lengthfor the reference wave number s0. In this way andwithin the paraxial approximation, one can analyti-cally represent its complex amplitude transmittanceby a quadratic phase factor, namely,

h~x, y! 5 expF2ips0

Z0~x2 1 y2!G . (1)

Next we reformulate the Fourier transformingproperty of a DL, as follows. Let an input transpar-ency with a wavelength-independent amplitudetransmittance t~x, y! be illuminated with a white-light point source S located at a normal-oriented dis-tance z, as shown in Fig. 1. The DL is located at adistance d from the point source. If we assume theusual Fresnel approximations to be valid for propa-gation over the distance d 2 z and take into account

10

the phase effect of the DL, it is easy to demonstratethat the monochromatic amplitude distribution U0~x,y; s! for the wave number s leaving the DL is given by

U0~x, y; s! 5 expFipS s

d 2 z2

s0

Z0D~x2 1 y2!G

3 *2`

`

* t~x9, y9!

3 expFipsS1z

11

d 2 zD~x92 1 y92!G3 expF2i2p

s

d 2 z~xx9 1 yy9!Gdx9dy9,

(2)

where some irrelevant constant factors have beenomitted. Finally, by use of the Fresnel diffractionformula once again and after some mathematical ma-nipulations, we can write the monochromatic com-plex amplitude over the transversal plane U1~x, y; s,d9! located at a distance d9 from DL as

1~x, y; s, d9! 5 expFips

d9 S1 2s

Ad9D~x2 1 y2!G3 *

2`

`

* t~x9, y9!

3 expHips

~d 2 z! Fdz

2s

A~d 2 z!G3 ~x92 1 y92!J 3 expF2 i2p

s2

Ad9~d 2 z!

3 ~xx9 1 yy9!Gdx9dy9, (3)

where the quantity A is given by

A 5s

d 2 z1

s

d92

s0

Z0. (4)

Fig. 1. Under white-light point source illumination the Fraun-hofer diffraction pattern of an arbitrary transparency provided bya DL is chromatically dispersed both axial and laterally. Thedifferent monochromatic versions form a frustum of a right-handcone whose apex coincides with the point source S.

September 1998 y Vol. 37, No. 26 y APPLIED OPTICS 6165

tqT

dtrod

D

6

From Eq. ~3! it is clear that the Fraunhofer diffrac-ion pattern of the input signal is achieved when theuadratic phase factor inside the integral is unity.his condition is verified when

dz

2s

A~d 2 z!5 0, (5)

or, if Eq. ~4! is taken into account, for a distanced9~s! such that

d9~s! 5Z0 ds

s0 d 2 Z0s . (6)

It appears that the transverse plane containing theFraunhofer diffraction pattern of the input transpar-ency is different for each wave number s. Moreover,in Eq. ~6! it is also possible to recognize that eachmonochromatic Fraunhofer plane coalesces, in termsof the geometrical-optics formulation, into the conju-gate plane through the DL of that containing thepoint source S.

On the other hand, substituting Eq. ~5! in for thelinear phase factor of Eq. ~3!, we obtain the scaling ofthe above Fourier transformation evaluated for eachwave number at the corresponding Fraunhofer plane,given by

x~s!

u5

y~s!

v5

zs

d9~s!

d5

Z0 zs0 d 2 Z0s

, (7)

where u and v are spatial frequencies.Inspection of Eqs. ~5! and ~7! reveals that the mono-

chromatic versions of the Fraunhofer diffraction pat-tern of the input transparency under broadbandspherical-wave illumination provided by a DL appearto be chromatically dispersed, both axially and later-ally. From Eqs. ~6! and ~7! it is important to recog-nize that the ratios x~s!yu@d 1 d9~s!# and y~s!yv@d 1d9~s!# are independent of s. Thus the same spatialfrequency is located in a different position in space foreach s, but they all lie along a straight line directedtoward the point source S. In other words, the set ofspatial frequencies with the same modulus r 5 ~u2 1v2!1y2 generates a diffraction volume with the shapeof a frustum of a right-hand cone whose apex coin-cides with S, as illustrated in Fig. 1.

