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chromatic hybrid refractive–diffractive lens withxtended depth of focus

ngel Flores, Michael R. Wang, and Jame J. Yang

A method for designing achromatic hybrid refractive-diffractive elements that can produce beams withlong focal depths while they preserve the entire aperture for capture of light and high transverseresolution is presented. Its working principle is based on the combination of a diffractive optical elementthat generates a long range of pseudonondiffractive rays and a refractive lens of opposite dispersion toform an achromatic hybrid lens. A hybrid lens with a fast f-number � f�1� that works in the entire visiblewave band �400–700 nm� was designed and fabricated. Simulation results demonstrate a factor-of-10improvement in depth of focus compared with that of a conventional f�1 lens, with matching 1-�m lateralresolution. Experimental results confirm the effectiveness of the proposed method through demonstra-tion of an achromatic hybrid lens with better than a factor-of-7 improvement in depth of focus and 1-�mtransverse resolution. © 2004 Optical Society of America

OCIS codes: 220.3620, 050.1970, 220.1000.

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. Introduction

ptical systems that simultaneously exhibit long fo-al depth and high lateral resolution find consider-ble applications in many fields, e.g., in microscopy,ptical alignment, imaging, and optical interconnec-ion. However, according to scaling and paraxial ap-roximations, conventional optical lenses obey theollowing well-known relations:

�X � k1��NA,

�Z � k2��NA2, (1)

here �X is the minimum resolvable feature size inhe transverse dimension, �Z is the depth of focus,nd � is the wavelength. In addition, NA representshe system’s numerical aperture and k1 and k2 areonstants that depend on the criteria adopted. Ac-ording to Eqs. �1�, increasing focal depth �Z simul-aneously enlarges minimum resolvable feature sizeX �decreasing the transverse resolution�, a well-nown trade-off in photographic and imaging usage.

A. Flores and M. R. Wang �mwang@miami.edu� are with theepartment of Electrical and Computer Engineering, University ofiami, Room 406, 1251 Memorial Drive, Coral Gables, Florida

3146. J. J. Yang is with New Span Opto-Technology, Inc., B-180,380 SW 72nd Street, Miami, Florida 33173.Received 28 April 2004; revised manuscript received 14 July

004; accepted 22 July 2004.0003-6935�04�305618-13$15.00�0

a© 2004 Optical Society of America

618 APPLIED OPTICS � Vol. 43, No. 30 � 20 October 2004

s a result, a large depth of focus requires smallumerical apertures, whereas high resolution de-ands large apertures. Thus conventional optical

lements cannot produce a beam with long focalepth and narrow lateral width concurrently. Theyan achieve increased depth of focus only throughperture reduction �decreasing NA�, which drasti-ally reduces the amount of light captured and theransversal resolution that the system can attain.

Over the years, many techniques to extend theepth of focus while preserving high lateral resolu-ion have been proposed. For example, the use ofxicons1,2 has been widely researched. These coni-al elements have been shown to achieve long depthf focus and high lateral resolution simultaneously.owever, it is difficult to fabricate axicons, which

oncentrate only a small fraction of energy into theocused beam, resulting in low light efficiency. Op-ical apodizers,3 elements that contain multipleransmitting rings with �� phase variations, havelso been widely investigated. Yet those elementsuffer from a decrease of optical power at the imagelane and from a decrease of transversal resolutionhat is due to obstructed aperture.

Other approaches consist of using computer-enerated holograms4,5 �holographic optical ele-ents� and diffractive optical elements6,7 �DOEs�

hat make use of pseudonondiffracting beamsPNDBs� or related techniques. PNDBs are charac-erized by nearly constant intensity distribution over

finite axial region and by a beamlike shape in the

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ransverse dimension. For monochromatic illumi-ation, these techniques exhibit high efficiency andood uniformity along the optical axis. However, be-ause of the high wavelength sensitivity of DOEs, forroadband illumination these elements suffer fromnacceptably high chromatic aberration. Wave-

ront coding digital restoration techniques have beenpplied with ample success to resolve the focal depth–esolution imaging problem, but these approaches re-uire additional signal and image processing, whichequire a large computing effort.8–10

In this paper we report a new method for designingchromatic hybrid refractive–diffractive lenses thatxtend the depth of focus �DOF� without sacrificinghe system’s transverse resolution. The extended-OF lens combines a specially designed DOE thatenerates a long range of pseudonondiffractive raysith a corresponding refractive lens to diminish any

hromatic aberrations in the desired spectral band.tilizing a hybrid refractive–diffractive device con-guration simultaneously preserves the favorableroperties of both the diffractive element �long focalepth� and the refractive lens �low chromatic aberra-ion and high energy concentration�.

The proposed method may be applied to variousptical wave bands for extension of focal depth. Thisesign will operate in the entire visible wavebandnd extend the DOF of a lens by a factor of 10 withoutecreasing any lateral resolution. Figure 1 showschematics of the proposed hybrid lens and of a con-entional lens for focusing a collimated imagingeam. From a geometrical optics viewpoint, ex-ended focal depth may be regarded as derived from

ig. 1. �a� Extended DOF hybrid refractive–diffractive lens sys-em and �b� conventional refractive lens system.

nonconventional lens with a longitudinally D

tretched focus of constant intensity distribution.uch an extended-DOF hybrid lens has been de-igned and fabricated to yield a fast f�1 lens withorrection of chromatic aberration in the visible spec-ral band. The hybrid lens has demonstrated signif-cant improvement in DOF while it retains the highransversal resolution displayed by conventional f�1enses. Such a lens has the potential for many ap-lications in imaging systems and optical microscopyo minimize the need to adjust focus in high-esolution settings.

