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3600 J. Opt. Soc. Am. A/Vol. 24, No. 11 /November 2007 Mendoza-Yero et al.

Focusing and spectral characteristics of periodicdiffractive optical elements with circular

symmetry under femtosecond pulsed illumination

Omel Mendoza-Yero,* Gladys Mínguez-Vega, Jesús Lancis, and Vicent Climent

GROC, Departament de Física, Universitat Jaume I, E12080 Castelló, Spain*Corresponding author: omendoza@uji.es

Received July 25, 2007; accepted September 21, 2007;posted September 26, 2007 (Doc. ID 85623); published October 25, 2007

The analytical solution is derived, within the Rayleigh–Sommerfeld formulation of diffraction, for the on-axisspectral irradiance of a broadband source after diffracting through a circular symmetric hard aperture. Byusing this solution, and within the paraxial approximation, we investigate several diffraction-induced effectsoriginated by binary diffractive optical elements made up of a set of annular apertures with equal areas andperiodic in the squared radial coordinate. In particular, the ability to focus femtosecond pulses is investigated.In addition, the analysis of the spectral modifier function associated with these elements allows us to simulatespectral shifts at focus positions. Finally, we introduce a relatively simple and low-cost technique to slice thespectrum of a broadband source in order to generate narrow bands or wavelength channels. © 2007 OpticalSociety of America

OCIS codes: 050.1220, 050.1970, 320.2250.

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. INTRODUCTIONingular optics is a well-established branch of physicshat studies the structure of monochromatic wave fieldsn the vicinity of points where the complex amplitude isero. At such points the phase of the wave is singular, ex-ibiting a very complicated behavior. For a spatially co-erent polychromatic field diffracted by a hard aperture,ramatic spectral changes near phase singularities wereredicted [1]. They included spectral switches and a two-obe spectrum, which were soon verified experimentally2]. The phenomenon of spectral switches previously dem-nstrated for spatially partially coherent light [3,4] can benderstood within the framework of polychromatic singu-

ar optics [5]. To indicate the significance of this noveliffraction-induced phenomenon, it has been a matter oftudy in high-numerical-aperture systems [6] and vecto-ial electromagnetic fields [7].

With the increasing number of new applications foremtosecond pulsed waves, it becomes necessary to under-tand the properties of the spectral changes in highly co-erent polychromatic fields. On this topic, the control oflueshifts and redshifts of the spectrum caused by diffrac-ion of a spatially coherent light at the edges of hardpertures may play an important role in the field of infor-ation encoding and transmission for optical telecommu-

ications [8,9]. The analytical expressions for the on-axispectral shifts and spectral switches of the power spec-rum of an ultrashort Gaussian pulse passing through aircular aperture were reported [10]. Furthermore, thepectral and temporal evolution of a Gaussian pulse dif-racted through a slit, near phase singularities, has beenecently investigated by means of a numerical simulationechnique [11,12].

On the other hand, because of the high repetition rate

1084-7529/07/113600-6/$15.00 © 2

nd the large spectral bandwidth of femtosecond pulsesmitted by mode-locked Cr4+:YAG lasers [13,14], theyhow potential applications in wavelength division multi-lexing (WDM) and/or optical time division multiplexingOTDM). The wavelength range of these pulses can over-ap with the second and third optical communication win-ows. In this context, the techniques used to slice a broadpectrum into several wavelength channels allow the costf multiwavelength transmitters for WDM or for multi-avelength OTDM to be reduced [15]. Initially, these

echniques were proposed for LED sources [16]. Theseechniques need spectral filters commonly based on fiberragg gratings and prisms [17,18], arrayed waveguideratings [19], acousto-optical modulators [20], or Fabry–erot etalons [21,22].In this paper, within the Rayleigh–Sommerfeld theory

f diffraction, an exact solution for the on-axis power spec-rum of an ultrashort pulse after diffracting through a setf N annular concentric hard apertures is obtained. Thisesult is particularly useful for investigating both the on-xis diffraction effects on the spectrum of a broadbandource and the focusing properties of ultrashort lightulses passing by binary diffractive optical elementsDOEs) with circular symmetry. Within this context, wehow how DOEs can be used to slice a broad spectrum.

The structure of the paper is as follows. In Section 2, aeneral expression for the spectral modifier function ofOEs with circular symmetry is derived, including some

omments on the well-known case of the circular aper-ure. The class of DOEs composed of annular apertures ofqual areas and also periodic in the squared radial coor-inate is treated with some detail in Section 3. In particu-ar, we explore their focusing characteristic under ul-rashort pulsed illumination and the ability of these

007 Optical Society of America

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Mendoza-Yero et al. Vol. 24, No. 11 /November 2007 /J. Opt. Soc. Am. A 3601

lements to cause spectral shifts or to slice the spectrumf a broadband source into several wavelength channels.he discussion of results appears in Section 4, whereashe conclusions are given in Section 5.

