In this paper we demonstrate how Bessel beam interferometry can be used to characterize the curvature of a reflecting surface. The approach is based on the fact that the intensity distribution produced by the coherent superposition of Bessel beams is a sensitive function of the relative phases between the constituting beams. We show how this phase sensitivity can translate into accurate measurements of the curvature of a wavefront. Experimental tests were made with a liquid mirror. We have also used Bessel beams to measure the precession angle of the liquid mirror. Our results show that Bessel beam interferometry is a very accurate tool for the optical testing of non-stationary surfaces and that it could be used as a general method of real-time, non-contact sensing. Bessel beam interferometry has the advantage of not requiring any reference arm that needs to be stabilized.
©2004 Optical Society of America
Durnin and al. have shown how to generate optical beams whose shape is well described by Bessel functions [1, 2]. These so-called Bessel beams are solutions of the free-space wave equation in circular cylindrical coordinates, just as plane waves are solutions of the wave equation in Cartesian coordinates. In theory, Bessel beams could propagate to infinity without any change in their transverse intensity distribution. As a result they are often called diffraction-free or nondiffracting beams. However, ideal Bessel beams cannot be produced because they are not square integrable. In practice, quasi-non-diffracting beams with a Bessel-like radial profile can be generated by using various methods such as an annular aperture located in the focal plane of a lens [1, 2], refractive axicons , holographic or diffractive optics elements [4, 5], to name a few. All such methods have the property of deflecting incident optical waves at a contant angle θ with respect to an axis of symmetry. The transverse intensity profile of the so-produced Bessel beams are not subject to diffractive spreading over a distance that can extend over several meters [2, 6]; that distance depends upon the size of a specific Bessel beam and the angle θ of the interfering waves that constitute it. The nondiffracting property of Bessel beams can be used in applications requiring laser beams with a large field depth. It has also been pointed out that the well-defined, narrow central lobe of Bessel beams of J0 type is well suited for applications requiring high-pointing accuracy . Bessel beams have found applications in nonlinear optics [8–10], in the manipulation of microparticles [11–13] or in optical coherence tomography . Intense Bessel beams from femtosecond systems were recently used to modify the index of refraction of a glass in such a way as to inscribe optical waveguides or microfluidics [15, 16].
In this paper we extend the field of applications of Bessel beams to the optical testing of reflecting surfaces. The interference pattern produced by the coherent superposition of Bessel beams is a sensitive function of the relative phases between the beams . This property allows using Bessel beams for interferometric measurements. Bessel beam interferometry has the advantage of not requiring any reference arm that needs to be stabilized. We show how Bessel beam interferometry can be used to monitor the radius of curvature of an optical wavefront. Experimental implementation of the method was realized using a liquid mirror of 1.4-meter diameter. We used a liquid mirror because it has a deformable surface whose radius of curvature is a function of the rotation speed of its container. Bessel beams were also used to monitor in real-time the tilt of a wavefront, hence providing information about the precession angle of the liquid mirror.
2. Principle of Bessel beam interferometry
In our experiment, we produce Bessel beams by using masks with thin concentric annular slits placed in the object plane of a Fourier transform lens (see Fig. 1).
The mask is illuminated with a monochromatic incident laser beam. It is known [1, 2] that, in the Fourier plane of the lens, each thin annular slit generates a beam whose field amplitude is well described by a Bessel function J0(αr). The spatial frequency α of the Bessel beam is given by α=πd/λf, where d the diameter of the slit, λ the optical wavelength and f the focal length of the lens. A mask with N concentric annular slits produces a superposition of N Bessel beams, each beam having its own values of spatial frequency and phase; the spatial frequency of a Bessel beam is fixed by the diameter of the slit producing it and its phase is specified by the local value of the incident beam phase at the position of the slit. In turn, the curvature of the wavefront of the incident beam determines the local value of its phase. As a result, the coherent superposition of these N Bessel beams produces an interference pattern that is a sensitive function of the wavefront of the beam incident to the mask . This feature is illustrated in Figs. 1(b) and 1(c) for the case of two Bessel beams interfering constructively or destructively. When the wavefront of the incident beam is tilted due to an off-axis propagation, the central spot of a Bessel beam is shifted laterally in the Fourier plane, as illustrated in Fig. 1(d). The purpose of this paper is to report experimental evidence that the phase sensitivity of Bessel beams how the phase sensitivity of Bessel beams can be exploited to perform measurements.
