1D and D laser line scan generation using a fibre optic resonant scanner David A. Roberts and Richard R. A. Syms Department of Electronic and Electrical Engineering, Imperial College of Science Technology And Medicine, Exhibition Road, London, SW7 BT, U.K. ABSTRACT Imaging the transverse vibrations of a resonating optical fibre cantilever is shown to be an effective method for generating 1D and D laser line scans. The resonant bending modes were excited by mounting a short length of 0.633µm single mode optical fibre on to a commercially available piezo-electric transducer. For an applied sinusoidal driving voltage of 0V pk-pk, the maximum displacement at the free end of an 11mm long fibre cantilever exceeded +/-1.0mm at resonance. The motion at the free end of the fibre was imaged and magnified using a mm diameter graded index lens and the resulting flying spot line scan was found to subtend a maximum arc of +/-19. Two distinct signal detection schemes were implemented and evaluated using standard bar code targets printed on plain paper. D Lissajous scans were demonstrated by exciting the orthogonal transverse bending modes of optical fibres with non-circular cross sectional areas. Both D-shaped fibres and circular fibres modified by reactive ion etching were investigated. The principles underlying the design of this Fibre Optic Resonant Scanner (FORS) are directly applicable to the design of an integrated optoelectronic micro-electromechanical scanning device. Keywords: Optical Scanning, Fibre Optics, Bar code readers, Micro-electro-mechanical systems. 1. INTRODUCTION Optical scanning has become an increasingly important technology for collecting data from a wide range of sources. By far the most common application is 1D bar code reading but there are many more applications either in current use or under development. For example, other applications include D bar code scanning 1, the internal inspection of pipelines and the ophthalmic imaging of the human retina 3. Bar code scanners can be broadly grouped into one of three categories; hand held light pens, light weight hand held or surface mounted devices or the large multi-line flying spot devices which are often found in supermarkets. Scanning is generally achieved by one of five methods. These are a) by providing multiple light sources/detectors, b) moving a single light source, c) moving the imaging lens, d) beam steering with either a galvanometer mirror, a spinning polygon mirror or a spinning holygon or e) moving the object to be scanned. For the purposes of the following discussion categories a and b will be described collectively as pre-objective scanners whilst categories d and e will be referred to as post-objective scanners. Advances in micro-engineered mechanical systems (MEMS) technology has stimulated significant interest in the development of MEMS based optical scanners. Kiang et al. 5 have reported an electro-statically driven micro-engineered torsion mirror scanner whilst Butler et al. 6 have demonstrated a similar device but driven with a thermal actuator. More recently Syms 7 has demonstrated a novel self assembled mirror device capable of deflections up to 16. All three of these examples used a postobjective torsion mirror to generate a 1D line scan. By using two surface micro-machined mirrors fabricated on separate substrates Hagelin et al. 8 were able to demonstrate a D raster scanning system. Francis et al. 9 have also demonstrated D scans by combining a semiconductor fan laser array with a Si micro-machined torsional mirror. Whilst providing the benefits of reduced size and the potential for mass production neither of these D scanning techniques are ideally suited for further levels of integration. A more compact D scanning solution has been reported by Ikeda et al. 10 and Goto. Here D scanning was achieved by exciting the bending and torsional resonant modes of the scanning mirror. By adopting a modular approach Ikeda et al. were able to design and demonstrate a device with the laser source, the D scanning mirror and the photodetectors all assembled within a self aligned and compact package. In this paper a hybrid pre-objective Fibre Optic Resonant Scanning (FORS) device is demonstrated and shown to be a viable and potentially simpler alternative to the post-objective micro-engineered scanners previously reported. The principles