. SPECTRAL MODIFIER FUNCTIONhe total on-axis spectral field A�z ,�� of a light pulse af-

er its propagation through a set of N concentric annularard apertures can be assessed from the inverse Fourierransform of the corresponding total diffraction field, thats,

A�z,�� =�−�

�

U�z,t�exp�i�t�dt. �1�

n this case, the total diffracted field U�z , t�, within theayleigh–Sommerfeld diffraction theory, is given by thexpression [23]

U�z,t� = �m=1

N z

�z2 + rim2

u0�t −�z2 + rim

2

c�

− �m=1

N z

�z2 + rom2

u0�t −�z2 + rom

2

c� . �2�

he symbol u0�t� holds for the temporal amplitude of thencident plane wave pulse, whereas rim and rom denotehe inner and the outer radii, respectively, of the mth an-ular aperture. From Eqs. (1) and (2) it is easy to showhat

A�z,�� = A0����m=1

N z

�z2 + rim2

exp�i��z2 + rim

2

c�

− A0����m=1

N z

�z2 + rom2

exp�i��z2 + rom

2

c� . �3�

n Eq. (3), A0��� represents the spectral field of the inci-ent pulse given by the inverse Fourier transform of u0�t�.herefore, the total spectral field is the result of the sumf the spectral fields corresponding to 2N boundary waveulses emitted at the edges of the annular apertures. Thepectral field of a boundary wave pulse is given by theultiplication of the spectral field of the incident pulse byspherical wave originated at the edge of the aperture. Inddition, the spectral field contributions from inner anduter edges of an annular aperture have opposite signs.

From Eq. (3), the spectral irradiance S�z ,��= �A�z ,���2ields

S�z,�� = S0���z2 �m=1

N

�n=1

N cos�

c�dim − din��dimdin

+

cos�

c�dom − don��domdon

− 2

cos�

c�dim − don��dimdon

� ,

�4�

here S ���= �A ����2 is the spectral irradiance of the in-

0 0

ident pulse and the terms d�i,o��m,n�=�z2+r�i,o��m,n�2 are

he distances from the edges of the annular apertures ton axial position z. Equation (4) is the main analytical re-ult of this work. The right hand term between curlyrackets is the so-called spectral modifier function�z ,��. This function can be interpreted as a spectral fil-

er that transforms the input signal S0��� into a modifiedutput spectral irradiance. Therefore, it allows us to pre-ict the on-axis spectral changes caused by the diffractionf broadband light by concentric circularly symmetricard apertures. It depends on the geometric features ofhe binary DOE and on the axial position considered.ote that for a fixed frequency component �=�0, the

unction S�z ,�0� coincides with the on-axis irradiance as-ociated with the diffraction of a monochromatic planeave. It is easy to see from the above ideas that the func-

ion M�z ,�� is suitable for both the analysis of on-axispectral shifts and the study of the focusing properties ofOEs with circular symmetry.In the simplest case of a circular aperture of radius a,

he spectral modifier function reduces to the form

M�z,�� = 1 +z2

z2 + a2 − 2z

�z2 + a2cos�

c�z − �z2 + a2�� .

�5�

or a fixed frequency component �=�0, the above func-ion oscillates between zero and four, showing axial posi-ions zl�0

(with l=1,2, . . .), at which M�zl�0,�0�=0. At

hese points, called phase singularities, the spectrumplits into two parts about �0. Within the paraxialpproximation it is easy to see that M�z ,��4 sin2 �a2 / �4cz�� and zl�0

=�0a2 / �2�cl�. The last expres-ion for M�z ,�� coincides with a result already reported5]. Further, frequencies of the spectrum given in the form+/−=�0±��, where �� denotes the frequency detuning,re omitted at the axial positions zl�+/−zl�0

±��a2 / �2�cl�. The linear dependence of zl�+/−on ��

eads to the spectral shift phenomenon. That is, as weove away from the circular aperture during a transition

rom zl�−to zl�+

, spectral irradiance goes from blueshiftso redshifts. This effect is also known as spectral switch.he frequency of these oscillations increases as we move

n toward the circular aperture. Studies on theiffraction-induced spectral shifts due to the passing ofroadband radiation through a circular hard apertureave been carried out [10].In order to gain further insight into the wide applica-

ion possibilities of Eq. (4), we focus our attention on theore interesting cases of binary periodic DOEs made of

ircular concentric hard apertures. Apart from the well-nown Fresnel zone plate (FZP), which is a particularase of the above structures, we found no publicationsbout their focusing properties with ultrashort pulses orhe associated spectral shifts originated by diffraction ef-ects.