. Design of Lenses with Extended Depth of Focus

diffractive optical element is a wave-front processorhat is capable of transforming light into many com-lex patterns that otherwise would be difficult to at-ain with conventional optics. DOEs offer severaldvantages compared with conventional optical ele-ents: They are thin, lightweight, and inexpensive

when they are mass produced�. Advances in de-ign, fabrication, and analysis of DOEs have madehese elements useful alternatives to refractive ele-ents in many optical systems.11–13

There are two major approaches to the design andimulation of long-focal-depth DOEs. One methodtilizes the geometric law of energy conservation forvaluating the desired phase transmittance withimple analytical solutions.14,15 This technique pro-uces poor performance results with minimal compu-ation time. We employ an iterative optimizationpproach in which an algorithm searches for the op-imal phase distribution to satisfy a desired outputntensity pattern. Several iterative optimizationechniques such as simulated annealing,16 and radi-lly symmetric iterative discrete on-axis encoding17,18

ave been widely reported. The latter technique inarticular has been shown to generate high-fficiency, fast-f-number diffractive lenses. Other it-rative methods such as phase retrieval �i.e., theerchberg–Saxton algorithm19,20 and the Yang–Gu al-orithm21 and its modified versions22� employ error-eduction methods to derive a phase distribution thatatisfies a desired intensity mapping. Although eachf these approaches has proved successful for numer-cal DOE design, the conjugate-gradient algorithm,23 aowerful technique for dealing with optimization prob-ems, was selected for the long-focal-depth DOE designecause of its high accuracy and fast convergence.Figure 2 shows a schematic of the optical system

or extended DOF in which the DOE is placed onnput plane P1 and Pz represents the output observa-ion plane. Letting u1 �r1� and u2 �r2� represent theeld distributions at the input and output observa-ion planes, we may express the corresponding waveunctions as

u1�r1� � �1�r1�expi1�r1��, (2)

u2�r2, z� � �2�r2, z�expi2�r2, z��, (3)

here 1 represents the phase distribution of the

OE, 2 expresses the output plane phase distribu-

20 October 2004 � Vol. 43, No. 30 � APPLIED OPTICS 5619

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ion, the input and output field amplitudes are giveny �1 and �2, and r1 and r2 denote the input andutput radial coordinates, respectively.In accordance with the Huygens–Fresnel principle,

utput wave function u2�r2, z� can also be representedn terms of the input wave function with the followinguperposition integral24:

u2�r2, z� � �r1 max

G�r2, r1, z�u1�r1�dr1, (4)

here transform kernel G�r2, r1, z� is expressed as

G�r2, r1, z� �2�r1

j�zexp� jkr01�. (5)

oreover, r01 represents the polar distance betweenhe aperture and observation planes:

r01 � z2 � r12 � r2

2 � 2r1 r2 cos� 1 � 2��1�2, (6)

here 1 and 2 correspond to the angles subtendedy the aperture and the observation planes, respec-ively. Considering a rotationally symmetric opticalystem and a binomial expansion of the square root,e can accurately approximate distance r01 as

r01 � z�1 �12

r12

z2 �18

r14

z4 �3

48r1

8

z8 � , (7)

here a third-order approximation has been used toccount for high-power, fast-f-number lenses that areot in the Fresnel domain. Note that, as we areoncerned mostly with generating a constant axialntensity at the output plane, and assuming that theeamlike profiles of PNDBs can be obtained automat-cally,23 we have simplified the radial coordinate inhe output plane by setting r2 to zero. Substitutingq. �7� into Eq. �5� yields the transform kernel, G.he composite diffraction pattern can then be con-tructed according to Eq. �4�.We emphasize that further simplification of the

ransform kernel is possible if the observation planeies in the Fresnel domain. Within this region therst two terms of Eq. �7� adequately approximate theinomial expansion. This condition is met if the

ig. 2. Rotationally symmetric optical system with DOE placed atnput plane P1.

igher-order terms of the expansion do not change m

620 APPLIED OPTICS � Vol. 43, No. 30 � 20 October 2004

he overall value of the superposition integral Eq.4��. In the Fresnel domain the transform kernelan be reduced to25