underlying the design and operation of the hybrid FORS device are directly applicable and compatible with the requirements of an integrated optoelectronic MEMS device. Both 1D and D laser line scans have been generated by exciting the resonant bending modes of a cantilever beam made from single mode optical fibre. A magnified image of the guided mode at the moving end of the fibre resulted in a line scan that subtended a maximum arc of +/-19. This displacement was large enough to allow standard bar codes to be read from distances as short as 50mm. The efficient detection of backscattered light from a distant target is another important aspect of integrated scanner design. Whilst optoelectronic integration is beneficial with respect to lowering manufacturing costs and reducing overall weight, it may result in lower signal to noise ratios because of the smaller numerical aperture of the detection system. A FORS device can be configured for efficient dual numerical aperture confocal detection 11 or for more conventional collection by means of a parabolic mirror. Dual numerical aperture (N.A.) detection is possible because the N.A. for the forward propagating beam is determined by the optical fibre whereas the N.A. for the backscattered signal is determined by the imaging lens. If the N.A.of the imaging lens is larger than that of the optical fibre, the backscattered signal propagating in the cladding modes of the fibre will be greater than that coupled into the backwards propagating guided mode. Both methods of detection have been implemented and compared.. FIBRE OPTIC RESONANT SCANNERS (FORS) The key components of a MEMS pre-objective scanning device are shown in figure 1. Light from a semiconductor laser diode is coupled into a single mode optical waveguide. The waveguide is supported by a cantilever beam which is free to resonate in one or two dimensions. A line scan of useful proportions is generated by using a lens to magnify the motion at the free end of the waveguide. The resonant modes of the cantilever are excited by a transducer which can either be formed on the surface of the device or attached to the base 1. A photodetector is necessary if the application requires detection of the light backscattered from the target. The important design criteria include the resonant frequency of the cantilever beam, the amplitude of the motion at its free end, the drive voltage of the transducer, the maximum magnification of the lens and the sensitivity of the detection system. In order to define the minimum performance requirements of an integrated FORS device we have constructed and evaluated the performance of a number of hybrid FORS devices. These were made using single mode optical fibre and inexpensive piezo-electric transducers..1 Hybrid FORS Devices Schematic diagrams showing two variations of the hybrid FORS device are shown in figure. The device shown in figure a was designed to evaluate signal detection using a parabolic mirror whereas the device shown in figure b was designed for dual numerical aperture confocal detection. In both cases a commercially available Kingstate KPE-165 piezo-electric transducer was used to excite the bending modes of the fibre optic cantilever. The piezo-electric transducer was removed from its plastic package and bonded to a 15mm diameter nitrile O ring using silicone rubber compound. More rigid mounting techniques were found to reduce the maximum displacement of the transducer at a given drive voltage. The transducer was rated for a maximum drive voltage of 0V pk-pk. Figure a shows the optical fibre supported by a 1mm 1mm 0.5mm piece of silicon wafer. A support with a small cross sectional area was preferable in order to minimize the damping of the transducer. However, to demonstrate dual N.A. detection it was necessary to replace the silicon support with a silicon photodetector. For ease of assembly a Siemens BPW3S surface mount photodetector was used throughout these experiments. This device had a 7mm photosensitive area and a total cross sectional area of approximately 17mm. In this configuration the maximum displacement at the tip of the fibre cantilever was found to be approximately half of that achieved when using the smaller silicon support. A short length of optical fibre, with its plastic jacket removed, was mounted to either the silicon support or the photodetector using index matching epoxy. This mode stripping arrangement facilitated leakage of the light propagating in the cladding modes and detection of the backscattered signal. Corning FS-SN-3 single mode silica fibre was used for all 1D scanning experiments. This fibre has a specified mode diameter of µm at 630nm, an N.A. of 0.1 and a cladding diameter of 15µm. Light from a collimated 650nm diode laser was coupled into the fibre using a microscope objective. The maximum output power at the free end of the fibre was 150µW. The resonant frequencies of the bending modes of the fibre cantilever are a function of its length, shape and material properties. A detailed analysis of the frequency response is deferred until section.. The amplitude of the motion at the free end of the fibre was measured as a function of transducer drive voltage and frequency by using a calibrated optical microscope. The contrast of the moving fibre tip was enhanced by end fire coupling white light into the fixed end of the fibre. The maximum scan length at a given image distance is determined by the amplitude of the fibre displacement and the magnification of the lens. However, the maximum allowable magnification is restricted by the necessity of maintaining