Now it is apparent that a strongly dispersive opti-cal element is required for obtaining incoherent su-perposition in a single plane with the samemagnification as the above set of chromatic planarobjects. First, from geometrical-optics reasoning,we recognize that, to reach the above goal, the dis-persive imaging element must be inserted at thesource plane to achieve at the image plane the sameimage size for all wavelengths simultaneously. Ofcourse, this fact implies converging spherical-waveillumination, and thus d , 0. Second, we choose asthe imaging element a second DL whose focal lengthis proportional to the wave number, as mentionedabove. The proposed optical architecture for achiev-ing an achromatic Fourier transformation is depicted


in Fig. 2. In Section 3 we employ the Fresnel–Kirchhoff diffraction integral to discuss, in mathe-matical terms, the behavior of our optical proposaland to validate the present approach.

3. Theoretical Analysis

To this end, for each monochromatic component ofthe broadband point source S the complex amplitude

istribution at the input plane is propagated throughhe optical system shown in Fig. 2. We start byecognizing that the diffracted field reaching the sec-nd DL ~DL2! can be evaluated from Eq. ~3! if we set9 5 2d. Moreover, if we denote by Z09 the image

focal distance of DL2 for the reference wave numbers0, the spectral light amplitude U2~x, y; s! leaving

L2 is

U2~x, y; s! 5 expF2ipSs

d1

s2

A9d2 1s0

Z09D~x2 1 y2!G

3 *2`

`

* t~x9, y9! 3 expHips

~d 2 z!

3 Fdz

2s

A9~d 2 z!G~x92 1 y92!J3 expFi2p

s2

A9d~d 2 z!~xx9 1 yy9!Gdx9dy9,

(8)

where

A9 5sz

d~d 2 z!2

s0

Z0. (9)

It is a straightforward matter to show that, in free-space propagation, the monochromatic field distribu-tion U3~x, y; s, D9! for the wave number s over the

Fig. 2. Air-separated DL doublet arranged for producing an ach-romatic Fourier transformation.

ttgt

T

h

transversal plane located at a distance D9 from DL2can be written as

U3~x, y; s, D9! 5 expF2ips

D9 S1 1s

BD9D~x2 1 y2!G3 *

2`

`

* t~x9, y9!expHips

~d 2 z!

3 Fdz

2s

A9~d 2 z!1

s3

BA92d2~d, 2 z!G3 ~x92 1 y92!J3 expF2i2p

s3

BA9dD9~d 2 z!

3 ~xx9 1 yy9!Gdx9dy9. (10)

In Eq. ~10! the quantity B is given by

B 5s

d1

s2

A9d2 1s0

Z092

s

D9. (11)

The irradiance distribution at the preceding planecorresponds to the Fourier transform of the functiont~x, y!, provided that

dz

2s

A9~d 2 z!1

s3

BA92d2~d 2 z!5 0. (12)

Constraint ~12! requires that the quadratic phase fac-tor inside the integral remain equal to unity for allvalues of the spatial coordinates x9 and y9. Insertingthe forms for A9 and B9 from Eqs. ~9! and ~11!, re-spectively, into Eq. ~12! and after a long but directcalculation, we conclude that the Fourier transform isachieved at a distance D9~s!, given by

D9~s! 51

1d

1s0

Z09s2

Z0s

s0 d2

. (13)

Again we note that, for each wave number s, theransverse plane containing the Fraunhofer diffrac-ion pattern of the input transparency is the conju-ate plane through the DL doublet of that containinghe source S.

In a similar way and considering the linear phasefactor in Eq. ~10!, we find that the scale factor of theFourier transform achieved at the above plane is

x~s!

u5

y~s!

v5

zZ0

s0 d2 D9~s!. (14)

he functional dependence of D9 and xyu or yyv on s,fixed by Eqs. ~13! and ~14!, respectively, indicatesthat both the position and the scale of the Fraunhoferdiffraction pattern of the input provided by the dif-fractive doublet are wavelength dependent. Sochromatic errors, both axial and transversal, appear.

