The axicon is a conical lens of growing technological im-portance [1] used to generate Bessel beams for opticaltrapping [2], to increase the depth of focus in imaging[3], for hyperopic refractive surgery [4], and for nonlinearoptical applications [5]. The focal length of an axicon isproportional to the off-axis distance of any ray incidentparallel to its optical axis, and thus a positive axicon willfocus an incident beam into an axial line image. Diver-gence of the incident beam shifts the image axially, sug-gesting a possible use in wavefront sensing. This hasindeed been explored in self-referenced interferometryto evaluate the refractive quality of axicons [6]. Actually,multifaceted prisms with large apex angles have alreadyfound sensing applications in astronomy [7] and ophthal-mology [8,9]. A common configuration employs an oscil-lating pyramid-shaped prism to obtain four pupil images,one for each facet, sequentially mapped onto a CCD cam-era. Aberrations divert the light, causing local brightnessdifferences among the images that would otherwise ap-pear equally bright when averaged for an oscillationperiod. An axicon resembles the pyramid, but with aninfinite number of facets forming the conical prism oraxicon lens.The point-diffraction interferometer makes use of a
tiny aperture to generate a quasispherical reference wavethat interferes with the wave of interest [10] where in-creased accuracy can be obtained with dual aperturesand phase shifting [11]. In turn, a larger hole may be usedto generate a less-divergent reference wave, extendingthe dynamic range, albeit at the cost of accuracy [12].In this Letter, an axicon is suggested for wavefront sen-sing in which the apex serves as a scattering center,creating an interferometric reference wave. The pro-posed scheme is analyzed with Fourier optics, and simu-lations are compared to experimental results obtainedwith standard ophthalmic trial lenses. The suitabilityand limitations of this approach for wavefront sensingare discussed.The experimental layout is shown schematically in
Fig. 1 together with examples of calculated intensitiesjψ ij2 for a perfectly sharp axicon with a large apex angleα ¼ 175°, refractive index n ¼ 1:51509 (BK7 glass), andillumination wavelength λ ¼ 632:8 nm. In the calculation,the field at the pupil has been limited by a 4:8mm circulariris in front of which trial lenses are inserted to generatecontrolled low-order aberrations. The field incident on
the CCD camera as modified by the axicon refractioncan be expressed by the convolution
ψ i ¼ �ψp expði �ΦWAÞ � FfexpðiΦaxiconÞg; ð1Þ
where the bar above the pupil field �ψp and wavefrontphase �ΦWA denotes a coordinate inversion (x → −xand y → −y). FfexpðiΦaxiconÞg is the Fourier transformof the paraxial linear radial-phase transmission functionof the axicon with ΦaxiconðrÞ ¼ −k0ðn − 1Þðtanðα=2ÞÞ−1r,and k0 is the wavenumber in air.
Taking advantage of the rotational symmetry of the ax-icon, the Fourier transform of its phase transmissionfunction can be rewritten as a Fourier–Bessel transform
Fig. 1. Schematic drawing (top) of the 4-f arrangement withtwo f25mm achromatic lenses and an axicon inserted to facil-itate wavefront sensing in a plane conjugate with the iris wheretrial lenses are inserted. Calculated intensity images (bottom) atthe CCD for different amounts of defocus (with correspondingimage cross sections) and cylinder expressed in diopters at theentrance pupil as obtained for a perfectly sharp 175° axiconapex.