sufficient resolving power in the image plane. For example, the minimum feature size in a standard Interleaved of 5 bar code is approximately 00µm. Since the guided mode diameter at the fibre tip is µm a magnification of 100 results in a flying spot with a beam waist that is comparable to the minimum feature size in the bar code. Clearly as the beam waist increases, the depth of modulation for the backscattered signal will decrease which makes decoding the information progressively more difficult. Off axis optical aberrations must also be considered when setting an upper limit for the magnification because the beam waist increases due to increasing aberration as the tip of the fibre moves away from the optical axis. The motion at the tip of the free end of the fibre was magnified using either a quarter pitch graded index (GRIN) lens or a molded plastic aspheric. The GRIN lens had a mm diameter, an N.A. of 0.37 and an effective focal length of.6mm. The aspheric lens had an outside diameter of 7.mm, a focal length of 3.3mm and an N.A. of 0.5. The maximum magnification used was approximately 80 at an image distance of 00mm. At this magnification a displacement of ±1.0mm at the free end of the optical fibre is expected to produce a scan length of ±80mm. Backscattered signals from standard bar codes printed on plain paper were amplified using a pre-amplifier with the BPW3S photodetectors connected in the transimpedance configuration. The signal was then further amplified and differentiated to produce trigger pulses for the digitizer circuit.. Calculation Of Resonant Frequencies The resonant frequencies ν, for a uniform cantilever beam of cross sectional area A and length L are given by equation 1. β EI = (1) πl ρa ν Where E, ρ and I are the Young s modulus, density and second moment of area respectively. In the following calculations E and ρ for silica were assumed to be 7.17 10 10 Nm - and.7 10 3 kgm -3 respectively 13. β is a constant which depends on the mode number and the boundary conditions at the ends of the beam. The values of β for the zeroth and first order modes of a beam with one end fixed and the other end free are 3.5 and. respectively. Analytical expressions for the second moment of area for simple geometrical shapes are given in standard books of tables 1. The second moment of area for circular and rectangular cross sections are given by equations and 3. πr I circular = () 3 bd I rec tan gular = (3) 1 where R is the radius of the circular fibre, b is the width of the rectangular beam and d is its thickness. The resonant frequencies for a 15µm diameter circular silica fibre cantilever were calculated for a range of lengths L and are plotted in figure 3. For cantilever lengths of between 5mm and 0mm the resonant frequency was between 3.6kHz and 5Hz. Experimental measurements of the resonant frequency for a range of cantilever lengths were found to be in excellent agreement with the theoretical values. An integrated FORS device will have dimensions that are considerably smaller than those of the hybrid device described above. The scaling laws for a system with a circular cross section are defined by equation and are such that the resonant frequency is proportional to the radius of the circular cross section R and inversely proportional to the square of the cantilever length L. β E R ν circular = () π ρ L For example, if the diameter of the beam is reduced by an order of magnitude to 1.5µm, the frequency response will remain in the range of 3.6kHz to 5Hz for cantilever lengths of between 1.6mm and 6.3mm. Clearly these are dimensions that could be implemented with moderately high packing density in wafer scale production. Another consideration relating to miniaturization is that the cross section of an integrated cantilever beam is more likely to be rectangular than circular. Equation 5 defines the relationship between the resonance frequencies for circular and rectangular cantilevers made from the same material and of the same length.
