Lens axicons: systems composed of a diverging aberrated lens and a perfect converging lens

Z. Jaroszewicz and J. Morales Vol. 15, No. 9 /September 1998 /J. Opt. Soc. Am. A 2383

Lens axicons: systems composed of a divergingaberrated lens and a perfect converging lens

Zbigniew Jaroszewicz and Javier Morales

Facultad de Ciencias, Universidad Nacional sede Medellın, P.O. Box 3840, Medellın, Colombia

Received December 24, 1997; revised manuscript received May 13, 1998; accepted May 19, 1998

A description is given of axicons obtained as a tandem of one diverging lens that has third-order sphericalaberration and one perfect converging lens. The asymptotic representation obtained with the help of the non-uniform stationary phase method allows the determination of such principal features of the focal segment asthe intensity distribution and the width of the central core along the optical axis. The obtained results arecompared with the numerical evaluation of the corresponding diffraction integral. The analyzed system con-tains four special cases: the doublet of both lenses, the setup of the perfect converging lens placed in the frontof the aberrated diverging lens, the setup of the aberrated diverging lens placed in the primary focal plane ofthe perfect converging lens, and the defocused Galilean telescope. A short discussion of the aberrated-lensbending factor as well as of choice of the most convenient system geometry is also included. © 1998 OpticalSociety of America [S0740-3232(98)02109-7]

OCIS codes: 050.1940, 120.362.

1. INTRODUCTIONAxicons, the elements that create a focal segment alongthe focal axis,1–3 are the subject of ongoing interest,mainly because of their utility for metrological applica-tions. In view of these applications, parameters of the fo-cal segment such as the axial intensity distribution andthe central-core width assume vital importance. Themost desirable characteristics have elements such as lin-ear axicons (constant central-core width),1,4–6 logarithmicaxicons (constant axial intensity),7,8 and Bessel beams(both parameters constant).9 Unfortunately, owing totheir complicated transmittance, the axicons mentionedabove are rather difficult to realize in practice.

In view of this situation the following problem can bestated: Is it possible to design axicons with satisfying pa-rameters by use of aberrated lens(es)? Such an approachseems to be the simplest and cheapest solution. Themost obvious example, a converging lens with sphericalaberration, represents the so-called backward-typeaxicon,7,10 where perfect crossing of rays leaving the ap-erture takes place; consequently, a great variation of rayangles occurs, leading thereby to a great variation of thecentral-core width. Nevertheless, this type of lens axiconhas found application in measuring techniques as thesource of the reference line.11,12

In light of the above discussion, only diverging lenseswith spherical aberration will be analyzed in this paper.The virtual focal segment created by the aberrated di-verging lens can be converted to a real focal segment, ofthe forward type, by adding a perfect converging lens.Since for metrological applications the long focal dis-tances are mainly of practical interest, throughout thiswork we apply a paraxial approximation. In addition,here we consider only systems that comprise a diverginglens that has third-order spherical aberration and a per-

0740-3232/98/092383-08$15.00 ©

fect converging lens, because the fifth-order spherical ab-erration leads to the appearance of a caustic point on theoptical axis and consequently to a change in the descrip-tion of the diffracted field.

Some special cases of lens axicons have already beenreported: A defocused Galilean telescope for which bothdesign13 and experimental results with numerical simu-lation of the corresponding diffraction integral14 were pre-sented, as well as a system that consists of a diffractiveelement, with codified spherical aberration, placed in theprimary focal plane of the converging perfect lens.15

The purpose of this paper is to consider the generalcase of the forward-type lens axicons in order to derive anintegral describing the complex amplitude distribution ofthe diffracted field, analyze the derived integral with thehelp of the nonuniform stationary phase method (SPM),and identify interesting special cases of the obtained so-lution. The nonuniform SPM proposed for theasymptotic evaluation of the diffraction integral allows usto obtain relationships both for the axial intensity distri-bution and for the width of the central core. In additionto the stationary wave, the edge waves are taken into ac-count, which in part permits us to include into the SPMresult the diffraction effects that are due to the aperturelimits. The results are compared with the numericalsimulations of the corresponding diffraction integrals.