It can be shown  that, in the Fourier plane of the lens at z=2f, the field Etot produced with a mask having N concentric thin annular slits is given by the following expression:
where dn is the diameter of the nth slit in the mask, An is a real number proportional to the amplitude of the beam incident on the nth slit in the mask, k is the wave number, R1 is the radius of curvature of the beam incident on the mask, αn is the spatial frequency of the nth Bessel beam and r is the radial coordinate (it is assumed herein that the beam has a parabolic wavefront; in principle, the result can be generalized to any wavefront of azimuthal symmetry). Examples of interference patterns that can be produced with a mask having two annular slits are shown in Fig. 2 (see graphics at the lower left). The computations were made for parameters typical of our experimental conditions: λ=632.8 nm, N=2, d1=3.13 mm, d2=4.36 mm and f=88.4 cm. It is evident that the overall profile of the interference pattern is quite sensitive upon the value of beam curvature 1/R1; clearly the value of intensity at center (r=0) varies considerably as a function of curvature.
When the mask has only two annular slits, the expression for the intensity Itot detected at the center (r=0) of the interference pattern and at z=2f can be reduced to the simple expression:
where and are constants proportional to the intensity of the beam incident on the mask at the position of the two annular slits. The curve at the top left of Fig. 2 shows how the intensity at the center of the interference pattern varies as a function of the curvature 1/R1. The periodic behaviour of the intensity at center as a function of the beam curvature is a signature of the self-imaging property of a pair of Bessel beams [17–19]. In general, self-imaging leads to a periodic behavior of the superposition of the Bessel beams as a function of the propagation distance z. For the case considered here, the periodic behaviour is predicted for a detector fixed at a steady value of z, but for an incident beam with a varying radius of curvature.
The sensitivity of the interference pattern to the wavefront curvature can be increased by increasing the argument of the cosine factor in Eq. (2). A first possibility consists in using a mask with an outer slit having a large diameter d2. In practice, the value of d2 is limited by the spatial extent of the beam incident on the mask. When d2 is larger than the beam diameter, the coefficient B2 of Eq. (2) becomes too weak with respect to B1 and the interference signal (last term of Eq. (2)) cannot be detected. A second possibility consists in replacing the inner slit by a hole with a small radius (this amounts to putting d1=0 in Eq. (2)). By doing so one obtains the interference between a Bessel beam and the classical Airy pattern. The graphics on the bottom right of Fig. 2 show that, under such conditions, the structure of the interference pattern becomes much sharper, with a narrowing of the central spot. The curve at the upper right of Fig. 2 shows that the intensity at center is much more sensitive to variations of the wavefront curvature 1/R1 than the curve at the upper left. One sees that, for the curve at the upper left, a change of the curvature of the incident beam by 0.07 m-1 is needed to move from maximum to minimum intensity; for the curve at the upper right only a change of 0.002 m-1 is needed.
3. Experimental setup
In this section, we first describe the experimental setup used to characterize the curvature of a reflecting surface by means of Bessel beam interferometry. We follow by recalling the principles of operation of the liquid mirror that is used to provide the reflecting surface. A liquid mirror has the interesting property that its radius of curvature can be adjusted to a high accuracy in a completely reversible way, making it well suited for the needs of our experiment.