ν rec tan gular ν circular = 3 d R (5) It follows that if the thickness of the rectangular beam is equivalent to the diameter of the circular beam then the resonant frequency increases by a factor of / 3 which represents an increase of only 15%. The cantilever in an integrated device could be a composite structure comprising a silicon cantilever beam and a silica based waveguide. Equation 1 shows that the ratio of the Young s modulus to the density of material used to fabricate the integrated cantilever will also determine the resonant frequency. Young s modulus and the density of silicon are 1.31 10 11 Nm - and.3 10 3 kgm -3 respectively which means the ratio of E/ρ is greater for silicon than for silica. The resonant frequency of a silicon beam is therefore approximately 5% greater than that for geometrically identical beam made from silica. The preceding analysis shows that a hybrid FORS device is expected to have a resonant frequency which is similar to that of an integrated device. This means that the bandwidth of the analogue signal processing circuits required to detect and decode the scanned information will be similar for both the hybrid and integrated FORS devices. The hybrid FORS device is therefore a useful vehicle for exploring the system response of the proposed integrated device. One disadvantage of using a structure that is circularly symmetric is that the resonance frequencies for orthogonal bending modes are identical. Consequently the locus of the fibre tip at resonance in a hybrid FORS device was often observed to be elliptical. This problem was resolved by introducing a degree of asymmetry to separate the resonant frequencies of the orthogonal modes. Two different methods for achieving the required asymmetry have been investigated. The first method was to modify the cross sectional area of the fibre by reactive ion etching for hours in an Ar/O /CHF 3 plasma. The second was to use a D-shaped optical fibre. In an integrated system, ellipticity can be avoided by ensuring that the aspect ratio of the cross sectional area is not equal to one. In order to determine the resonant frequencies for a D-shaped cantilever beam, the second moment of area must be calculated for motion parallel and perpendicular to the plane defined by the flat surface of the D-section. The second moments of area for motion parallel I ll and perpendicular I to the fibre flat are defined by equations 6a and 6b. I ll = y da (6a) I = x da (6b) For a beam with a semi-circular cross section I ll is defined by equation 7a and is equivalent to the second moment of area for the circular section divided by two. Calculation of the second moment of area for motion in the plane perpendicular to the flat surface is complicated by the need to integrate about the centroid of the section. For a semi-circular section the centroid is displaced from the plane of the flat by a distance of 0. times the radius of section 1. I for a semi-circular section is defined by equation 7b. πr I ll = (7a) 8 I = 0.1098R (7b) The cross sectional area of the D-fibre used in the D scanning experiments is shown schematically in the inset of figure. Here the flat surface of the D is offset from the core of the fibre by approximately 5µm. It follows that a semi-circular cross section does not accurately represent the shape of a D-shaped optical fibre. Let us define a shape factor α which is given by the ratio of the distance from the core to the flat surface r, divided by the radius of the semicircular section, R. Equations 7a and 7b together with equation then define the limiting values of I ll and I for optical fibres with shape factors ranging from 0 (semi-circular) to 1 (circular). The second moments of area for D-shaped cross sections are given by equations 8a and 8b 1 where the angle θ is defined in the inset of figure. The cross sectional area of the D-section is given by equation 9. I ll = R 1 3 (3θ 3sinθ cosθ sin θ cosθ ) (8a) I R = θ sinθ cosθ + sin 3 6 16sin θ θ cosθ 9( θ sinθ cosθ ) (8b)
( θ sinθ cosθ ) A = R (9) Figure shows graphs of k/r versus shape factor α, where k is the square root of (I/A). If both of the orthogonal bending modes are excited at once it is possible to generate a D Lissajous scan. The shape and stability of such a scan is determined by the frequency ratio of the orthogonal modes which can be determined by specifying fibre with the appropriate shape factor. For D-fibre with a shape factor of 0.08, the ratio of the resonance frequencies for parallel and perpendicular bending modes is expected to be 1.78:1 3.1 Resonant Frequencies 3. RESULTS The high correlation between the theoretical and measured values of the resonant frequencies for the circular fibre has already been shown in figure 3. The results of a similar analysis for the D-shaped fibre are shown in figure 5. Here the experimental values for the resonance frequency were found to be slightly higher than predicted. The reason for this small discrepancy may be because the surface tension forces that act on the D-fibre when it is drawn, cause the flat surface to bow slightly and for the corners of the fibre to become more rounded. 3. 