2. ANALYSIS OF THE SYSTEMCONSISTING OF THE DIVERGINGABERRATED LENS AND THE PERFECTCONVERGING LENSLet us consider the general geometry of our problem, aspresented in Fig. 1. The aberrated diverging sphericallens characterized by the focal length f1 , the aberration

1998 Optical Society of America

2384 J. Opt. Soc. Am. A/Vol. 15, No. 9 /September 1998 Z. Jaroszewicz and J. Morales

coefficient b, and the inner and outer radii of the apertureR1 , R2 is described by the transmittance

t~r ! 5 H exp@ik~r2/2 f1 1 br4!# r P ~R1 , R2!

0 otherwise.

This lens is illuminated by a plane wave and is placed ata distance l in front of a perfect converging lens. The dif-fraction field illuminating the second lens can be writtenas an axially symmetrical Fresnel integral:

U~r8, z ! 5 CU0kil E0

`

t~r !J0~krr8/z !exp~ikr2/2l !rdr,

(1)

where J0 is the zero-order Bessel function of the firstkind, k 5 2p/l; l is the wavelength of the plane wave il-luminating the transmittance; r is the radial coordinate ofthe diverging lens plane; r8 is the radial coordinate of theconverging lens plane; U0 is the amplitude of the illumi-nating wave; and C 5 exp@ik(l 1 r82/2l)# is the phase fac-tor, usually irrelevant in considerations that do not in-volve the interference of diffraction fields. After this fieldis multiplied by the transmittance of the perfect converg-ing lens t1(r8) 5 exp@2ikr82/2 f#, where f is its focallength, and after repeated use of the Fresnel diffractionintegral and changing of the integration order, the result-ing field at a distance z behind the second lens, followingthe identity16

E0

`

J0~bx !J0~cx !exp~iax2!xdx

5i

2aJ0~bc/2a !exp@2i~b2 1 c2!/4a# (2)

can be written as follows:

U~r, z ! 5 U0kf

i~lf 1 zf 2 zl !

3 expH ikF l 1 z 1r2~f 2 l !

2~lf 1 zf 2 zl !G J3 E

0

`

t~r !J0S krrflf 1 zf 2 zl D

3 expH 2ikF r2~z 2 f !

2~lf 1 zf 2 zl !G J rdr, (3)

Fig. 1. Illustration of the geometry of a system composed of anaberrated diverging lens placed in front of a perfect converginglens.

where r is the radial coordinate of the observation plane.One can see that the obtained integral in its general

shape can be expressed as

I~k ! 5 ER1

R2

g~r !exp@ikf~r !#dr, (4)

where

g~r ! 5 J0S krrflf 1 zf 2 zl D r,

f~r ! 5r2

2 F 1f1

2~z 2 f !

~lf 1 zf 2 zl !G 1 br4.

For such an integral an asymptotic evaluation can befound by means of the nonuniform SPM3,17:

I~k ! 5 F 2p

kuf 9~rs!uG1/2

g~rs!exp(i$kf~rs!

1 sgn @ f 9~rs!#p/4%)

1g~r !

ikf8~r !exp@ikf~r !#U

R1

R2

for rs P ~R1 , R2!,

I~k ! 5 F p

2kuf 9~rs!uG1/2

g~rs!exp@i sgn f 9~rs!p/4#

3 exp@ikf~rs!# for rs 5 R1 ∨ R2 ,

I~k ! 5g~r !

ikf8~r !exp@ikf~r !#U

R1

R2

for rs ¹ ^R1 , R2&,

(5)

where r 5 rs is the stationary point determined by thecondition

f8~rs! 5 0. (6)

The first part of the first expression of Eq. (5), which isvalid inside the integration range, is a stationary wave,whereas the second expression describes two edge wavesthat originate from the limits of the aperture. The sec-ond expression gives the result for the stationary pointslying at the limits of the aperture and is equal to half ofthe stationary-wave amplitude in these points. The lastexpression describes the field outside the integrationrange and is composed of two edge waves. Axicons ana-lyzed throughout this work have only one stationarypoint; from the point of view of geometrical optics thismeans that a given point on the optical axis receives onlyone bundle of rays arriving from one corresponding infini-tesimal annulus in the axicon’s aperture. Equation (5)contains only the first term of the stationary wave (pro-portional to k21/2) and the first term of the edge waves(proportional to k21). Higher terms, proportional tok23/2,k25/2,..., in the case of the stationary wave and tok22, k23,..., in the case of the edge waves, can be ne-glected because of the great value of the wave vector k inthe optical wavelength range.17 Moreover, only the sta-tionary points of the first kind are considered in theanalysis (i.e., such that the second derivative in the sta-tionary point differs from zero).17 Let us add that the