3.1 Bessel beam interferometry
The setup we have used to characterize the surface of a liquid mirror based on Bessel beam interferometry is illustrated in Fig. 3. The Gaussian beam from a He-Ne laser (λ=632.8 nm) is relayed by a set of mirrors and a lens up to the liquid mirror. An iris is used as a spatial filter and acts as a point source to provide the beam that illuminates the liquid mirror. The beam is expanded by a converging lens (L0) of focal length f0=25.4 mm. A tilted flat mirror deflects the beam in the vertical direction, towards the liquid mirror. In the plane of the liquid mirror, the beam has a spot size (FWHM) of 20 cm. A beam splitter deflects part of the beam reflected by the liquid mirror towards a mask with two thin concentric annular slits. The diameters of the slits are d1=3.13 mm and d2=4.36 mm. The mask is placed at the focus of lens LT with a focal length f=88.4 cm and a diameter of 2.54 cm (the same parameters were used for the computations shown in Fig. 2). The width of each annular slit was Δd=16 µm; this was found to be thin enough to produce Bessel beams of excellent quality. A CCD camera is placed at the other focal plane of the lens in order to observe the interference pattern due to the superposition of the two Bessel beams. The value of R1, the radius of curvature of the beam incident on the mask, can easily be related to the radius of curvature Rm of the liquid mirror through ABCD ray matrices .
3.2 Liquid mirror
Liquid mirrors take advantage of the fact that the surface of a liquid is very smooth and takes the shape of an equipotential surface so that one readily gets an optical quality surface [21, 22]. For example, the fact that the surface of a liquid rotating in a gravitational field takes the shape of a parabola has been used to make inexpensive mirrors having excellent surface quality. The technology is young but its performance is well documented by laboratory tests [23–25] as well as by astronomical and lidar observations [26–28]. The basic liquid mirror setup can be found in ref. . It basically consists of a thin layer of mercury resting on the surface of a rotating solid container having an approximately parabolical shape. The radius of curvature Rm of the liquid surface is given by Rm=g/ω2, where g is the acceleration of gravity and ω the angular velocity of the container put on a turntable. In practice, to dampen environmental perturbations, it is necessary to use a thin layer of mercury (of the order of 1 millimeter). The thin layer obviously limits the range of variation of the radius of curvature obtainable for a given container shape. As a consequence, given the particular container that we used, we were limited to radii of curvature ranging from 2.00 m to 2.02 m. The diameter of the liquid mirror was 1.4 m. One key feature of liquid mirrors is that their surface can be deformed in a totally reversible way, showing no hysteresis (as opposed to deformable solids).
The surface of a liquid mirror is subject to three main sources of defects: vibrations that cause surface ripples, levelling errors of the mirror that induce coma-like large-scale perturbations, fluctuations in the speed of rotation that vary the radius of curvature. Alignment errors are not a serious issue; the mirror can easily be levelled to the required fraction of an arcsecond. Surface ripples are minimized through the damping effects of thin layers. In our setup fluctuations of the radius of curvature are minimized by the use of both a frictionless air bearing and a synchronous electromotor driven by a power supply stabilized by a crystal oscillator. Optical shop tests have shown that the surface of a well-tuned liquid mirror follows a paraboloid of revolution to an accuracy of λ/20 or better [24, 25].
4. Experimental results
In our experiment, we have observed the variation of intensity at the center (r=0) of the superposition of two Bessel beams for values of liquid mirror radius of curvature Rm ranging from 1,9996 m to 2,0200 m (see Fig. 4). The results clearly show that the experimental curve agrees very well with theory.
Interference patterns produced by the superposition of two Bessel beams are shown in Fig. 5, for different values of the radius of curvature of the liquid mirror. We clearly see that the intensity at center is maximum for Rm=1.9996 m and Rm=2.0167 m, and minimum for Rm=2.0100 m. The central spot becomes asymmetric when Rm is larger than 2.0100 m. This asymmetry is due to a precession of the liquid mirror. The precession generates real-time phase fluctuations that limit the precision of Bessel beam interferometry for the measurement of mirror curvature. The precession was characterized by using the well-defined peak of a Bessel beam produced with a mask having a single thin annular slit (d1=3.13 mm). The same setup as shown in Fig. 3 was used; the speed of rotation of the liquid mirror was stable and the lens LT was removed to avoid misalignment effects. We have observed that, during a complete revolution of the liquid mirror, the center of curvature of the liquid mirror follows a cardioid-like path having P - V amplitudes of 45 µm and 40 µm along two orthogonal axes (Fig. 6)). This corresponds to precession angles of 22.5 µrad and 20 µrad (precision of ±1 µrad). We had found a similar behavior in previous optical-shop tests obtained with a scatter-plate interferometer, albeit with a smaller P - V amplitude [23–25]. The larger amplitude in this set of measurements was due to the poorer dynamical balancing of the mirror used in the present work.