G�r2, r1, z� �2� exp�i2�z���

�i�z

� exp� i��z

�r22 � r1

2��J0�2�r2 r1

�z �r1,

(8)

here J0 denotes a zero-order Bessel function of therst kind.Subsequently, note that performing numerical

imulations requires that the continuous functionsresented above be sampled and converted into dis-rete form. Thus, in discrete form, Eqs. �2� and �4�an be expressed as

u1,m � �1,m exp�i1,m�, m � 1, 2, . . . , M, (9)

u2,l,z � �m�1

M

Gl,m,zu1,m, l � 1, 2, . . . , L, (10)

here M and L represent the number of samplingoints along the input and output observation planes,espectively. Hence the goal for designing the DOEith extended DOF is to determine phase distribu-

ion 1 that can transform an input amplitude pat-ern �ul,m� into the desired field distribution �u2,l,z�ith constant value �u20� along the optical axis. As-

uming that the total number of observation planesz are along the z axis, the estimated difference be-

ween the desired and the actual field distributions23

E � �q�1

Nz

W�q���l�1

L ��20�l �

� �m�1

M

G1,l,m,z�1,m exp�i1,m��2 , (11)

here a weighting factor W�q� that satisfies normal-zing condition ¥q�1

Nz W�q� � 1 has been introduced.s a result, the DOE design algorithm entails finding

he optimal phase 1 to minimize the error function,, as calculated by Eq. �11�.Employing the conjugate-gradient method yields

hase distribution 1 with the following iteration al-orithm:

1�k�1� � 1

�k� � ��k�d�k�, k � 0, 1, 2, 3, . . . ,(12)

here 1�k�, ��k�, and d�k� denote the phase, the step

ize, and the search direction, respectively, in the kthteration. The conjugate-gradient algorithm is anterative technique that requires an initial input forhe unknown variable, 1, and updates the variablet the kth iteration according to Eq. �12�. The geo-

etric law of energy conservation is used to set the

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esired amplitude �20, and, although a random initialhase l

�0� can be used to start the iteration process,logarithmic phase function is used14:

1 � �1�2a ln�d1 � ar2� � const., (13)

here

a � �d2 � d1��R12

nd d1, d2, represents the interval of constant axialntensity and R1 represents the clear DOE aperture.he logarithmic phase function derived from the geo-etrical law of energy conservation is also known to

enerate a uniform intensity distribution along theptical axis, thus allowing the algorithm to yield aore accurate solution with faster convergence.he numerical iteration process terminates eitherhen error E reaches a small predesignated value orhen the number of iterations exceeds a given cycle.nce the phase distribution for a long DOF is ob-

ained by use of the conjugate-gradient algorithm,he approximate surface-relief profile, t�r�, of theOE is acquired from the following phase–thickness

elationship:

t�r� ���r�

2��n � 1�. (14)

. Achromatization by Use of a Hybrid Element

OEs are planar elements that consist of zones thatetard the incident light wave by modulation of theefractive index or the surface profile. The lightmitted from different zones interferes and formshe desired wave front. Inasmuch as these phe-omena are strongly dependent on the wavelengthf light, DOEs are generally restricted to monochro-atic applications. To combine the advantages of

efractive optics �low dispersion, high energy con-entration� and diffractive optics �ability to imple-ent optical functions that are difficult to attainith conventional optics� we designed a hybrid

efractive–diffractive lens. The hybrid lens main-ains the long DOF described above while it signif-cantly reduces chromatic aberrations for widepectral band inputs.Chromatic aberration is caused by the dependence

f a lens’s refractive index on wavelength or on dis-ersion. If collimated light of broad spectral band-idth �i.e., white light� is considered, red, green, andlue light that passes through the lens will focus � fr,

g, fb� at different positions along the optical axis, asemonstrated in Fig. 3�a�. The focal length of a con-entional lens is defined as

1f ���

� n��� � 1�� 1R1

�1R2

�tn��� � 1�

R1 R2 , (15)

here t represents the lens’s thickness and n char-

cterizes the lens material’s refractive index. In the u

roposed hybrid configuration a plano–convex refrac-ive lens is selected for easy DOE integration. For alano–convex lens the focal length is defined as

1f ���

� n��� � 1�� 1R1

� . (16)

herefore the wavelength dependence of the materialndex causes the three images to be dispersed relativeo one another. The property of refractive-indexariation with wavelength is called material disper-ion and is represented by Abbe number V. In theisible spectrum the Abbe number of a refractive lenss calculated as

Vr �nd � 1nF � nc

, (17)

here nF, nd, and nc correspond to refractive indicest 486.1, 587.6, and 656.3 nm, respectively. Notehat in the visible spectrum Vr is always a positiveumber.Chromatic aberration has been known to be cor-

ected through the use of achromatic doublets, forhich the combination of positive and negative lensesith different refractive indices removes dispersionffects. The drawbacks to such methods are that the

ig. 3. Chromatic aberration of �a� a refractive lens and �b� aiffractive lens.

se of two distinct optical materials is required and

20 October 2004 � Vol. 43, No. 30 � APPLIED OPTICS 5621

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hat difficult positioning and packaging are necessaryor the curved elements. In general, the correctionf chromatic aberration by use of two elements inontact can be satisfied under the following con-traints:

P � P1 � P2,

P1

V1�

P2

V2� 0, (18)

here Pi is the power �inverse focal length� of the ithens, P is the total system power, and Vi is the Abbeumber of the correcting lens. Likewise, it has beeneported that chromatic aberration can also be cor-ected through the use of hybrid refractive–iffractive elements.25,26 Hybrid achromats utilizehe dispersion properties of diffractive elements,hich are opposite those of refractive elements Fig.�b�� to diminish dispersion effects. Unlike refrac-ive achromats, these hybrid devices require only oneype of refracting material, and it is not so difficult toeproduce the curvatures. The Abbe number of aiffractive element is given as