corresponding to a hollow cone of wave vectors that inthe absence of aberrations form a ring of light incident onthe CCD. Divergence of the incident light contained inΦWA changes the focusing within this ring and can poten-tially be used for wavefront sensing. The numerical re-sults shown in Fig. 1 for the special cases of sphericaldefocus and cylinder, respectively, confirm sensitivityto the magnitude as well as to the sign of the aberrationterm. However, the variation is not symmetric with re-spect to zero diopters because of the axicon refraction.For very large apex angles, the hollow cone of refractedwave vectors narrows down, with Eq. (1) reduced toψ i ¼ �ψp expði �ΦWAÞ.Recognizing the paraxial potential of projecting the
(inverted) pupil wavefront to the image plane by an es-sentially flat axicon, the proper axicon apex may be usedas a scattering center to generate an interferometric re-ference wave that can substantially enhance the usabilityof the approach. In such a case, a quasiplanar referencewave ψapex originating in the pointlike apex scattering bythe axicon interferes with the light ψ i in the plane of theCCD. This approach resembles the point-diffraction in-terferometer [10], albeit without attenuation of theprobed wave field ψ i and with the tiny aperture of thepoint-diffraction interferometer replaced by the axiconapex. Its scattering strength, which determines the con-trast of recorded interferograms, is given by the dielec-tric contrast and apex size. A blunt axicon will increasethe strength of the reference wave but also deterioratethe attainable resolution [12], since the scattered fieldeventually differs from a plane wave across the observa-tion plane. The size and incidence of the focused lightspot on the axicon apex will impact the contrast andshape of the generated interferograms, and thereforecareful alignment is needed.Figure 2 shows experimentally recorded CCD intensity
images with different amounts of defocus, induced bytrial lenses in the entrance pupil, of an expanded26-mm-wide (1=e2) He─Ne laser beam (632:8 nm wave-length) using the aforementioned 175° BK7 axicon.The width of the sampled beam has been limited to4:8mm by the iris at the entrance pupil, whose edge isbrought to a focus in the image plane and fills most ofthe CCD. The experimental images show a characteristicrotationally symmetric pattern of defocus and superim-posed high-frequency interference patterns caused by im-purities and the fact that the axicon is not antireflexcoated. For comparison, calculated interferograms jψ i þψapexj2 with different amounts of defocus have been in-cluded. The best correspondence between experimentsand theory was observed with simulations in whichthe axicon being used was assumed to be practically flatand with a 6-μm-wide apex plateau (corresponding tothree pixels in the Fourier plane, i.e, at the apex) indu-cing a phase shift of −3π=2, as found by numerical fitting.Examination of the axicon with an optical microscopehas indeed confirmed the presence of an approximately6-μm-wide apex defect (slightly smaller than the Airy disc
of an unaberrated incident beam). The numerical inter-ferometric results are found to be slightly wider thanthe experimentally observed images, possibly from smalldifferences in the limiting iris. Experimentally observeddefocus images at �1:00D contain approximately 2.5 in-terference fringes from the center outward, correspond-ing to �0:98D, in good agreement with expectations andcorresponding to an accuracy of ∼λ=17. For larger aber-rations more light intersects the axicon farther from theapex, reducing the contrast and complicating the inter-pretation as a consequence of refraction.
As a step toward astigmatism, cylindrical lenses havebeen considered, and results are shown in Fig. 3 togetherwith corresponding numerical simulations. Some devia-tion may be observed for very small cylindrical defocus,with the predicted central brightness being attenuated inthe experimental results. The number of interference
Fig. 2. Interferometric images of wavefront aberrationsrecorded with different amounts of defocus (top) and corre-sponding simulations (bottom) with a blunt axicon apex gener-ating a suitable interferometric reference wave.
Fig. 3. Interferometric images of wavefront aberrations re-corded with different amounts of cylindrical defocus (top)and corresponding numerical simulations (bottom) with a bluntaxicon apex generating a suitable interferometric referencewave.