1D Scanning Figure 6 shows the measured displacement at the free end of the fibre as a function of drive frequency and drive voltage for a hybrid FORS device. The fibre cantilever was mounted on a silicon support and was 10.7mm long. The resonant frequency of the fundamental mode was 863Hz and the corresponding Q factor was 150. The maximum displacement of the free end of the fibre was found to be in excess of ±1.1mm for drive voltage of only 0V pk-pk. A near perfect 1D scan was ensured by etching the fibre for hours in an Ar/O /CHF 3 plasma. With the total displacement in excess of mm the minor axis of the ellipse, defined by the locus of the fibre tip, was only 6µm long. This corresponded to an aspect ratio in excess of 300:1. The maximum drive voltage required to scan the full mm entrance pupil defined by the GRIN lens was found to be 16V pk-pk. At this maximum drive voltage the line scan subtended an arc of +/-19 which at an image distance of 00mm resulted in a linear scan line approximately 10mm long. The lower of the two oscilloscope traces shown in figure 7 is the derivative of the signal received using a hybrid FORS device with a 5mm diameter parabolic mirror for signal collection and an image distance of 150mm. The target was a test pattern of 1.75mm wide bars and spaces printed on plain white paper. The upper trace shows the output from the digitizer circuit and is a perfect facsimile of the original test pattern. With scan lengths in excess of 35mm it was possible to read bar codes defined using industry standard symbologies such as Interleaved of 5 and Code 39. These are typically up to 35mm and 70mm long respectively. Figure 8 shows a standard Interleaved of 5 test pattern and the digitized output from the hybrid FORS device. The image distance was again 150mm and the agreement between the original pattern and the digitized output was excellent. Whilst the use of a parabolic mirror to focus the backscattered signal on to the photodetector increases the overall size of the scanner it does permit operation with considerable depth of field. Stable oscilloscope traces of the digitized backscattered signal were recorded without refocusing for image distances as small as 50mm. At an image distance of 00mm the trace was recognizable but less stable. Figure 9 shows the digitized signal for the centre portion of the bar code shown in figure 8 but this time sensed using cladding mode detection. The image distance for this scan was 50mm and the length of the scan was limited by the loading affect of the photodetector on the piezo-electric transducer. The backscattered light was collected using the f=3.3mm molded aspheric lens described above. This lens was preferred because it reduced the optical interference encountered when using a GRIN lens. Whilst using the GRIN lens it was found that a scanning etalon was formed between the moving end of the fibre and the front surface of the GRIN lens. For long image distances the magnitude of optical interference associated with the scanning etalon was comparable to the magnitude of the backscattered signal. The larger entrance pupil and the greater working distance of the aspheric lens resulted in a larger signal to noise ratio and stable decoding as illustrated in figure 9. 3.3 D Scanning D scanning was successfully demonstrated by exciting the orthogonal bending modes of a 15µm diameter D-fibre with a shape factor of 0.08. Figure 10 shows the motion at the tip of an 11.mm long D-fibre cantilever resonating at 0Hz (zeroth
order perpendicular mode) and at 693Hz (zeroth order parallel mode). The video caliper lines define a 300µm 300µm square area. The ratio of the frequencies for the parallel and perpendicular modes was 1.7 which was in good agreement with the value of 1.78 predicted above. Figure 11a shows the D scan achieved by driving the piezo-electric transducer with a signal which comprised frequency components corresponding to the zeroth order perpendicular and parallel resonant bending modes. Figure 11b shows a similar scan but with the driving frequency for the perpendicular mode reduced to 396Hz. The frequency ratio was thereby increased to 1.75:1 which allowed the Lissajous nature of the scanning motion to be more readily observed.. CONCLUSIONS Imaging the transverse vibrations of a resonating optical fibre cantilever is an effective method for generating 1D and D laser line scans. The general principles applied to the design of the hybrid FORS device are applicable to a fully integrated MEMS scanning device. Efficient detection of the backscattered signal is readily achieved using a simple parabolic mirror arrangement which also provides for good depth of field. Cladding mode detection has been shown to be a viable alternative for applications where the maximum size of the scanning device is constrained. Exciting the orthogonal bending modes of non-circular optical waveguides represents a simple method for generating D line scans which could be used to read D bar codes, project simple D bit map images or for inspection scanning systems. 