. PERIODIC DIFFRACTIVE OPTICALLEMENTSereafter, we assume that the paraxial approximationolds; so d �z+r2 / �2z� and d d =d d

�i,o��m,n� �i,o��m,n� im in om on

=tr

AtIr−snW(c[−o

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fc

FtPd=M

pl=FMhfIta

AOTtl

Apt�cigz=at

aospdtsmfrn=dvcr

3602 J. Opt. Soc. Am. A/Vol. 24, No. 11 /November 2007 Mendoza-Yero et al.

dimdon�z2. Then, after some mathematical manipula-ions of Eq. (4), the spectral modifier function M�z ,�� isewritten in the form

M�z,�� = 4�m=1

N

�n=1

N

sin��ron2 − rin

2 �

4cz �sin��rom2 − rim

2 �

4cz ��cos��ron

2 − rom2 �

4cz−

��rim2 − rin

2 �

4cz � . �6�

t this point, a DOE composed of N annular hard aper-ures of identical areas �0 is considered �ron

2 −rin2 =�0 /��.

n addition, if we also assume periodicity in the squaredadial coordinate, the following relation is satisfied: ron

2

rom2 =rin

2 −rim2 =��n−m��0 /�. The parameter ��1 repre-

ents the ratio of the area given within the bounds of theth and the mth inner (or outer) ring radii to the area �0.e refer to this kind of binary mask as periodic DOE

PDOE). We note that the spectral modifier function of aonvergent FZP [which can be obtained from Eq. (4) of24]] is a particular case of Eq. (6) when ri�m,n�=p �m ,n�1�1/2 and ro�m,n�=p �m ,n�−1/2�1/2, where p2 is the periodf the zone plate transmittance.

Now, for design purposes we set ri1=0. Then the outernd the inner radii of the hard apertures within theDOE are determined by the formulas rom ��0�� /��1/2 m− �1−1/���1/2 and rim= ��0�� /��1/2 m1�1/2, respectively. In according with this, the period p2

f the PDOE is �0� /�. In the case of �=2, these relationsoincide with those for the FZP previously mentioned. Inig. 1 we plot the amplitude transmittance of the PDOEsith �=2, �=3, and �=4, using the same spatial magnifi-

ation scale in all cases.Based on the above assumptions, the spectral modifier

unction given by Eq. (6) transforms into the followingompact result:

M�z,�� = 4 sin2� ��0

4�cz�sin2�N

���0

4�cz�sin2����0

4�cz�. �7�

irst, the slowly oscillating right-hand term in Eq. (7) de-ermines the energetic content of the different foci of theDOE, whereas the trigonometric quotient allows us toerive their axial locations. These are given by zn���0 / �4�2cn� (with n=1,2, . . .). In this way we find�zn�=4N2 sin2�n� /��. For n=m� (with m=1,2, . . .), the

Fig. 1. PDOEs with (

ositions of the foci coincide with zero values of the enve-ope corresponding to axial phase singularities. When �2, Eq. (7) reduces to the spectral modifier function of aZP, which is commonly expressed in the form [9]�z ,��=sin2�N��0 /2�cz� / cos2���0 /4�cz�. On the other

and, if N=1, Eq. (7) reduces to the spectral modifierunction of a circular or annular aperture of area �0 [5].n accordance with the above analysis, if the position ofhe principal focus is denoted z1= f1, the foci of the FZPre given by f1 / �2m+1�.

. Focusing Femtosecond Pulses by Periodic Diffractiveptical Elementshe spatial irradiance distribution at the position z fromhe PDOE is evaluated numerically by means of the fol-owing integral:

I�z,�0� =�0

�

M�z,��S0���d�. �8�

t this point, it is convenient to analyze first the focusingroperties of the PDOEs under monochromatic illumina-ion. In mathematical terms S0���=Smon�D��−�0�, whereD�� is the Dirac function of argument and Smon is aonstant with the same dimensions as S0���. In Fig. 2, therradiance function I�z ,�0� is plotted for �=3 and �=4, to-ether with that for a FZP, within the axial interval from1=0.56 m to z2=2 m. For this plot, �0=11.78 m2, �03� /4�1015 Hz (equal to a wavelength of �0=800 nm),nd N=9. The selection of this interval allows us to showhe second, third, and fourth foci of the FZP.

