Characterization of a refractive logarithmic axiconIlya Golub,1,* Brahim Chebbi,2 Dagan Shaw,1 and Dariusz Nowacki1

1School of Advanced Technology, Algonquin College, Ottawa, Ontario K2G 1V8, Canada2School of Engineering, Laurentian University, Sudbury, Ontario P3E 2C6, Canada

*Corresponding author: golubi@algonquincollege.com

Received June 4, 2010; revised July 21, 2010; accepted July 22, 2010;posted July 27, 2010 (Doc. ID 129190); published August 13, 2010

We show that it is feasible to design and manufacture a refractive logarithmic axicon that generates a quasi-diffraction-free/Bessel beam with nearly constant beam size and intensity over a predetermined range. The noveloptical element was characterized with both coherent and incoherent light, and good correspondence with the pre-dicted behavior of the intensity distribution and spot size was found. The energy flow was also found to be nearlyconstant over most of the designed range. Logarithmic axiconsmay find applications in situations where large depthof field and uniform axial intensity/energy distributions are important. © 2010 Optical Society of AmericaOCIS codes: 260.0260, 220.0220, 070.3185, 120.0120, 230.0230, 070.7345.

Extension of depth of field (DOF) or focal range is one ofthe major thrusts in the development of modern imagingsystems [1–5]. The proposed methods, however, areeither complex/expensive or suffer from loss of the inci-dent light energy, reduction of the image resolution, ordecreased quality of system transfer function. Axicon[6,7] is an optical element that focuses light into afocal line segment and is, thus, well suited for imagingwith extended DOF. This property to generate a quasi-diffraction-free/Bessel beam [8] renders axicons to findapplications in diverse fields, such as optical tweezers,microscopy, super-resolution, optical coherence tomo-graphy (OCT), nonlinear optics, etc. [9–15]. However,a classical axicon exhibits a linear growth of the on-axisintensity for an incident plane wave. This disadvantage—nonuniform longitudinal intensity distribution—can beovercome by using logarithmic axicons (LA) by engineer-ing the phase of the optical element [16–19].So far, only a holographic version of a logarithmic

axicon was experimentally demonstrated [17]. This LAhas a very large chromatic dispersion and small effi-ciency. Moreover, a question was raised concerningthe uniformity of lateral resolution and energy flow alongthe focal line of the LA [18,19]. To address these issues,we designed, procured, and characterized a refractivelogarithmic axicon for the first time to the best of ourknowledge. Our measurements show that such an LAgenerates a beam of nearly constant lateral size and con-stant intensity/energy flow over most of the predeter-mined range.Uniform axial intensity distribution can be achieved

with a spatial light modulator (SLM) [20] as well as bya tandem of a beam former and a nonlinear axiconcombined with an aberration corrector [21]. While anSLM-based system is versatile, its efficiency is only afew percent, and it is very chromatic and expensive. Thetandem system requires micrometer precision alignmentbetween the elements and is bulkier than an LA. Thus arefractive LA has many advantages compared with theseschemes, particularly in applications using incoherentlight like OCT [12], imaging, and illumination. While LAcan produce interference-caused oscillations in theintensity distribution, they can be smoothed out byapodization and the use of incoherent light.

We concentrate on a forward LA that, compared to abackward LA, has better lateral resolution uniformity[18]. For a plane incident wave, the expression forthe forward LA retardation function (the phase retarda-tion is obtained by multiplying this function by 2π=λ)is [16–18]

φðrÞ ¼ −

12a

lnðf 1 þ ar2Þ þ const; ð1Þ

where a ¼ ðf 2 − f 1Þ=R2, f 1 and f 2 are the beginning andthe end of the focal range of the LA, and R is the radius ofthe axicon. We chose f 1 ¼ 50 mm, f 2 ¼ 150 mm, and R ¼6:25 mm, meaning that a plane wave incident on such anelement will be focused into a line segment extendingfrom 50 to 150 mm. To calculate the sag (thickness) ofthe optical element, the retardation function [Eq. (1)] isdivided by (n − 1), n being the refractive index of the LA.The logarithmic axicon was manufactured by B-Con En-gineering Inc., using the diamond turning of a plasticZeon E48R with n ¼ 1:52.

Figure 1 shows the sag of the designed forward axiconwhile comparing it with the sags of two lenses havingfocal lengths f 1 ¼ 50 mm and f 2 ¼ 150 mm (dashed anddotted curves, respectively).