SPM has already been successfully applied in the analysisof other types of axicons, both linear4–6,8 andlogarithmic.3,8,18

The expression for the stationary point, found by apply-ing Eq. (6) to the phase function of the diffraction integral[Eq. (4)], is given by

rs 5 $@~z 2 f !/~lf 1 zf 2 zl ! 2 1/f1#/4b%1/2; (7)

hence on the basis of Eq. (5) the intensity of the station-ary wave is described by

ISt~r, z ! 5I0pkf 2

4b~lf 1 zf 2 zl !2 J02$krf @~ f1z 2 f1f 2 lf

2 zf 1 zl !/4bf1~lf 1 zf 2 zl !3#1/2%. (8)

For r 5 0 the axial intensity of the stationary wave canbe found,

ISt~0, z ! 5 I0pkf 2/4b~z 2 t !2~ f 2 l !2, (9)

where 1/t 5 1/f 2 1/l is the image position of the plane ofthe aberrated lens. It is worth noting that the axial in-tensity does not depend on the focal length of the aber-rated lens.

Next, the position of the first zero of the transversaldistribution is equal to

r0 5 cl0@4bf1~ f 2 l !3/f 2~ f1 1 l 2 f !#1/2

3 @~z 2 t !3/~z 2 s !#1/2, (10)

where 1/s 5 1/f 2 1/(l 1 f1) is the paraxial focus of thewhole setup and c 5 0.3827 is a constant. On derivingthe above expression, one finds that the minimum widthr0,MIN and its position on the focal segment zMIN , aregiven by

r0,MIN 5 cl0@27bf1~ f 2 l !3~s 2 t !2/f 2~ f1 1 l 2 f !#1/2,

zMIN 5 3s/2 2 t/2. (11)

The particular values of aperture radii and focal seg-ment position impose additional conditions on the set ofparameters f1 , f, b, and l that characterize the tandem ofboth lenses. These conditions result from ray-tracingequations applied to both lenses. Ray-tracing equationsfor the radial geometry of our problem and within theparaxial approximation (sin u ' tan u) take the followingform:

dw

dr5 2sin u ' 2

rz~r !

, (12)

where w is the phase function of the analyzed element andz(r) is the distance at which the ray crosses the opticalaxis. For the first lens, the ray-tracing equations statethat

1/f1~r ! 5 1/f1 1 4br2, (13)

where f1(r) is the focal length corresponding to the givenvalue of r. In the case of the second lens the inclinationof the incoming ray should be taken into account, whichresults in the expression

1/z 5 1/f 2 1/@l 1 f1~r !#. (14)

Setting r 5 0, we determine the value of paraxial focusz 5 s [Eq. (10)]. For the annular-aperture axicon of radii

R1 and R2 that form the focal segment between z 5 d1and z 5 d2 , on substituting Eq. (13) into Eq. (14) one ob-tains a set of two equations that connect the parametersof the lenses:

1

d1,25

f1 1 l 2 f 1 ~l 2 f !4bf1R1,22

f~ f1 1 l ! 1 lf4bf1R1,22 . (15)

The application of the annular aperture has the pur-pose of removing the rays parallel to the optical axis,which are responsible for broadening of the focal segmentnear and at the paraxial focus.8,10 For the aberrated lensthe value R1 5 R2/31/2 has been proposed for the centralstop.13 This value resulted from the requirement thatthe edge inner ray cross the focus of the outer edge ray atas great a distance as possible. In subsequent examplesthe value resulting from the condition

R1 5 R2d1 /d2 (16)

is assumed, which on the one hand allows us to minimizethe changes of the central-core width (in these cases inwhich it possesses a minimum) and on the other to com-pare results with the previous ones obtained for othertypes of axicons.3,8,10,18

On the basis of the obtained solution, four interestingspecial cases of the lenses’ mutual position can be distin-guished. Three of the cases correspond to l 5 0 ( f15 f 1 D), l 5 f ( f1 5 D), and l 1 f1 5 f 1 D, where Dis the distance between the position of the secondary focalpoint of the aberrated diverging lens and the primary fo-cal point of the converging lens (Fig. 1). The first twocases lead to focal segments that are characterized by ex-treme values of axial intensity and central spot widthcompared with other mutual positions of the lenses, andthe third case corresponds to the defocused Galilean tele-scope, which is the most widespread case in practice.13,14

The fourth case worth mentioning refers to the placementof the converging lens in front of the diverging lens; theresult obeys l 1 f1 5 f 1 D, where D is now the distancebetween the position of the primary focal point of the ab-errated diverging lens and the secondary focal point of theconverging lens (Fig. 2). This case describes a reversibly

Fig. 2. Illustration of the geometry of a system composed of aperfect converging lens placed in front of an aberrated diverginglens. The lower half refers to the lack of aberration and showsmutual position of both lenses, whereas the upper half illustratesthe formation of the focal segment in the aberration’s presence.


illuminated defocused Galilean telescope and turns out tobe described by the same type of solution as the case l5 0. All four solutions are the subject of more detailedanalysis in the following subsections.