From Fig. 2, one can see that for �=3 and �=4, therere two and three foci, respectively, in place of each focusf the FZP. In the case of �=4, the central focus has theame axial position and height as that of the FZP. Theeak values of secondary foci shown in Figs. 2(a) and 2(b)iffer from the focus of the FZP by 25% and 50%, respec-ively. In general, the focusing features of the PDOEs aretrongly related to the value of the parameter �. Underonochromatic illumination there are �−1 modulated

oci in the vicinity of focal points associated with the cor-esponding FZP. However, the total on-axis energy doesot depend on the parameter �, that is E��0

�M�z ,��dzN��0 / �2c�. Consequently, the total on-axis energy is re-istributed. Therefore, in the case of PDOEs with an evenalue of the parameter �, the spatial resolution of theirentral focus is improved with respect to that of the cor-esponding FZP.

, (b) �=3, and (c) �=4.

a) �=2

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fStstfitpTtdssTspmtc

ioriwp=twF

ctg

curves

F�

Mendoza-Yero et al. Vol. 24, No. 11 /November 2007 /J. Opt. Soc. Am. A 3603

After the above discussion, we consider a plane waveulse with temporal amplitude u0�t�=exp�−i�0t�exp−t2 / �4 0

2�� and standard deviation width of 0=3 fsequivalent to an amplitude full width at half-maximumf 10 fs) that illuminates the PDOE. The term �0 denoteshe central frequency of the incident pulse. The spectralrradiance of the incident pulse was S0�����−�

� u0�t�exp�i�t�dt�2=4� 02 exp −2 0

2��−�0�2�.In Fig. 3, the spatial irradiance distribution of PDOEs

ith �=3 and �=4 are shown, for the same axial intervaliven in Fig. 2, for both monochromatic illumination andncident femtosecond pulse illumination of 0=3 fs. Foromparison, the reported focusing behavior of a FZP un-er femtosecond pulse illumination [25,26] is also shownn Fig. 3. It is clear from Fig. 3 that the secondary foci ofhe PDOEs are almost indistinguishable for femtosecondllumination owing to the superposition of the differentiffraction patterns associated with each monochromaticomponent of its broad spectrum. In addition, the differ-nce between maximum and minimum values of the axialrradiance decreases, giving rise to a loss of contrast. Inact, there are no longer axial zeros of the irradiance dis-ribution within the interval given in Fig. 3. Note that be-ause of the presence of the additional foci, the minimumalue of the axial irradiance of the PDOEs is increased.he envelope of the axial irradiance distribution for fem-osecond illumination is irregular, and it changes in accor-ance with the variations of positions and transversalizes of the foci for monochromatic illumination.

. Diffraction-Induced Spectral Shifts by Periodiciffractive Optical Elementse start this subsection by analyzing the on-axis spectralodifications due to the diffraction of a femtosecond pulse

y a PDOE. The on-axis spectral irradiance is determined

Fig. 2. Irradiance function versus the axial coordinate z. Solid

ig. 3. Normalized irradiance versus the axial coordinate z. So=3, and (c) �=4; diamonds, case of femtosecond pulse illuminat

rom S�z ,��=M�z ,��S0���. The initial Gaussian spectrum0��� is assumed to have a full width of 4/ 0. Consequen-

ially, for representation purposes the spectral interval iset to be �0−2/ 0����0+2/ 0. In Fig. 4, several spec-ral changes, as we move from left to right, of secondaryoci given in Figs. 2(a) and 2(b) are shown. All the spectran Fig. 4 are normalized to unity, and the insets located atheir top right-hand parts indicate the considered axialosition. The initial spectrum S0��� is plotted in Fig. 4(g).he spectra in Figs. 4(a) and 4(f) are redshifted, whereas

hose given by Figs. 4(c) and 4(d) are blueshifted. In ad-ition, Figs. 4(b) and 4(e) show multiple splitting. Ithould be pointed out that, except for the spectrum de-cribed in Fig. 4(b), spectra are plotted at foci positions.herefore, one can perform the experimental test withpectrometers of low sensitivity. The central frequency ofeaks, except for the case given in Fig. 4(b), can be deter-ined by the simple relation �= �q /n��0. Here, q is an in-

eger positive number, and n denotes the subscript asso-iated with the axial position of a focus.