Fig. 1. Sag (thickness) versus radius of a forward logarithmicaxicon (solid line) and lenses with f ¼ 50 mm (dashed curve)and f ¼ 150 mm (dotted curve).

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0146-9592/10/162828-03$15.00/0 © 2010 Optical Society of America

In measuring the on-axis intensity distribution, weused a 0:534 μm He–Ne laser beam expanded telescopi-cally to have a quasi plane wave when using its centralpart and a Thorlabs OSL1 white-light source. To complywith a requirement on the finite spatial coherence of thewhite-light source [22], a 50 μm aperture at source wasused and the beam was collimated to a nearly planewave. As in [22], it was found that for such an aperture,while the LA produced only two detectable fringes/rings,the beam exhibits the properties of a quasi-diffraction-free one, i.e., it has a central spot propagating over a dis-tance exceeding the Rayleigh range by several orders ofmagnitude. For a laser source, the fringes were much bet-ter resolved. A 20× microscope objective was used toenlarge the central spot. Figure 2(a) compares the mea-sured longitudinal intensity distribution for a forward LAfor both a white-light source (solid curve) and a coherentone (dotted curve). The white-light source produces abroader curve due to its larger divergence. A more salient

feature is the coherent-source-generated interference. Asexpected, the curve is smoother when using an inco-herent light source. In addition, these interference fringescan be suppressed by apodization [19,20]. A simulation ofthe propagation of the beam produced by our LA basedon the Fresnel diffraction integral [Fig. 2(b)] shows agood correlation with the experimental results [Fig. 2(a)].Figure 2(c) presents the measured longitudinal intensitydistribution for a 5° base angle classical axicon illumi-nated by a white-light source. While the logarithmicaxicon has a nearly constant intensity over most of thefocusing range, 50 to 150 mm, the ordinary axicon hasa linear on-axis intensity that varies by a factor of threeover the same range—more than 1 order of magnitudelarger than the intensity variation for a LA.

The number of discernible fringes for a white-lightBessel beam is given by [22]

N ¼ Int

�1:67ω=ΔFWHM − 1=2

2

�; ð2Þ

where ΔωFWHM is the FWHM bandwidth of the source offrequency ω. In our case,ΔωFWHM ¼ 200 nm, resulting inmaximum of two detectable fringes whose structure ispartially washed out. Thus, to measure accurately thequasi-Bessel beam size, we used the He–Ne laser. Thetransverse intensity distributions for an ordinary axiconand an LA are shown in Figs. 3(a) and 3(b) with a cali-bration scale. The measured quasi-Bessel beam spot size(distance between the first two minima) of the LA asfunction of distance is depicted in Fig. 4. It is evident thatthe spot size, after decreasing drastically at the beginning

Fig. 2. (Color online) (a) Measured on-axis intensity distribu-tions of beams produced by a forward logarithmic axicon with asag shown in Fig. 1 for a white-light source (solid curve) and fora laser (dotted curve). (b) Calculated on-axis intensity distribu-tions of a forward logarithmic axicon with the same parametersas in Fig. 1. (c) Measured on-axis intensity distribution for aclassical, 5° base angle, axicon.

Fig. 3. (Color online) Transverse intensity distributions ofbeams produced by (a) classical axicon and (b) logarithmicaxicon with a calibration scale.

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of the focusing range, is practically constant from ∼7 cmon. From this result and the measurement of near-uniform on-axis intensity over the same range [Fig. 2(a)],we conclude that the energy flow is constant over most ofthe designed range, i.e., from ∼70 to ∼150 mm.In conclusion, we have demonstrated a refractive

logarithmic axicon that generates a quasi-nondiffracting/Bessel beamwith nearly constant beam size and constantintensity over a predetermined range. Such beams mayfind applications in imaging, illumination, atmosphericpropagation, OCT, nonlinear optics, and other situationswhere large depth of field and uniform on-axis intensitydistribution are important. Moreover, this work showsthat it is feasible to design and implement simple/inexpensive refractive optical elements by reverse engi-neering of desired on-axis intensity distributions to cor-responding aspheric element shapes.

This work was supported by an Ontario Centers ofExcellence grant.

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Fig. 4. (Color online) Quasi-Bessel beam spot size of aforward logarithmic axicon as a function of distance fromthe axicon.

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