10

Furthermore, if Eq. ~14! is taken into account, it canbe recognized that D9~s! and x~s!yu or y~s!yv arerelated by a constant factor.

Achromatic correction can be achieved if the deriv-ative of the function D9~s! or equivalently of the scalefactor with respect to the wave number vanishes at acertain design wave number s0. In this way D9~s!

as a stationary value. This approach guarantees

Fig. 3. Plot of the functional dependence on s of the secondaryspectrum associated with the Fraunhofer diffraction pattern pro-vided by ~a! our proposed DL doublet ~solid curve! and ~b! a con-ventional nondispersive objective ~dashed line!. In both cases weassume white-light illumination.

Fig. 4. Graphical representation of the maximum value CAM ofthe function CA~s! versus the spectral bandwidth Ds of the illu-minating source for ~a! the DL doublet ~solid curve! and ~b! anachromatic objective ~dashed line!. The extreme wave numberswere chosen in such a way that s0, given by Eq. ~24!, is in all casesequal to 1.75 mm21.


dhl

ot

l

et

rwtctoi

atfi

6

that the final diffraction volume constituted by themonochromatic versions of the Fraunhofer diffractionfield is folded about the transversal plane located ata distance D90 5 D9~s 5 s0! from DL2. In mathe-matical terms, we require

]D9(s)]s

Us5s0

5]x(s)

]sU

s5s0

5]g(s)

]sU

s5s0

. (15)

After some simple algebraic manipulations, Eq. ~15!leads to the constraint

Z09 5 2d2

Z0, (16)

which links the focal length for the reference wavenumber s0 of both DL’s with the separation betweenthem. Finally, by combining Eqs. ~13! and ~16!, we

etermine that the achromatic picture of the Fraun-ofer diffraction pattern of the diffracting screen is

ocated at a distance D09 from DL2 such that

D09 5d2

d 2 2Z0. (17)

Equation ~16! demonstrates that the image focallengths of DL1 and DL2 have opposite signs. More-ver, to obtain a real achromatic Fourier transforma-ion, i.e., D09 5 0 from Eqs. ~16! and ~17!, we infer that

DL1 and DL2 should be diverging and converginglenses, respectively, such that the ratio uZ09yZ0u be-ongs to the interval @0, 4#. Another choice of the

above ratio leads to a virtual achromatic Fraunhoferdiffraction pattern, and in this case an additionalrefractive objective is required to produce a real im-age and then to yield access to the spectral content ofthe input object.

On the other hand and from a practical point ofview, we wish to emphasize that, as indicated in Eq.~14!, the scale factor of the achromatic Fourier trans-form is linearly related to the distance z between theinput transparency and DL2. Thus, by simply se-lecting the position of the input aperture along theoptical axis of the system, we can alter the magnifi-cation of the achromatic Fraunhofer diffraction pat-tern up to a prescribed value.

4. Secondary Spectrum

As a result of the achromatic correction, low residualchromatic aberrations, both axial and transversal,remain. For evaluating the secondary spectrum,the field amplitude at the achromatic Fraunhoferplane of the DL pair is calculated for each wave num-ber of the illuminating source. Using polar coordi-nates, replacing D9 with D09 in Eq. ~10!, and takinginto account Eqs. ~16! and ~17! make it a straightfor-


ward matter to show that the amplitude distributionat issue, U4~r, u; s, D09!, can be written as

U4~r, u; s, D09! 5 expF2i2pE~s!

b2 r2G *0

`

*0

2p

t~r9, u9!

3 expFi2pF~s!

a2 r92G3 expF2i2p

G~s!

abr r9

3 cos~u 2 u9!G r9dr9du9. (18)

In Eq. ~18! a and b stand for the maximum radialxtent at the input and the output planes, respec-ively, and the functions E~s!, F~s!, and G~s! are,

respectively,

E~s! 512 S b

D09D2

3 3D09s 11

d 2 zdzs 1 ~d 2 z!Z09s0

21

Z09

~2s 2 s0!

s2 4 ,

F~s! 5 212 Sa

zD2 s

d 2 zdz

21

Z09

s~2s 2 s0!