5. ACKNOWLEDGEMENTS The authors gratefully acknowledge Dr. A Holmes, Dr. K. Leaver and Dr. E. Yeatman for many helpful discussions. Funding for this project was provided by the U.K. EPSRC. 6. REFERENCES 1. N. Normand and C. Viard-Gaudin, A Two Dimensional Bar Code Reader, Proceedings of the 1 th IAPR International Conference On Pattern Recognition, Conference C Signal Processing, 3, pp.01-03., 199. H. Goto, Si Micromachined D Optical Scanning Mirror And Its Application To Scanning Sensors, Optical MEMS and Their Applications, IEEE/LEOS Summer Topical Meeting, pp. 17-18, 1996. 3. A.G.H. Podoleanu, and D.A. Jackson, Combined Optical Interference Tomograph And Scanning Laser Ophthalmoscsope, Electron. Lett., 3, pp. 1088-1090, 1998.. G.F. Marshall, Optical Scanning, Marcel Dekker Inc., New York, 1991. 5. M.-H. Kiang, O. Solgaard, R.S. Muller and K.Y. Lau, Micromachined Polysilicon Microscanners For Barcode Readers, IEEE Photonics Technology Letters, 8, No. 1 December 1996. 6. J.T. Butler, V.M.Bright,and J.R.Reid, Scanning And Rotating Micromirrors Using Thermal Actuators, Proc. SPIE, 3131, pp.13-1, 1997. 7. R.R.A. Syms, Operation Of Surface Tension Self Assembled 3D Micro-opto-mechanical Torsion Mirror Scanner, Electron. Lett., 35, pp. 1157-1158, 1999. 8. P.Hagelin, K.Cornett and O. Solgaard, Micro-machined Mirrors In A Raster Scanning Display System, Optical MEMS, IEEE/LEOS Summer Topical Meeting, pp. 11/109-10, 1998. 9. D.A. Francis, M.-H. Kiang, O. Solgaard, K.Y. Lau, R.S. Muller and C.J. Chang-Hasnain, Compact D Laser Beam Scanner With Fan Laser Array And Si Micro-machined Microscanner, Electron. Lett., 33, pp. 113-115, 1997. 10. M. Ikeda, H. Totani, A. Akiba, H. Goto, M. Matsumoto and T. Yada, PZT Thin-film Actuator Driven Micro Optical Scanning Sensor By 3D Integration Of Optical And Mechanical Devices, 1 th IEEE Conference On Micro-Electro- Mechanical Systems, pp.35-0, 1999. 11. D.A. Roberts, R.R.A. Syms, A.S. Holmes and E.Y. Yeatman, Dual Numerical Aperture Confocal Operation Of Moving Fibre Bar Code Reader, Electron. Lett., 35, pp. 1656-1658, 1999. 1. K.Yamada and T. Kuriyama, A Novel Asymmetric Silicon Micro-Mirror For Optical Beam Scanning Display, 11 th IEEE Conference On Micro-Electro-Mechanical Systems, pp. 110-115, 1998. 13. K.E. Petersen, Micromechanical Light Modulator Array Fabricated On Silicon, Applied Physics Letters, 31, No. 8, pp. 51-53, 1977. 1. Young, W.C., Roark s Formulae For Stress And Strain, International Edition For Stress And Strain, McGraw-Hill, pp. 6-69, 1989.
GRIN Lens Waveguide Photodetector Silicon Substrate Laser Cantilever Figure 1: Schematic diagram showing the proposed integrated FORS device. Parabolic Mirror Back Scattered Signal Optical Fibre Photodetector Silicon Support Piezoelectric Transducer O Ring GRIN Lens a) b) Figure : Schematic diagrams of hybrid FORS devices: a) large numerical aperture detection, b) cladding mode detection.
000 3500 Theory Experiment Resonant Frequency (Hz) 3000 500 000 1500 1000 500 Zeroth Order Mode First Order 0 0 5 10 15 0 5 Length (mm) Figure 3: Resonant frequency vs. length for a fibre cantilever with a 15µm diameter circular cross section. y 0.6 θ0.55 0.5 R r x Parallel Mode 0.5 k/r 0. y 0.35 θ 0.3 0.5 Perpendicular Mode R r x 0. 0 0.1 0. 0.3 0. 0.5 0.6 0.7 0.8 0.9 1 Shape Factor (r/r) Figure : Square root of I/A, normalized with respect to R, for a D-shaped optical fibre as a function of the shape factor.
000.0 1750.0 Resonant Frequency (Hz) 1500.0 150.0 1000.0 750.0 500.0 P erpen dicula r Mo de P aralle l Mod e 50.0 0.0 0 5 10 15 0 5 Length (mm) Figure 5: Resonant frequency vs. length for a 15µm diameter D shaped optical fibre with a shape factor of 0.08. 100 1000 0V 16V 1V 8V Amplitude (um) 800 600 00 00 0 80 85 850 855 860 865 870 875 880 885 890 Frequency (Hz) Figure 6: Amplitude of the displacement at the tip of the fibre vs. drive frequency and voltage for a 10.7mm long etched fibre mounted on a silicon support.
Figure 7: Digitized and derivative backscattered signals from a test target with 1.75mm wide bars and spaces. Figure 8: Digitized output from hybrid FORS device and original Interleaved of 5 bar code target. Figure 9: Digitized output from hybrid FORS device using cladding mode detection.
Figure 10: Optical micrographs showing the motion at the tip of an 11.mm long D-shaped optical fibre resonating at 0Hz (zeroth order perpendicular mode) and at 693Hz (zeroth order parallel mode). a) b) Figure 11: a) The motion at the tip of a D-shaped optical fibre excited by applying a drive signal to the piezo-electric transducer which comprised sinusoidal frequency components corresponding to the zeroth order perpendicular (0Hz) and parallel (693Hz) resonant modes. b) Lissajous figure achieved with the frequency ratio adjusted to 1.75.