The precession of the axis of rotation of the mirror is due to the fact that the air-lubricated bearing on which the container rests has a coning error. In our setup, the coning error is worsened by an off-centered load. This precession produces a wobbling mirror. As a consequence, the wavefront sensed by the annular slit of the mask wobbles with an amplitude of the order of 500 nm and a period equal to the period of rotation of the turntable. This periodic perturbation was detected by using a mask with one annular slit (diameter d2=20.2 mm and thickness Δd=20 µm) and one hole at center (dh=1.97 mm). We observed interference patterns with clear maximum or minimum values at center, as a function of the phase of the motion of the liquid mirror during one revolution (see Fig. 7). The contribution of the Airy pattern coming from the hole at the center of the mask is insensitive to variations of the wavefront, which are of the order of 0.1 nm peak-to-valley at the position of the hole. This is totally negligible in our case for the wavefront variation detected by the annular slit is three orders of magnitude larger. The interference pattern observed at 0 degree shows a clear maximum at the center; that pattern corresponds to the theoretical intensity distribution numbered iv in Fig. 2(b). Due to the displacement of the cardioid, the intensity distribution somehow loses its symmetry, as seen in the interference pattern at 90 degrees. However, an intensity minimum appears when the container has completed half a revolution (i.e., at 180 degrees). This observation is in agreement with theory, considering the wavefront distorsion caused by the precession of the mirror.
We have shown that Bessel beam interferometry can be used for the measurement of the curvature of a wavefront (a precision of the order of the λ/20 was achieved) and of its tilt (within 1µrad). The use of a mask with a single thin annular slit and a centered pinhole has been shown to maximize the sensitivity of the measurements. A mask having a single slit can be used for tracking the center of curvature of a reflecting surface. The method could also be used in transmission to characterize lenses and lens systems. Bessel beam interferometry is simple to implement and inexpensive. It also bears the advantage of being self-referenced, e.g., it does not require the presence of a stabilized reference arm as in many interferometric setups. This feature makes its alignment easy and relaxes the requirements on mechanical vibrations. Bessel beam interferometry is very well suited for the real-time monitoring of small changes in the parameters of an optical wavefront.
If Bessel beam interferometry has already produced measurements with a precision of λ/20, (as inferred from Fig. 4), its accuracy could be refined further. Improvements in the experimental setup (thinner slits, higher numerical aperture of the optics) and numerical analysis of the interference patterns could lead to a much higher accuracy in wavefront characterization. The accuracy of Bessel beam interferometry could also be improved by using phase-shifting or multi-wavelength techniques. It would also be possible to use Bessel beam interferometry for absolute measurements of wavefront curvature by using sequentially more sensitive masks.
The method of Bessel beam interferometry described in this paper assumes that the incident beam possesses azimuthal symmetry and that its wavefront is parabolic. Such conditions were prevailing during our experimental measurements. Inhomogeneities of the beam under analysis were averaged out by sampling it using annular slits. The method could be used with azimuthally symmetric beams having nonparabolic wavefronts, by sampling the beam at different radial positions with a number of annular slits. We are presently considering how the method could be used to characterize arbitrary wavefronts deviating from azimuthal symmetry; Bessel beam interferometry could then provide global information on the wavefront curvature by doing an angular averaging of the wavefront phase.
This work was supported by grants from Natural Sciences and Engineering Research Council (NSERC), Formation de Chercheurs et Aide à la Recherche (FCAR), and Canadian Institute for Photonic Innovations (CIPI). We thank the reviewers for their instructive comments.
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