A. Doublet of the Aberrated Diverging Lens and thePerfect Converging Lens (l 5 0)The special case l 5 0 means that the two lenses areplaced together, forming a doublet. Consequently thetransmittance of the whole axicon can be given in an ex-plicit form. Its phase function is equal to

w~r ! 5 2r2/2s 1 br4 (17)

where 1/s 5 1/f 2 1/f1 . Eq. (15) now takes the shape

1/s 2 1/d1,2 5 4bR1,22 . (18)

Hence the particular values for s and b depend only onfocal-segment coordinates and the aperture of the axicon:

b 5 ~d2 2 d1!/4d1d2~R22 2 R1

2!,

s 5 d1d2~R22 2 R1

2!/~d2R22 2 d1R1

2!. (19)

The stationary point simplifies to the following form:

rs 5 @~z 2 s !/4bzs#1/2. (20)

The asymptotic result within the focal segment will begiven by an interference pattern of three waves; that is,one stationary wave and two edge waves:

I~r, z ! } IEl 1 IEu 1 ISt 1 2~IElIEu!1/2 cos FElEu

1 2~IStIEl!1/2 cos FStEl

1 2~IStIEu!1/2 cos FStEu , (21)

where

IEu 5 I0

d22

~d2 2 z !2 J02S krR2

z D ,

IEl 5 I0

d12

~d1 2 z !2 J02S krR1

z D ,

ISt 5 I0pkJ02$kr@~z 2 s !/4bsz3#1/2%/4bz2 (22)

are intensities of the upper edge wave, the lower edgewave, and the stationary wave, respectively. The phasefactors of the interference terms from Eq. (21) are in turnequal to

FEuEl 5 k@~R22 2 R1

2!~1/2z 2 1/2s ! 1 b~R24 2 R1

4!#,

FStEl 5 k@R22~1/2z 2 1/2s ! 1 bR2

4 1 ~s 2 z !2/16bz2s2#

2 3p/4,

FStEu 5 k@R12~1/2z 2 1/2s ! 1 bR1

4 1 ~s 2 z !2/16bz2s2#

2 3p/4. (23)

Outside the focal region the intensity is equal to

I~r, z ! } IEl 1 IEu 1 2~IElIEu!1/2 cos~FElEu 2 p!.(24)

At the ends of the focal segment I 5 ISt(R1,2)/4. In Fig.3(a) axial intensity calculated numerically from the corre-sponding diffraction integral is shown, and in Fig. 3(b) the

distribution obtained according to Eqs. (21)–(24) isshown. In Fig. 3(c) the difference between the two plotsis drawn. Particular values of the radii of the annularaperture and the position of the focal segment that wereassumed during plot preparation are the same as in otherpublications on the subject3,8,10,18 so that they can be com-pared with other examples; i.e., R1 5 2.5 mm, R25 5 mm, d1 5 100 mm, and d2 5 200 mm (which re-sults in s 5 85.71 mm and b 5 6.667 3 1025 mm23).The corresponding diffraction integral was of the follow-ing form:

Fig. 3. Axial intensity of the doublet consisting of an aberrateddiverging lens and a perfect converging lens (R1 5 2.5 mm, R25 5 mm, b 5 6.667 3 1025 mm23, s 5 85.71 mm, d15 100 mm, d2 5 200 mm, and l0 5 632.8 nm): (a) numericalevaluation of the diffraction integral (the curve corresponds tothe intensity of the stationary wave), (b) distribution resultingfrom the stationary phase method according to Eqs. (21)–(24), (c)difference between the two distributions.