~s 2 s0!2

,

G~s! 5 2SazDS b

D09D Z09s0

1 2~s 2 s0!

2

s2 Sd 2 zdz

Z09s0

s1 1D .

(19)

First, we note that in general the achromatic Fou-ier transform suffers from a quadratic phase error,hich indeed results in wavelength dependence. In

his paper we neglect the above phase curvature be-ause we are interested in only an achromatic Fourierransform in irradiance. Second, in Eq. ~18! we rec-gnize the quantity F~s! in the quadratic phase factornside the integral as the defocus coefficient W20 of

the aberration function in units of wavelength, whichis employed profusely in the description of opticalimaging systems.13 In this way we define the defo-cus chromatic aberration ~DCA! as

DCA~s! 5 F~s!. (20)

Similarly, the quantity G~s! in the linear phase factorin Eq. ~18! is related to the coefficient W11 of theberration function in units of wavelength. Hencehe transversal chromatic aberration ~TCA! is de-ned as

TCA~s! 5 G~s! 2 G~s0!. (21)

From Eqs. ~19! we recognize that both the defocusand the transversal chromatic aberrations havesmaller values as the distance z in modulus becomes

w

c

greater. For the remainder of this section, and withno lack of generality, we restrict our analysis of bothfunctions to the most unfavorable situation in whichuzu reaches its minimum value, i.e., z 5 d. In thiscase the transparency is against DL1. Under theabove assumption, the secondary spectrum results in

DCA~s! 5 212 Sa

dD2

Z09s0CA~s!,

TCA~s! 5 2SadDS b

D09D Z09s0CA~s!, (22)

here the chromatic aberration ~CA! is given by

CA~s! 5~s 2 s0!

2

s0~2s 2 s0!. (23)

Thus aside from some constants both chromaticerrors have an identical functional dependence on the

Fig. 5. Irradiance distribution at the achromatic Fraunhofer plangrating, is located in two different axial positions: ~a! Gray-level~b! Irradiance profile along a horizontal line in ~a! for each RGB comirradiance distribution when z 5 2200 mm. ~d! RGB irradiance

10

wave number. From a practical point of view it isconvenient to choose the reference wave number s0 insuch a way that CA~s1! 5 CA~s2!, where s1 and s2denote the effective end wave numbers of the incidentlight. In this way, in each transversal plane insidethe final diffraction volume we obtain, with the samescale factor, the Fraunhofer diffraction pattern fortwo different wave numbers. From Eq. ~23! thisvalue of s0 is

s0 5 2s1s2

s1 1 s2. (24)

Equation ~24! is equivalent to selecting the referencewavelength l0 as the arithmetical mean between theextreme wavelengths l1 and l2. In Fig. 3 a plot ofCA versus the wave number s is shown as a solidurve. In this plot we assume that s1 5 sc 5 1.52

mm21 and that s2 5 sF 5 2.06 mm21, which corre-

he setup shown in Fig. 2 when the input, a 2-D square diffractionre of the Fraunhofer irradiance distribution when z 5 2283 mm.ent of the incident light. ~c! Gray-level picture of the Fraunhoferle along a horizontal line in ~c!.

e of tpictu

ponprofi


F

wod

m

fdtZ

6

spond, respectively, to the Fraunhofer lines labeled Cand F for the visible spectrum. In this way from Eq.~24! we obtain s0 5 1.75 mm21. For comparison thefunctional dependence evaluated in an identical way,on the wave number of the transversal chromaticaberration associated with the Fraunhofer diffractionpattern provided by a conventional refractive lens isshown as a dashed line in Fig. 3. In this case

TCA~s! 5 2SafDSb

fD fs0CA~s!, (25)

where f is the image focal length of the lens and nowCA~s! is

CA~s! 5s 2 s0

s0. (26)

For our optical arrangement it is obvious that themaximum value of the function CA~s!, denoted byCAM, is achieved at both extreme wave numbers.