In this context one might realize that diffraction-nduced spectral modifications that take place at the focif PDOEs can slice the initial spectrum into several nar-ow bands or channels. In order to illustrate the abovedea, we plot the normalized spectral irradiance of PDOEsith �=2 and �=4 at the axial position z=12.67 cm. Thisosition corresponds to the monochromatic focus for ��0 obtained with n=37 and n=74, respectively. Note

hat this result is also valid with a LED used as a sourcehenever the spectrum corresponds to the one shown inig. 4(g).Visual inspection of Fig. 5 clearly identifies 7 spectral

hannels at the focus of the FZP, whereas in the case ofhe PDOE with �=4 this number is multiplied by three toive 21 channels. It can be proved that the greater the

, (a) �=3 and (b) �=4; diamonds, case of the Fresnel zone plate.

ves, monochromatic illumination using PDOEs with (a) �=2, (b)

lid curion.

ntntshn

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3604 J. Opt. Soc. Am. A/Vol. 24, No. 11 /November 2007 Mendoza-Yero et al.

umber of annular apertures of a PDOE, the shorter ishe spectral width of these channels. In addition, theumber of channels within the spectral width depends onhe axial position considered, being greater for a focus po-ition closer to the PDOE. The central frequency andeight of each channel can be tuned through PDOE engi-eering.

. ANALYSIS OF RESULTShen an ultrashort light pulse impinges a PDOE, itsonochromatic foci are spatially extended along the axial

irection. Hence a single PDOE is not good for tightly fo-using ultrashort light pulses, as Fig. 3 shows. However,he ability of PDOEs to originate an extended depth of

Fig. 5. Spectra in the focus located at z

Fig. 4. Spectra at different axial positions (indicated b

eld with broadband sources can find application in oph- 6

0

halmology. On the other hand, a suitable combination ofDOEs and refractive optical elements has been demon-trated to significantly compensate the above-mentionedhromatic dispersion [27].

The spectral characteristics exhibited by the PDOEs al-ow us to observe blueshift and redshift effects at someoci positions, where the intensity of light is high enougho carry out experiments with a good signal-to-noise ratio.

In most practical cases, the paraxial approximationatisfies the accuracy requirements of our measurements,f they are not performed too close to the PDOE. For in-tance, in the case of PDOEs with �=3 and �=4, �=�0,nd within an axial interval from z=0.1 m to z=0.16 m,he axial irradiance functions corresponding to the secondight-hand terms in Eq. (4) and Eq. (7) are plotted in Figs.

cm of PDOEs with (a) �=2 and (b) �=4.

nsets) for the third focus of PDOEs with �=3 and �=4.

=12.67

(a) and 6(b), respectively. The heights of irradiance

ig. 6. Differences between axial irradiances derived within Fresnel (solid curves) and Rayleigh–Sommerfeld (diamonds) regimes, fora) �=3 and (b) �=4 when �=� .

y the i

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5WtstttosestPnPcbwal

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1

1

1

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1

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2

2

2

2

2

2

2

2

Mendoza-Yero et al. Vol. 24, No. 11 /November 2007 /J. Opt. Soc. Am. A 3605

eaks associated with the Rayleigh–Sommerfeld solutionecrease as we move in toward the PDOEs, whereas theorresponding ones for the Fresnel solution do not vary.ain differences appear about 10 cm from the PDOEs,hich have outermost ring radii of 1.94 and 2.24 cm, re-

pectively.

. CONCLUSIONSithin the Rayleigh–Sommerfeld formulation of diffrac-

ion, we derived the analytical solution for the on-axispectral irradiance of a broadband light after diffractionhrough a set of concentric transparent rings with arbi-rary spatial distribution. This result allows us to studyhe axial focusing properties and the spectral changesriginated by the diffraction of broadband light througheveral DOEs. The particular case of DOEs that havequal annular aperture areas and are periodic in thequared radial coordinate were discussed with some de-ail. Blueshift and redshift effects at different axial foci ofDOEs were shown. The use of ultrashort pulsed illumi-ation leads to a loss in the focusing abilities of theDOEs because of the dependence of the diffraction pro-ess on the wavelength of the incident radiation. On theasis of the diffraction-induced effects caused by PDOEs,e suggest a novel technique for slicing the spectrum ofn ultrashort light pulse or a LED into several wave-ength channels.

CKNOWLEDGMENTShis research was funded by the Conselleria de Empresa,niversitat i Ciència, Generalitat Valenciana, Spain, un-er the project GV/2007/128. Partial financial supportrom the Spanish Ministerio de Educación y Ciencia is ap-reciated, including grants from the FIS2007-62217/roject and from the program “CONSOLIDER” under theroject “SAUUL, CSD2007–013.” Omel Mendoza-Yerohanks the Convenio UJI-Fundació Caixa CastellóBancaixa) (grant 07i00530) for covering costs of the re-earch.

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