fringes observed across the image for þ1:00D is, how-ever, again in good agreement with expectations.Figure 4 shows spherocylinder results obtained with
combinations of adjacent spherical and cylindrical lensesnear the limiting iris. Good agreement is observed be-tween experimental and numerical results, but with someminor deviation for the case of −0:50D sphere, þ0:25Dcylinder. As an example, consider the −0:50D sphere,þ1:00D cylinder combination resembling crossed cylin-ders and ideally producing an astigmatic wavefront.Approximately 1.5 interference fringes are containedfrom the center and outward, corresponding to about�0:58D, in fair agreement although possibly with aslightly larger cylinder value than expected.A remaining issue is the dynamic range of the axicon
used. Figure 5 shows experimental examples with in-creased defocus and crossed cylinders. For larger aber-rations, more light intersects the axicon farther from theapex, reducing the contrast and complicating the inter-pretation of the interferograms due to refraction. Thusestimates beyond �3D defocus become noisy, althoughan improved setup may extend the range slightly. Alter-natively, an axicon with a larger apex area may be used,similar to the use of larger holes in the point-diffractioninterferometer [12]. The impact of refraction may only beneglected in the limit that all rays intersect the apex atoff-axis distances ≪2λ=ðn − 1Þðπ − αÞ, with the apexangle expressed in radians.In conclusion, an axicon used for wavefront sensing
has been analyzed as a proof of concept experimentallyand numerically. The sensor bears similarities to the tra-ditional point-diffraction interferometer, as well as to thepyramidal wavefront sensor (but with rotational symme-try). Interferometric wavefront sensing has been demon-strated in which light scattering by the axicon apex
generates a suitable interferometric reference wave, in-creasing substantially its usability. When aberrationsare small, the focused light will be localized to a plateaunear the imperfect axicon apex, and the sensor perfor-mance is similar to that of a point-diffraction interferom-eter. For larger aberrations, the refraction caused by theaxicon becomes dominant, making a sensing applicationmore complex but feasible, as shown in Fig. 1. The linearoperational range of the sensor can be improved by re-ducing the refraction using a larger apex angle or liquidimmersion, or by focusing onto it more tightly.
A deformable mirror could advantageously be used inplace of trial lenses to better control the amount of in-duced aberrations, including higher orders. The exactwavefront phase may be extracted from the recorded in-terferograms and projected onto individual Zernike poly-nomials for a circular pupil. Finally, axial oscillations(comparing a plane before and after the focus) may alle-viate the sensitivity to the quality of the axicon apex, butthis is beyond the scope of the present study.
This research was funded by Science FoundationIreland under grants 07/SK/B1239a and 08/IN.1/B2053.
References
1. J. H. McLeod, J. Opt. Soc. Am. 44, 592 (1954).2. V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and
K. Dholakia, Nature 419, 145 (2002).3. J. Sochacki, A. Kolodziejczyk, Z. Jaroszewicz, and S. Bará,
Appl. Opt. 31, 5326 (1992).4. D. P. S. O’Brart, C. S. Stephenson, K. Oliver, and J. Marshall,
Ophthalmology 104, 1959 (1997).5. T. Wulle and S. Herminghaus, Phys. Rev. Lett. 70, 1401
(1993).6. D. Kupta, P. Schlup, and R. A. Bartels, Appl. Opt. 47, 1200
(2008).7. R. Ragazzoni, J. Mod. Opt. 43, 289 (1996).8. I. Iglesias, R. Ragazzoni, Y. Julien, and P. Artal, Opt. Express
10, 419 (2002).9. E. M. Daly and C. Dainty, Appl. Opt. 49, G67 (2010).10. R. N. Smartt and W. H. Steel, Jpn. J. Appl. Phys. 14, 351
(1975).11. H. Medecki, E. Tejnil, K. A. Goldberg, and J. Bokor, Opt.
Lett. 21, 1526 (1996).12. E. Acosta, S. Chamadoira, and R. Blendowske, J. Opt. Soc.
Am. A 23, 632 (2006).
Fig. 4. Interferometric images of wavefront aberrations re-corded with �0:50D of sphere and cylinder of þ0:25D andþ1:00D (top) and corresponding numerical simulations (bot-tom) with a blunt axicon apex generating a suitable interfero-metric reference wave.
Fig. 5. Measurements of increased spherical defocus þ3, þ5,and þ10D, and (right) one example of crossed þ3D= − 3Dcylinder.