rom Eqs. ~23! and ~24! CAM yields

CAM 5

SDs

s0D2

1 1 F1 1 SDs

s0D2G1y2 , (27)

here Ds stands for the effective spectral bandwidthf the illuminating source, i.e., Ds 5 s2 2 s1. Theependence of CAM on Ds for s0 5 1.75 mm21 is

shown by the solid curve in Fig. 4. As above, thevariation of Ds of the maximum value of the trans-versal chromatic aberration given by a refractive lensis shown by the dashed line in the same figure.From Eq. ~26! we obtain

CAM 5 CA~s2! 2 CA~s1! 5Ds

s0. (28)

Fig. 6. White-light Fraunhofer diffraction pattern of a 2-D diffrac~a! gray-tone representation and ~b! RGB irradiance profile along


Examination of the plots in Fig. 4 reveals that thechromatic compensation carried out with our pro-posal is a factor of the order of 10 higher even forwhite light ~Ds 5 0.5 mm21! if all other multiplicativeconstants that appear in Eqs. ~22! and ~25! are equal.

In Eq. ~22! and within the paraxial approximation,it is possible to recognize the ratios ~ayd! and ~byD09!as the half-angles subtended from the point source bythe maximum radial extent of the input and the out-put, respectively. In particular, for the choice of~ayd! 5 ~byD09! 5 0.0436, which corresponds to amaximum angular extent of 62.5°, and for Z09 5 100

m and s0 5 1.75 mm21, the resulting absolute valueof the maximum defocus chromatic aberration is 3.8for a light source whose spectral content spreads overthe entire visible spectrum. Furthermore, with thesame selection for the geometrical parameters a spec-tral range of Ds 5 135.7 mm21 ~Dl 5 44.2 nm! isenough to achieve a maximum defocus chromatic er-ror of less than 1y4. In accordance with the Ray-leigh quarter-wavelength rule, the separated DLdoublet behaves in this case like a broadband opticalFourier transformer that is completely free fromchromatic error.

5. Experimental Verification

To verify experimentally the behavior of our sepa-rated DL doublet, we constructed the optical arrange-ment shown in Fig. 2 by following the prescription inEq. ~16!. A converging white-light beam createdrom a high-pressure xenon arc lamp impinged on theiffracting screen. For the practical implementa-ion we chose two DL’s with image focal distances of0 5 2200 mm and Z09 5 100 mm for the wavelength

l 5 514 nm. Both lenses are four-level diffractiveelements constructed with multimask-level photore-sist technology, with the diffraction efficiency for theprincipal focal length limited to approximately 70%.With a choice of s0 in accordance with Eq. ~24!, i.e.,s0 5 1.75 mm21, the axial separation between the

rating provided by a conventional achromatic refractive objective:rizontal line.

tion ga ho

~o

ao

DL’s was d 5 2127.2 mm. In this way the achro-matic Fraunhofer diffraction pattern of the inputtransparency was achieved at a distance of D09 533.2 mm from DL2 @see Eq. ~17!#, and the maximumvalue of the chromatic error @fixed by Eqs. ~22! and

Fig. 7. Same as for Fig. 5 but for three different axial positions oz 5 2200 mm, and ~e! z 5 2160 mm.

10

27!# was 15.0 ~a maximum angular extent of 65°ver both the input and the output plane is assumed!.Figure 5 shows a gray-level picture of the irradi-

nce distribution at the achromatic Fraunhofer planef our DL pair when a two-dimensional ~2-D! square

uble circular aperture as the input object: ~a! z 5 2283 mm, ~c!
f a do

fiactfcmbmclvsplst

pcDadIYarnptlaFcp

ctd

6

diffraction grating is employed as an input transpar-ency. Two different axial locations for the gratingwere considered. These images were recorded byplacement of a color CCD camera at the real achro-matic Fraunhofer plane of the optical setup. Theplots in Figs. 5~b! and 5~d! show the irradiance pro-