I~r, z ! 5 I0~k/z !2U ER1

R2

exp@ikf~r !#J0~krr/z !rdrU2

,

(25)

where f(r) 5 r2(1/2z 2 1/2s) 1 br4 and was calculatedwith standard numerical procedures. The approximaterepresentation for the Bessel function follows that in thebook by Abramowitz and Stegun.19

The observations from the obtained results are the fol-lowing: The distribution of the axial intensity decreasesproportionally to z22, and in general its change is greaterthan for other types of axicons3,8,18; in turn the width ofthe central peak is maintained with quite good accuracy(Fig. 4). The width of the central core will vary accordingto the formula

r0 5 cl0@4bsz3/~z 2 s !#1/2. (26)

The minimum width r0,MIN and its position on the focalsegment zMIN are given by

r0,MIN 5 cl0~27bs3!1/2,

zMIN 5 3s/2. (27)

For our particular example, zMIN 5 128.57 mm andr0,MIN 5 8.153 mm. Since r0,MAX 5 r0(d1,2) 5 9.687 mm,we get the maximum variation of the central core(r0,MAX 2 r0,MIN)/r0,MAX equal to 15.83%, which is ap-

Fig. 4. Radius of the central spot along the focal segment of thedoublet consisting of an aberrated diverging lens and a perfectconverging lens (R1 5 2.5 mm, R2 5 5 mm, b 5 6.6673 1025 mm23, s 5 85.71 mm, d1 5 100 mm, d2 5 200 mm,and l0 5 632.8 nm): (a) for the stationary wave only, (b) nu-merical evaluation of the diffraction integral.

proximately three times higher than the result obtainedin the case of the logarithmic axicon with analogousparameters3,8,18).

One can see that except for the direct neighborhood ofthe focal segment’s ends, where the assumption about theisolation of the stationary point becomes violated, theagreement between the numerical solution and the SPMevaluation is good. The range of the singularity regionscan be treated approximately as the double distance DMIN

between the nominal end of the focal segment and thefirst neighboring local minimum of the axial intensity dis-tribution obtained according to the SPM. To determinethe position of the minimum, we make some simplifyingassumptions. First, if we consider the vicinity of one ofthe ends of the focal segment, then the interference termof both edge waves, the interference term of the otheredge wave with the stationary wave, and the intensity ofthe other edge wave are neglected. Second, if we takeinto account that the phase of the remaining interferenceterm varies slowly near the end of the focal segment com-pared with the rapidly growing edge wave, the phasevalue will be approximated by its value at the end of thefocal segment, which is equal to 21/21/2. Then Eq. (21) issimplified to the following form:

Inorm 5 1 1 A1/2 /D2 2 ~2A1,2!1/2/D, (28)

where A1,2 5 I0d1,22 /ISt(D 5 0), D 5 ud1,2 2 zu, and

Inorm5 I/ISt(D 5 0). The minimum, found by the deri-vation of Inorm with respect to D, is given by

D1,2MIN 5 ~2A1,2!1/2 5 d1,2

2 S 8b

pkD 1/2

5 K1,2d1,2S d1,2

2pD1,2R D 1/2

,

(29)

where D1,2R 5 pR1,2

2 /l0 is the Rayleigh distance, i.e., acommonly assumed limit for the Fresnel zone of the el-ement’s aperture, and K1,2 5 (d1,2

3 /dG2 dA)1/2 is a numeri-

cal constant of order one [dG 5 (d1d2)1/2, dA 5 (d11 d2)/2 are the geometrical and arithmetic means, re-spectively, of the focal segment ends]. The value of DMIN,although derived under many simplifying assumptions,allows us to obtain a qualitative measure of the singular-ity region. In general, DMIN decreases when the focalsegment becomes the smaller part of the Fresnel zonerange. In our example K1 5 0.577 (D1

MIN 5 1.30 mm),and K2 5 1.633 (D2

MIN 5 5.23 mm). Values found nu-merically from I(0, z) determined by the SPM are 1.02mm and 3.92 mm, respectively.

It is also desirable to find the bending factor that leadsto the required amount of spherical aberration. To dothis we use formula (9e) from the book by Jenkins andWhite,20 combined with Eq. (18):

b 51

32 f 13n~n 2 1 !

Fn 1 2

n 2 1q2 1 4~n 1 1 !pq

1 ~3n 1 2 !~n 2 1 !p2 1n3

n 2 1G , (30)

where n is the refractive index of the lens, q is the bend-ing factor, and p is the position factor (which in the case ofplane-wave illumination is equal to 21). Resolving Eq.(30) for q, one obtains


q1,2 5n 2 1n 1 2 $2~n 1 1 ! 6 @32bf 1

3n~n 1 2 !