Vd ��d

�F � �c, (19)

here �F, �d, and �c represent wavelengths of 486.1,87.6, and 656.3 nm, respectively. Thus in the vis-ble spectrum the Abbe number of a DOE is a �neg-tive� constant, independently of the DOE’s material.When one is designing a hybrid lens with extendedOF, only the total desired power �P� has to be spec-

fied. As the lens manufacturer provides Vr, and asd is constant, Eqs. �18� reduce to a simple two-quations–two-unknowns �P1, P2� problem set.olving Eqs. �18�, we obtain the individual powers ofhe refractive and diffractive lenses that are requiredor eliminating chromatic aberration. To design forhybrid lens that extends the DOF a certain distance

z, one should design the DOE by means of theonjugate-gradient algorithm to provide a constantxial intensity along the following range:

1Pnear_hyb

�1P

��z

2,

1Pfar_hyb

�1P

��z

2, (20)

here Pnear_hyb and Pfar_hyb correspond to the near-nd far-field hybrid powers within the extended focalange. Inserting Eqs. �20� into Eqs. �18� yields theequired DOE constant intensity range:

Pd_near � Pnear_hyb � Pr, Pd_far � Pfar_hyb � Pr,

fd_near �1

Pd_near, fd_far �

1Pd_far

. (21)

ere Pd_near and Pd_far represent the near- and far-eld diffractive powers within the region of constant

ntensity. In addition, fd_near and fd_far correspondo the long-DOF near- and far-field diffractive focalengths, respectively. Attaching the DOE to the ap-

ropriate power refractive lens �Pr� generates the de- o

622 APPLIED OPTICS � Vol. 43, No. 30 � 20 October 2004

ired power hybrid lens with extended focal range �zlong the optical axis.Highlighting Eqs. �18�, we note that because gen-

rally Vr �� Vd the power of the diffractive element isuch lower than the refractive power. Table 1 lists

he corresponding refractive and diffractive-numbers required for obtaining certain achromaticybrid lenses with SF11 as the refractive-lens mate-ial. Table 1 affirms that the designed DOE lies inresnel domain for most hybrid lens combinations.he low-power diffractive lenses that are evidentlyeeded for even fast high-power hybrid lenses to bechieved enable us to design our long-focal-depthOEs without having to resort to rigorous diffraction

heory. The use of scalar diffraction theory �as de-ailed in Section 2� leads to fast convergence timesnd is highly accurate in the Fresnel–Fraunhofer do-ain.Furthermore, the hybrid design technique allows

xcellent flexibility in refractive material selection.OEs with long DOFs can be specifically designed to

ombine with numerous refractive materials. Like-ise, a program has been developed to input the de-

ired hybrid power, the desired spectral band, andhe properties of the refractive material to be used.he program generates the refractive power andOE surface-relief profile coordinates �by use of a

onjugate-gradient algorithm� that are necessary toxtend the depth of focus by a factor of 10 about theesired hybrid power. For example, to design a UVybrid lens with quartz as the refractive material,ne can design a DOE based on the optical propertiesf quartz. Similar DOEs can be designed for visiblend infrared hybrid lenses as well.

. Fabrication of Long-Focal-Depth Hybrid Lenses

he proposed DOE is a phase filter element. Nu-erous techniques such as diamond turning, photo-

ithography, and laser writing have been developedor fabrication of DOEs. Likewise, we have demon-trated laser generation of gray-level masks and aechnique for the fabrication of phase-only DOEs byne-step direct etching on glass masks for practicalurface-relief profiles.27 Laser direct writing onigh-energy-beam-sensitive glass produces a gray-

evel mask; varying the laser intensity radiation onhis glass generates a corresponding gray-level trans-ittance pattern. Subsequently, direct etching of

he gray-level mask plate by use of diluted hydroflu-

Table 1. Required Refractive �SF11 Glass� and Diffractive f-NumbersNeeded to Achieve Corresponding Hybrid Lenses

Desired HybridLens f-Number

RequiredRefractivef-Number

RequiredDiffractivef-Number

f�1 f�1.1 f�8.5f�2 f�2.3 f�17f�5 f�5.7 f�42f�10 f�11.3 f�85

ric acid results in the desired DOE surface-relief

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rofile. The direct etching creates a one-steplignment-free process that can support a large num-er of phase levels for the fabrication of high-fficiency quasi-continuous surface profile DOEs.tching calibration is performed to quantify the re-

ation between etching depth and laser-written trans-ittance. The optimal surface profile for the

xtended DOF DOE derived from the conjugate-radient algorithm is then input to a laser-writingachine. One then microscopically aligns the fab-

icated DOE with the refractive lens to construct theybrid extended DOF lens.

. Hybrid Refractive–Diffractive Lens with Extendedepth of Focus

o illustrate the effectiveness of the proposed hybridxtended-DOF lens we designed and fabricated a hy-rid lens with a fast f-number, f�1, that works in thentire visible wave band �400–700 nm�. A plano–onvex refractive lens made from SF11 glass waselected. SF11 is a flint glass with excellent chem-cal resistivity and adequate transmission in the vis-ble wave band. Its refractive index is 1.7847 at the87.6-nm design wavelength, and its Abbe number Vrs 25.76. The high dispersion property of SF11 isxploited in the hybrid design to complement thearge dispersive nature of the diffractive element.