les ~for each RGB component of the incident light!long a horizontal line obtained by integration of theorresponding 2-D irradiance distribution with a ver-ical slit. It is well known that the Fourier trans-orm of the amplitude transmittance of a 2-D gratingonsists of a series of regularly spaced diffractionaxima. From Figs. 5~a! and 5~c! it is clear that, for

oth axial positions, the locations of the diffractionaxima are nearly wavelength independent. For

omparison a gray-level representation of the white-ight Fourier transform of the same 2-D grating pro-ided by a conventional spherical refractive lens ishown in Fig. 6~a!. For this situation the irradiancerofile for each RGB component along a horizontaline is depicted in Fig. 6~b!. These plots show thetrong dependence on wavelength of the locations ofhe diffraction maxima.

White-light-achromatized Young’s fringes and Airyatterns are illustrated in Fig. 7. Here a doubleircular aperture is inserted at the input plane of ourL doublet. Three different axial positions of theperture were considered. The resulting RGB irra-iance profiles are plotted in Figs. 7~b!, 7~d!, and 7~f !.t is remarkable that the spacing between theoung’s fringes at the achromatic output plane islmost the same for all wavelengths of the incidentadiation. Thus many dark fringes can be recog-ized. Also, the position of the first zero of the Airyattern is easily detected. A gray-level representa-ion and the horizontal RGB profiles of the white-ight Fourier transform of the same double circularperture as provided by a refractive lens are shown inigs. 8~a! and 8~b!, respectively. Note the severehromatic-blurring effects that distort the diffractionattern.

Fig. 8. Same as for Fig. 6 but for a dou


The experimental results described above are con-lusive. The achromatic and scale-tunable Fourierransforming capabilities shown by our proposed DLoublet are notable.

6. Conclusions

In summary broadband dispersion compensation forthe Fraunhofer diffraction pattern of an arbitraryinput transparency has been carried out by means ofa separated DL doublet. Although the optical archi-tecture is unable in general to produce an exact can-cellation, we have demonstrated by using Fresneldiffraction theory that the conditions for achieving awhite-light achromatic optical Fourier transform canbe accomplished if the separation between the DL’s isproperly chosen. To this end the input must be il-luminated with a white-light spherical convergingwave front, the lens pair DL1 and DL2 must haveimage focal lengths opposite in sign, and DL2 must beinserted at the transverse plane containing thebroadband virtual point source.

Furthermore, we have determined and verified ex-perimentally that the position of the input along theoptical axis permits the variation of the scale factorof the Fourier transformation. Nevertheless, theachromatism is preserved. Results of laboratory ex-periments concerning the achromatization and thescale tunability of the Fraunhofer diffraction patternof two archetypical diffracting apertures have clearlyshown both abilities.

A simple analytical expression for evaluating thechromatic behavior of our proposed method has beenobtained by use of wave-front aberration theory. Forwhite light the residual chromatic aberration associ-ated with our achromatic Fourier transformer is ap-proximately 10 times lower than that of a refractiveFourier transform lens. Moreover, our configurationeffectively behaves as a chromatically unaberratedsystem for a certain limited spectral bandwidth ~ap-proximately 50 nm!.

The flexibility and very easy implementation na-

ircular aperture as a diffracting screen.
ble c

c

4. C. G. Wyne, “Extending the bandwidth of speckle interferom-
ture of the proposed setup are other practical issuesto be considered. In this respect our proposed setupdoes not require any dispersive or glass objective, andthus it can be used for other ranges of the electro-magnetic spectrum outside the visible region, say, forinstance, with soft x-rays. In this region it is verydifficult to use conventional refractive optics becausethe refractive indices of the materials are slightly lessthan unity. Zone plates were proposed some yearsago to tackle this subject in the field of x-ray micros-copy. In this way achievable resolutions of the orderof 10 nm have been demonstrated.14 These applica-tions could now be extended to broadband x-raysources with the achromatic device we have proposed.
This research was supported by the Direccion Gen-eral de Investigacion Cientıfica y Tecnica ~grantPB93-0354-C02-02!, Ministerio de Educacion y Cien-ia, Spain.

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achromatic fourier transforming properties of a separated diffractive lens doublet: theory and...

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