2 n2~4n 2 1 !/~n 2 1 !2#1/2%. (31)

Now, placing the chosen value of f1 , one can calculate thevalue of bending factor q that is necessary for obtainingthe required value of b. For instance, for f15 18.42 mm,b 5 6.667 3 1025 mm23 and n 5 1.5 q2 is equal to 0; i.e.,an equiconcave diverging lens is sufficient.

In conclusion, to produce long focal segments, which isusually the case, and not exceed reasonable values of thebending factor, both lenses composing the doublet shouldbe of rather short focal length.

The practical advantage of the discussed solution is thepossibility of its realization in a compact form, which isadvantageous if one takes into account the problems ofthe mutual positioning of both elements in other cases.The considered case is distinguished among all others ad-mitted by the general solution by the smallest change inthe central-core width and the greatest change in theaxial intensity distribution.

B. Perfect Converging Lens Placed in Front of anAberrated Diverging Lens (l 1 f1 5 f 1 D)Another approach is to separate the doublet and illumi-nate the diverging lens with a converging ideal sphericalwave front produced by a perfect lens that is in fact a Gal-ilean telescope illuminated reversibly (Fig. 2). This case,although not covered by the general solution, turns out tobe described by the same formulas as in the previous case,with f replaced by f 2 l as the focal length of the idealconverging wave front in the plane of the aberrated lens.The present arrangement offers the possibility of chang-ing the length of the focal segment by varying the sepa-ration distance l. This relationship can be found by deri-vation of d1,2 with respect to l, which on the basis of Eqs.(18) and (30) is equal to

]d1,2

]l5 2

d1,22

~ f 2 l !2 H 1 1 NA1,22 F n 1 1

n~n 2 1 !q

13n 1 2

2npG J . (32)

It can be seen that for small values of NA (numericalaperture) the influence of the second expression inside thecurly braces, i.e., the change in the aberration that is dueto the change in the illuminating wave-front curvature,can be neglected in comparison with 1, although on theother hand the required bending factor then increases.

However, the sensitivity of the focal-segment param-eters with respect to the separation distance l of thelenses, especially for long focal segments, can be exces-sive. To decrease this sensitivity, the optical power ofthe converging lens can be split into two elements, e.g.,one forming a doublet with the diverging lens and the sec-ond being relatively weak, which would permit us to con-trol the parameters of the focal segment by relativelylarge axial displacements of the weak element andthereby facilitate the adjustment.

C. Aberrated Diverging Lens Placed in the PrimaryFocal Plane of a Perfect Converging Lens (l 5 f )From Eq. (8) it can be seen that for l 5 f, axial intensitybecomes constant. This version, for f1 → `, i.e., with adiverging aberrated lens replaced by a diffractive elementcontaining only the aberration phase function, was ana-lyzed in Ref. 15. As above, the parameters s (now 1/s5 1/f 2 1/(f 1 f1) and b can be determined from Eq. (15)and are given by

b 5 ~d2 2 d1!/4f 2~R22 2 R1

2!,

s 5 d1,2 2 aR1,22 . (33)

The paraxial focus s is equal to 66.67 mm. In turn, incontrast to the previous solution, the aberration coeffi-cient b cannot be determined only from the aperture andfocal-segment parameters; it depends on the focal length fof the transforming lens.

The width of the central core exhibits the greatestvariation as compared with the other cases. Substitutingl 5 f into Eq. (8), one obtains the expression for thecentral-core width

r0 5 cl0f 2@4b/~z 2 s !#1/2. (34)

It should be added that such behavior of the central coremakes the considered setup less convenient for metrologi-cal purposes, but on the other hand it is particularly use-ful for imaging with an increased depth of focus, since thesize of the image is kept constant along the whole focalsegment, in contrast to other axicons.15

In the present case the condition expressed by Eq. (16)no longer leads to equal angles of rays at both ends of thefocal segment (see Fig. 1). For a comparison of thepresent case with the previous one to be made possible—for example, by starting the focal segment with the samewidth—a relation between the aperture edges in theplanes of both lenses should be found. From elementarygeometry we obtain

R1,28 5 R1,2d1,2 /f. (35)