For a conventional SF11 f�1 refractive lens theOF is approximately 2.6 �m, with a diffraction-

imited beam spot size of �1 �m. The focal length ofhe f�1 hybrid lens was designed to be 3.0 mm. Tochieve a factor-of-10 times DOF improvement in thisase, i.e., 26-�m depth of focus, we should set its focalange from 2.987 to 3.013 mm. With the focal lengthf the hybrid system set as fhybrid � 3 mm, we utilizedqs. �18� to obtain the focal lengths of the diffractivend refractive lenses as fd � 25.4 mm and fr � 3.4m, respectively. Employing the conjugate gradi-

nt method as discussed in Section 2, we designed aOE with a long DOF �focal range, 24.6–26.0 mm�.he simulated on-axis intensity distribution of theesigned long-focal-depth DOE is illustrated in Fig.�a�. When it is combined with the appropriateower refractive lens, the optical system should ex-ibit an extended focal depth about the desired sys-em focal length, fhybrid. To show the factor-of-10mprovement in DOF that the hybrid lens provides,e also show simulated on-axis beam intensity dis-

ributions for both the hybrid f�1 lens �solid curve�nd the conventional f�1 lens �dotted curve� in Fig.�c�.The simulated phase function �r� required for pro-

ucing the DOE with extended DOF is shown in Fig.�b�. This function can be converted into surface-elief profile t�r� by use of Eq. �14��, which will betilized for fabrication of the DOE. A quasi-ontinuous, high-efficiency diffractive lens was thenabricated with our laser direct-write technique.27

The point-spread imaging characteristic of theong-focal-depth DOE was then experimentally ana-yzed. Figure 5 shows the experimental arrange-

ent for measuring the focusing performance of the c

OE. An expanded collimated He–Ne laser beam at632.8-nm wavelength was used to illuminate the

ample. The focused spot was projected onto a

ig. 4. �a� Simulated on-axis intensity distribution along the zxis of the designed DOE, �b� corresponding simulated phase pro-le of the designed DOE, and �c� simulation of the on-axis intensityistribution along the z axis of the combined refractive–diffractiveybrid f�1 lens �solid curve� and the conventional f�1 SF11 lensdotted curve�.

harge-coupled device �CCD� image sensor by a mi-

20 October 2004 � Vol. 43, No. 30 � APPLIED OPTICS 5623

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roscope objective lens �60��. A 60� objective wasmployed in the experimental arrangements to com-ensate for the limited CCD sensor resolution of 7.4

per pixel. The objective and the CCD device werehen mounted upon a three-dimensional translationtage. A submicron-sensitive differential microme-er with 0.5-�m resolution was used to sweep thebjective lens and the CCD camera across the z axisnd analyze the focusing performance of the DOE.igures 6 and 7 show several pictures of the focusedpot quality and the transverse intensity distributionf our fabricated DOE along the z axis. Utilizing theiffractive depth-of-focus criterion of 81% peak inten-ity that constitutes the focal range, we measured theiffractive element’s extended DOF to be 1.33 mm,ufficiently close to the designed DOE value of 1.4m. There is an error of 5% inherent in the wet

tching process.Although simulation and experimental results ver-

fy the DOE’s long DOF, the device will follow spec-fications only at the design wavelength ��d�. Forxample, a simulation of the on-axis intensity distri-utions behind the DOE for three arbitrary wave-engths in the visible spectrum �� � 0.47, 0.53, 0.62m� is shown in Fig. 8�a�. Even though the DOExtends the DOF at each wavelength there is severehromatic aberration and reduced efficiency, as ex-ected. The same simulation with three arbitraryavelengths in the visible waveband was performedith our hybrid lens. As shown in Fig. 8�b�, the

hromatic aberration has been significantly reducednd the factor-of-10 improvement in the DOF com-ared with that of a conventional f�1 lens was pre-erved. Likewise, the simulation was performed forconventional f�1 lens, shown in Fig. 8�c�, illustrat-

ng the dispersive behavior of conventional lenses asell.In addition to nearly achromatic extended DOF

roperties, the f�1 hybrid lens also maintains theigh transverse resolution that is inherent in f�1

enses. As determined from Eqs. �1�, the resolutionf a conventional f�1 lens is approximately 1 �m.imilarly, Eqs. �1� affirm that increasing the DOF tenimes �to 26 �m� reduces the resolving power of theystem to �4 �m. Nevertheless, simulation resultseveal that our hybrid lens can simultaneously ex-end the DOF without sacrificing the large aperture

ig. 5. Experimental arrangement for measuring the focusing pe�1 lenses.

rformance of a long-focal-depth DOE and both hybrid and conventional

NA� and the consequent high transverse resolution D

624 APPLIED OPTICS � Vol. 43, No. 30 � 20 October 2004

ig. 6. Beam spot images observed at different planes from theOE lens at �a� 24.6, �b� 25.0, �c� 25.4, and �d� 25.93 mm. A long

OF is demonstrated.

odacat

Fabsizes have been obtained by use of a 60� objective magnification.

