EOS Topical meeting on Diffractive Optics 1
Diffraction analysis of lens axicons
A. Burvall, A. Goncharov, and J.C. Dainty Applied Optics, Department of Experimental Physics
National University of Ireland, Galway, Ireland anna.burvall@nuigalway.ie
1 Introduction
Axicons, as shown in Fig. 1(a), are optical elements that produce long and narrow focal lines along the optical axis, rather than the conventional focal point created by a lens. The focal line can be used for e.g. alignment, extending the focal depth of existing methods, or particle trapping and transportation. Axicons are mainly produced as refractive glass cones, or as diffractive gratings consisting of concentric circles. They can also be constructed from ordinary lenses or lens systems, referred to as lens axicons. There are several lens axicon designs, using spherical or aspheric surfaces to produce the necessary amount of spherical aberration.
Numerical analysis of axicons, mostly generation of axial and transverse intensity distri- butions, is generally done by insertion of the thin-element approximation into the Fresnel diffraction integral. For lens axicons, and for large-angle refractive axicons, the thin-element approximation is no longer accurate. We use instead a hybrid method based on finite ray- tracing, followed by Fresnel diffraction analysis at the last surface[1]. For off-axis points, the Fresnel approximation is not valid[2] and the Fresnel-Kirchoff diffraction integral is used. This method allows us to generate axial and transverse intensity distributions for almost any lens axicon system. We present diffraction analysis for the telephoto axicon, concentrating on its off-axis properties.
d1 d2 R2
R1
z x
y
z z Intensity
d1
d2 D1/2
D2/2
d1 d2
ρ
(a) (b)
Figure 1: (a) Principle of the axicon. (b) Principle of the telephoto lens axicon.
2 The telephoto lens axicon
The telephoto axicon is a singlet lens axicon with spherical surfaces, as illustrated in Fig. 1(b).
The double-pass, achieved using reflective coatings on parts of the surfaces, produces negative spherical aberration. If the surfaces are made concentric or nearly concentric about the aperture centre, most off-axis aberrations are avoided. Figure 2 shows the axial and transverse intensity distributions produced by such a lens, evaluated numerically by the hybrid raytracing and diffractive method. Ideally, both the axial intensity and the line width should be uniform.
As this is not possible, even for a refractive cone axicon, the best possible compromise is
normally chosen. For the lens used in Fig. 2, intensity and line width variations are small for
a lens axicon with spherical surfaces.
2 Diffractive Optics ’05 Warsaw
x [mm]
y [mm]
0.005 0 0.005 0.01
0.01
0.005
0
0.005
0.01
(a) (b)
40 50 60 70 80
0 500 1000 1500 2000 2500
z [mm]
Intensity [a.u.]
Figure 2: (a) Axial and (b) transverse intensity distributions (z=55 mm) for a telephoto lens axicon made from BK7 with D
1=10 mm, D
2=20 mm, R
1=−14.800 mm, R
2=−26.104 mm, and thickness 10.000 mm.
3 Off-axis properties of the telephoto lens axicon
The main advantage of the telephoto lens is its off-axis properties. Normal axicons suffer from off-axis aberrations, mainly astigmatism, that cause the focal line to broaden and take on an asteroid shape[2]. If all surfaces of the telephoto lens axicon are concentric about the centre of the aperture, these aberrations are removed and the focus retains its narrow, circular shape.
Since the possible angles are limited to approximately ±5
◦by vignetting, the second surface was allowed to deviate from the concentric curvature to improve the axial properties. The result is shown in Fig. 3(a) and (b), for 3
◦and 5
◦respectively. Figure 3(c), for comparison, contains the intensity produced by a corresponding refractive cone axicon at 5
◦off-axis angle.
x [mm]
y [mm]
0.01 0.005 0 0.005 0.01
3.505
3.5
3.495
3.49
x [mm]
y [mm]
0.02 0 0.02
8.79 8.78 8.77 8.76
x [mm]
y [mm]
0.05 0 0.05
8.76 8.78 8.8 8.82 8.84