Hence, assuming R18 equal to 2.5 mm and f 5 s (i.e., thelength of the diverging lens is supposed to be infinite, be-cause the aberration coefficient can therefore be thesmallest), from Eqs. (16) and (35) we get values of R1 ,R2 , and R28 equal to 1.667, 3.333, and 10 mm, respec-tively. As one can see, this solution cannot serve to pro-duce long focal segments, since it requires either a focallength comparable to the focal segment or to the great ap-erture of the converging lens (now the distance D is equalto f1 and cannot be treated as a small quantity as in theremaining cases). In addition, the variation of the cen-tral core turns out to be considerably greater than in theprevious cases: r0,MAX 5 r0(d1) 5 9.687 mm and r0,MIN5 r0(d2) 5 4.843 mm, yielding (r0,MAX 2 r0,MIN)/r0,MAXequal to 50%. Moreover, in the present case the aberra-tion coefficient required for creating the same focal seg-ment is greater than before [compare Eqs. (19) and (33)].Even in the most favored case, when the focal length ofthe aberrated lens f1 tends to infinity and f 5 s, the valueof b is much greater than for the previous arrangementand equals 67.5 3 1025 mm23, after the radii of the aper-ture are changed according to Eq. (35). For these rea-


sons a detailed discussion of focal-segment characteristicswill be omitted; moreover, such discussion can be foundelsewhere.3,15

D. Defocused Galilean Telescope (l 1 f1 5 f 1 D)This particular version of the lens axicon corresponds tol 1 f1 5 f 1 D; i.e., the secondary focal point of the aber-rated diverging lens is placed at a distance D from the pri-mary focal point of the converging lens and was in fact thefirst proposed version of the forward-type lens axicon.13

Its experimental realization and numerical simulationhave been also reported recently.14 The parameters ofthe focal segment are described by solutions found for thegeneral case [Eqs. (9) and (10)]. For long focal distances(which is usually the case of interest), the following ap-proximations are valid:

D ! f1 ,f,t,l ! s,z,d1,2 . (36)

Therefore the axial intensity [Eq. (9)] and the central-corewidth [Eq. (10)] can be approximated with good accuracyby expressions similar to those found in the doublet’scase:

ISt~0, z ! ' I0

pk

4bz2 S f

f1D 2

,

r0 ' cl0S f1

f D 2

@4bsz3/~z 2 s !#1/2. (37)

Analogously, as in Subsection 2.B, the excessive sensitiv-ity of the focal-segment coordinates with respect to theseparation change of the system components can be de-creased by splitting the optical power of the converging el-ement into two, with one part being much weaker thanthe other.

On the other hand, in contrast to the case described inSubsection 2.B, the aberration coefficient is not influencedby the separation change, because the diverging lens is il-luminated by a plane wave and the position factor re-mains constant.

3. CONCLUSIONSWe have presented the general case of the forward-typelens axicon consisting of a third-order spherical aberra-tion diverging lens and a converging perfect lens. The in-tegral describing the diffracted field behind the axiconwas evaluated asymptotically with the help of the nonuni-form stationary phase method (SPM), and the axial inten-sity distribution and the central-core width of the focalsegment were found. Four special cases were chosen formore detailed analysis because of the parameters of theirfocal segments. Three of the cases turned out to be use-ful in metrological applications: a defocused Galileantelescope, a perfect converging lens placed in front of anaberrated diverging lens, and a doublet. The last ex-ample, the doublet, was analyzed with the help of a moredetailed SPM, with edge waves taken into account. Al-though the SPM is an asymptotic approach, the resultsnevertheless turned out to be surprisingly good. Only inthe direct vicinity of the focal segment’s ends, where theassumption about the stationary point isolation becomesviolated, did singularities appear; moreover, their extent

is usually negligible compared with the length of thewhole focal segment. The description offered by the non-uniform SPM is quite simple, and the diffraction fieldwithin the scope of this approach becomes the interfer-ence of the stationary wave with edge waves.

The analysis indicates clearly the poorer quality of thelens axicon that consists of a diverging aberrated lens anda converging perfect lens in comparison with other kindsof axicons: Neither the axial intensity nor the width ofthe focal segment is constant. However, changes in thecentral core width are quite moderate for some cases (e.g.,the doublet), and constant axial intensity can be achievedfor others. The poorer performance of refractive lens axi-cons is compensated by the higher efficiency of focal seg-ment formation as well as by easier and cheaper construc-tion.

As we conclude our presentation of the setup consistingof aberrated and perfect lenses, it should be added thatemploying the projecting converging lens with a speciallydesigned amount of spherical aberration has been sug-gested, which would allow us to achieve the long focal seg-ments with a slowly varying position of the first zero.21

Since the lens with third-order spherical aberration canbe shown to produce, at a certain distance, a complex am-plitude distribution that in first approximation is aspherical wave front that has fifth-order spherical aberra-tion, this case will be treated separately in a forthcomingpublication, with regard to the doublet consisting of third-and fifth-order-spherical-aberration lenses.