Farf�1 SF11 lens.

f conventional fast-f-number lenses. A three-imensional plot in the region of interest was gener-ted �see Fig. 9� to demonstrate the simultaneousonstant intensity distribution along the optical axisnd the high lateral resolution of 1 �m that the sys-

ig. 7. Transverse intensity distribution from the fabricated DOEt �a� 24.6, �b� 25.0, �c� 25.4, and �d� 25.93 mm from the lens. Theeam remains in focus from 24.6 to 25.93 mm. Note that spot

ig. 8. Simulated focused on-axis beam intensity distributionlong the z axis for three arbitrary wavelengths: �a� before ach-omatization, �b� after achromatization, and �c� for a conventional

em generates.

20 October 2004 � Vol. 43, No. 30 � APPLIED OPTICS 5625

hbhal�auFsib

ahsb�t f

ccdiSe

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Fcmm from the lens. The measured DOF is 2.6 �m.

5

After confirming the functionality of our proposedybrid lens through simulation, we fabricated a hy-rid lens, and the PSI characteristics of both theybrid and the conventional f�1 lenses were observednd compared. A plano–convex spherical SF11 f�1ens with a 3-mm focal length from Edmunds OpticsPCX45-118� was employed for the experimentalnalysis of a conventional f�1 lens. Once again, wetilized the experimental arrangement detailed inig. 5 to analyze the focusing performance of theample lenses across the optical axis. The ratio ofntensity versus axial distance for the fabricated hy-rid sample was recorded and is plotted in Fig. 10.Experimentally acquired images of the beam spot

long the optical axis for the conventional and theybrid f�1 lenses are shown in Figs. 11 and 12, re-pectively. Experimental results show that the hy-rid lens maintains a focused beam spot for an20-�m on-axis range. For a traditional f�1 lens

he beam spot remains in focus for �2.6-�m. There-

ig. 9. Three-dimensional simulation plot demonstrating simul-aneous factor-of-10 DOF improvement and 1-�m transverse res-lution.

ig. 10. Variation in on-axis focus spot intensity of the fabricated

pybrid refractive–diffractive lens, demonstrating the long DOF.

626 APPLIED OPTICS � Vol. 43, No. 30 � 20 October 2004

ore a better-than factor-of-7 improvement in DOFompared with conventional f�1 lenses has been ac-omplished experimentally. Laser speckles that areue to the monochromatic nature of the laser beamncidence cause parts of the noise shown in Fig. 12.uch noise is significantly reduced when an incoher-nt light source is used, as shown in Fig. 13.In addition, the on-axis intensity fluctuation shown

n Fig. 10 can be attributed in part to the error thats inherent in the DOE wet etching process and to theropagation nature of the PNDB. Deviation fromhe expected simulated results �factor-of-10 improve-ent in DOF� is also possibly due to the microscopic

lignment of the diffractive and refractive portions ofhe lens. The slight misalignments may lead to off-xis aberrations, which additionally reduce the effi-iency of the hybrid lens. The concentricity of theOE with the refractive lens needs to improve

hrough use of a proper alignment instrument. Im-roved dry etching and alignment techniques shouldield a more-accurate DOE and better hybrid lens

ig. 11. PSIs acquired experimentally at the focal plane by aonventional f�1 lens at �a� 2.999, �b� 3.000, �c� 3.001, and �d� 3.002

erformance.

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The experimentally acquired beam spot resolutionsor both lenses were analyzed as shown in Figs. 14�b�nd 14�c�. A ray-tracing software �Zemax� simula-ion plot of the plano–convex SF11 f�1 lens’ spot sizet the focal plane is also included Fig. 14�a��. Theearly equal resolution of 1 �m �the actual widthith a 60� objective is 60 �m for approximately 1-�m

esolution� generated by the hybrid lens illustrateshat the hybrid lens preserves the high transverseesolution. Thus the high resolution of a conven-ional f�1 lens was achieved while the depth of focusas extended concurrently.The improvement in DOF by use of the hybrid lens

s accomplished in principle through the introductionf some small sidelobes similar to that of the PNDB.s the central lobe diverges after the initial focus, theidelobes converge to offset such a diverging effect

ig. 12. PSIs acquired experimentally at the focal plane by ourybrid f�1 lens at �a� 2.990, �b� 2.997, �c� 3.005, and �d� 3.01 mmrom the lens. The measured DOF is �20 �m.

nd thus result in an extended depth-of-focus behav-

or. These additional sidelobes, shown in Fig. 14�c�,re in agreement with the behavior of the PNDB.28

t is true that the additional sidelobes may degradehe image quality. These sidelobes, however, are inggregate significantly smaller than the main centralobe of the reduced aperture refractive lens of theame DOF, as confirmed through the diffraction-imited simulation results presented in Fig. 15. Thedvantage of using the hybrid lens for DOF improve-ent is thus obvious.

ig. 13. Image of a portion of a U.S. Air Force resolution targetaken with the fabricated hybrid f�1 lens. The target is illumi-ated with a white-light source and separated by color filters.