Z. Jaroszewicz is on leave from the Institute for AppliedOptics, Kamionkowska 18, 03-805, Warsaw, Poland.

REFERENCES1. J. H. McLeod, ‘‘The axicon: a new type of optical element,’’

J. Opt. Soc. Am. 44, 592–597 (1954).2. L. M. Soroko, ‘‘Axicons and meso-optical imaging devices,’’

in Progress in Optics, E. Wolf, ed. (Elsevier, New York,1989), Vol. 27, pp. 109–160.

3. Z. Jaroszewicz, ‘‘Axicons, design and propagation proper-ties,’’ Vol. 5 of Research and Development Treatises, M.Pluta, ed. (Polish Chapter of the International Society forOptical Engineering, Warsaw, 1997).

4. J. Fujiwara, ‘‘Optical properties of conic surfaces. I. Re-flecting cone,’’ J. Opt. Soc. Am. 52, 287–292 (1962).

5. M. V. Perez, C. Gomez-Reino, and J. M. Cuadrado, ‘‘Diffrac-tion patterns and zone plates produced by thin linear axi-cons,’’ Opt. Acta 33, 1161–1176 (1986).

6. A. Vasara, J. Turunen, and A. T. Friberg, ‘‘Realization ofgeneral nondiffracting beams with computer-generated ho-lograms,’’ J. Opt. Soc. Am. A 6, 1748–1754 (1989).

7. J. Sochacki, A. Kolodziejczyk, Z. Jaroszewicz, and S. Bara,‘‘Nonparaxial designing of generalized axicons,’’ Appl. Opt.31, 5326–5330 (1992).

8. A. T. Friberg, ‘‘Stationary-phase analysis of generalizedaxicons,’’ J. Opt. Soc. Am. A 13, 743–750 (1996).

9. J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, ‘‘Diffraction-free beams,’’ Phys. Rev. Lett. 58, 1499–1501 (1987).

10. L. R. Staronski, J. Sochacki, Z. Jaroszewicz, and A. Kolodz-iejczyk, ‘‘Lateral distribution and flow of energy in uniform-intensity axicons,’’ J. Opt. Soc. Am. A 9, 2091–2094 (1992).

11. A. C. S. van Heel, ‘‘Modern alignment devices,’’ in Progressin Optics, E. Wolf, ed. (Elsevier, New York, 1960), Vol. 1,pp. 288–329.

12. R. B. Gwynn and D. A. Christensen, ‘‘Method for accurateoptical alignment using diffraction rings from lenses withspherical aberration,’’ Appl. Opt. 32, 1210–1215 (1993).


13. R. M. Herman and T. A. Wiggins, ‘‘Production and uses ofdiffractionless beams,’’ J. Opt. Soc. Am. A 8, 932–942(1991).

14. T. Aruga, ‘‘Generation of long-range nondiffracting narrowlight beams,’’ Appl. Opt. 36, 3762–3768 (1997).

15. J. Rosen, B. Salik, and A. Yariv, ‘‘Pseudo-nondiffractingbeams generated by radial harmonic functions,’’ J. Opt. Soc.Am. A 12, 2446–2457 (1995).

16. I. S. Gradshtein and I. M. Ryzhik, Tables of Integrals, Se-ries, and Products (Academic, New York, 1980), Chap. 6.

17. J. Stamnes, Waves in Focal Regions (Hilger, Bristol, 1986),pp. 91–135.

18. Z. Jaroszewicz, J. F. Roman Dopazo, and C. Gomez-Reino,‘‘Uniformization of the axial intensity of diffraction axiconsby polychromatic illumination,’’ Appl. Opt. 35, 1025–1031(1996).

19. M. Abramowitz and I. A. Stegun, eds. Handbook of Math-ematical Functions (Dover, New York, 1972), pp. 355–389.

20. F. A. Jenkins and H. E. White, Fundamentals of Optics(McGraw-Hill, London, 1984), pp. 152–162.

21. R. M. Herman and T. A. Wiggins, ‘‘High-efficiency diffrac-tionless beams of constant size and intensity,’’ Appl. Opt.33, 7297–7306 (1994).

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Lens axicons: systems composed of a diverging aberrated lens and a perfect converging lens