Additionally, the light transmitting efficiency of

20 October 2004 � Vol. 43, No. 30 � APPLIED OPTICS 5627

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5

oth lenses was numerically and experimentallynalyzed. The light efficiency of the proposed hy-rid lens is similar to that of other optical elementshat employ nondiffracting techniques for genera-ion of constant axial intensity. Experimentaleasurements of the central spot encircled energy

or a conventional aberrated and our hybrid ex-ended DOF f�1 lenses yielded 1.64% �2.16% fromhe Zemax simulation� and 2.77% efficiency, respec-ively. These results indicate that our hybridxtended-DOF lens has higher efficiency than aimilar f�1 conventional lens. The reason is thathe aspherical �logarithmic� phase profile of theOE compensates for some of the spherical aberra-

ig. 14. �a� Zemax simulation plot of the transverse resolution ofn SF11 f�1 lens. Measured transverse resolution for �b� a con-entional f�1 lens and �c� the hybrid f�1 lens. Note that spot sizesere obtained with a 60� objective magnification.

ion that is inherent in conventional refractive r

628 APPLIED OPTICS � Vol. 43, No. 30 � 20 October 2004

enses, thus leading to better efficiency than for aonventional spherical lens.To compare imaging quality we tested the achro-atic performance of the fabricated lens and com-

ared it with that of a conventional f�1 lens. Ahite-light source was used to illuminate a U.S. Airorce resolution target, and images were taken withoth lenses. Three 10-nm-bandwidth color filterscentral wavelengths at 656, 532, and 487.6 nm� weresed to generate the red, green, and blue illumina-ion, respectively, and the numeral 5 was imaged.he results for a traditional f�1 lens are presented inig. 16, and, as predicted by Fig. 8�c�, the effects ofhromatic aberration can be clearly observed.

The chromatic performance of the fabricated hy-rid lens �Fig. 13�, however, shows excellent improve-ent over that of the conventional lens alone, with

nly a slight focal shift observed, as expected from ourimulation results. Unlike other reported long-ocal-depth–high-resolution systems that depend ononochromatic illumination, the proposed hybrid

ens with extended DOF and high transverse resolu-ion works over a broad waveband in the visible spec-rum. To the best of our knowledge this is the firstime that a nearly achromatic hybrid lens with anxtended DOF has been developed.Finally, we verified the imaging depth of field en-

ancement by having both the hybrid and the con-entional f�1 lenses image an object placed at variousxed distances from the lenses. For this experimenthe hybrid–conventional lens was used to project tar-et images directly onto the CCD sensor array. TheOF improvement was examined through imaging

omparison of the three-bar pattern that appears inhe Air Force resolution target.

To demonstrate the simultaneous DOF improve-ent with high resolution, we imaged the highest-

ig. 15. Diffraction-limited simulation results demonstrating aomparison of resolution between extended-DOF and conventionalenses. The small-aperture lens �dotted curve� is designed with

esolution segment of the target: Group 7, element 6

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228.10 line pairs�mm�. Experimental results showhat the three-bar pattern appears resolved when theamera �with the hybrid lens� is placed at distances of.72–5.85 mm from the object �see Fig. 17�. For aimilar system that uses a conventional f�1 imagingens, experimental results given in Fig. 18 show thathe camera resolves the pattern at a limited distance of.75–5.77 mm from the object. When the Rayleighesolution criterion of 73.5% midpoint intensity wasmployed between the peak intensities of the imaged

ars, the traditional imaging lens produced a 0.02-mm

OF. By comparison, the hybrid lens produced a.13-mm DOF. As a result, nearly a factor-of-7 im-rovement in DOF was experimentally accomplishedor the highest-resolution target sector. We empha-ize that, although the DOF enhancement presentedas accomplished for a high-resolution target portion,

imilar results were obtained for the low-resolutionectors of the U.S. Air Force target.

. Conclusions

technique for designing achromatic hybridefractive–diffractive lenses that can extend theepths of focus of conventional lenses while they con-erve the aperture for equivalent transverse resolu-ion has been developed. The working principle isased on a specially designed diffractive optical ele-ent that modulates the incident light wave to pro-

uce a constant axial intensity distribution within aiven long focal range. When it is combined with aorresponding refractive lens, an achromatic hybridens with a long focal depth and unaltered transverseesolution can be achieved.

ig. 16. Image of a portion of the U.S. Air Force resolution targetaken with the conventional f�1 lens. The target is illuminated

ig. 17. Focus-free images of a 228-line pair�mm resolution tar-et when the hybrid f�1 imaging lens was used. Clear images

We employed the design technique to obtain a hy-

20 October 2004 � Vol. 43, No. 30 � APPLIED OPTICS 5629

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5

rid f�1 lens with a factor-of-7 improvement in DOF,-�m transverse resolution, and efficient operation inhe entire visible wave band. The flexibility of theybrid design technique also allows DOEs with longOFs to be designed for any number of refractive ma-

erials. Thus, custom development of hybridxtended-DOF lenses can easily be achieved. Im-roved etching and alignment techniques that yieldore-accurate surface-relief profiles could result in

actor-of-10 improvement in DOF, as demonstratedhrough numerical simulations. As the proposedethod performs well in the most strenuous case � f�1:

ast, high-power lens with large aperture�, it shouldork well for higher-f-number lenses. By minimizing

ocus adjustment of optical imaging systems, we expecthat this achromatic hybrid lens with long DOF andigh transverse resolution will benefit many practicalptical systems.

This project was supported in part by the Nationalcience Foundation.

eferences1. J. H. Mcleod, “Axicons and their uses,” J. Opt. Soc. Am. 50,

166–169 �1960�.

630 APPLIED OPTICS � Vol. 43, No. 30 � 20 October 2004

2. J. W. Y. Lit and R. Tremblay, “Focal depth of a transmittingaxicon,” J. Opt. Soc. Am. 63, 445–449 �1973�.

3. H. Wang and F. Gan, “Phase-shifting apodizers for increasingfocal depth,” Appl. Opt. 41, 5263–5266 �2002�.

4. A. Vasara, J. Turunen, and A. Friberg, “Realization of generalnondiffracting beams with computer-generated holograms,” J.Opt. Soc. Am. A 6, 1748–1754 �1989�.

5. N. Davidson, A. A. Friesem, and E. Hasman, “Holographicaxilens: high resolution and long focal depth,” Opt. Lett. 16,523–525 �1991�.

6. W. Cong, N. Chen, and B. Gu, “Generation of nondiffractingbeams by diffractive phase elements,” J. Opt. Soc. Am. A 15,2362–2364 �1998�.

7. R. Liu, B. Dong, and B. Gu, “Implementation of pseudo-nondiffracting beams by use of diffractive phase element,”Appl. Opt. 37, 8219–8223 �1998�.

8. G. Hausler, “A method to increase the depth of focus by twostep image processing,” Opt. Commun. 6, 38–42 �1972�.

9. E. R. Dowski and W. T. Cathey, “Extended depth of fieldthrough wave-front coding,” Appl. Opt. 34, 1859–1866 �1995�.

0. H. B. Wach, E. R. Dowski, and W. T. Cathey, “Control ofchromatic focal shift through wave-front coding,” Appl. Opt.37, 5359–5367 �1998�.

1. V. A. Soifer, ed., Methods for Computer Design of DiffractiveOptical Elements �Wiley, New York, 2002�.

2. B. Kress and P. Meyrueis, Digital Diffractive Optics �Wiley,West Sussex, England, 2000�.

3. H. P. Herzig, ed., Micro-Optics: Elements, Systems, and Ap-plications �Taylor & Francis, London, 1997�.

4. J. Sochacki, A. Kolodzieiczykm, Z. Jaroszewic, and S. Bara,“Nonparaxial design of design of generalized axicons,” Appl.Opt. 31, 5326–5330 �1992�.

5. A. Friberg, “Stationery-phase analysis of generalized axicons,”J. Opt. Soc. Am. A 13, 743–750 �1996�.

6. N. Yoshikawa and T. Yatagi, “Phase optimization of a kino-form by simulated annealing,” Appl. Opt. 33, 863–868 �1994�.

7. W. Hudson Welch, J. E. Morris, and M. R. Feldman, “Iterativediscrete on axis encoding of radially symmetric computer-generated holograms,” J. Opt. Soc. Am. A 10, 1729–1738 �1993�.

8. H. Haidner and G. M. Morris, “Wavefront quality of optimizeddiffractive lenses,” Pure Appl. Opt. 6, 191–202 �1997�.

9. O. Bryngdahl, “Geometrical transformations in optics,” J. Opt.Soc. Am. 64, 1092–1099 �1974�.

0. R. Gerchberg and W. Saxton, “A practical algorithm for deter-mination of phase from image and diffraction plane pictures,”Optik 35, 237–246 �1972�.

1. G. Yang and B. Gu, “On the amplitude-phase retrieval problemin the optical system,” Acta Phys. Sin. 30, 410–413 �1981�.

2. J. Fienup, “Phase-retrieval algorithms for a complicated opti-cal system,” Appl. Opt. 32, 1737–1746 �1993�.

3. R. Liu, B. Dong, G. Yang, and B. Gu, “Generation of pseudo-nondiffracting beams with use of diffractive phase elementsdesigned by the conjugate-gradient method,” J. Opt. Soc. A 15,144–151 �1998�.

4. J. W. Goodman, Introduction to Fourier Optics �McGraw-Hill,San Francisco, Calif., 1968�, p. 65.

5. N. Davidson, A. A. Friesem, and E. Hasman, “Analytic designof hybrid diffractive-refractive achromats,” Appl. Opt. 32,4770–4774 �1993�.

6. C. Londono and P. Clark, “Modeling diffraction efficiency ef-fects when designing hybrid diffractive lens systems,” Appl.Opt. 31, 2248–2252 �1992�.

7. M. R. Wang and H. Su, “Laser direct-write gray-level maskand one-step etching for diffractive microlens fabrication,”Appl. Opt. 37, 7568–7576 �1998�.

8. M. R. Wang, C. Yu, and A. J. Varela, “Efficient pseudo-nondiffracting beam shaping using a quasicontinuous-phase

ig. 18. Images of a 228-line pair�mm target pattern with a

diffractive element,” Opt. Eng. 40, 517–524 �2001�.