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We present a technique for mapping the complete 3D spatial intensity profile of a laser beam from its fluorescence in an atomic vapour. We propagate shaped light through a rubidium vapour cell and record the resonant scattering from the side. From a single measurement we obtain a camera limited resolution of 200 × 200 transverse points and 659 longitudinal points. In constrast to invasive methods in which the camera is placed in the beam path, our method is capable of measuring patterns formed by counterpropagating laser beams. It has high resolution in all 3 dimensions, is fast and can be completely automated. The technique has applications in areas which require complex beam shapes, such as optical tweezers, atom trapping and pattern formation.
© 2013 Optical Society of America
1. Introduction
Laser beams with increasingly intricate complex profiles have become interesting for a range of applications. Particles from the micron size range down to single atoms can be trapped by the dipole forces produced from light beams [1, 2]. These forces have been exploited in optical tweezers [3], allowing micro manipulation of beads and biological matter. Atoms can be trapped by the same forces and have the further advantage that their strong resonances allow tuning of the force as a function of the detuning.
With the advent of spatial light modulators (SLMs) it has become possible to generate a wide variety of beam profiles, expanding the possibilities for trapping and guiding. Complex 3D beam shapes have been proposed [4–6] and implemented in atom trapping [7,8] and optical tweezers [9–12]. In order to verify the accuracy of the beam generation however, traditional methods require stepping a camera in the beam path followed by reconstruction [13]. This method suffers from several drawbacks: The beam axis distance can be difficult to measure accurately, the process is manual and slow, and beam structures created by interference of counter-propagating laser beams cannot be imaged. Here we present an alternative method based on tomographic beam reconstruction from fluorescence in a thermal gas.
Laser induced fluorescence techniques have been used in a different context in liquids and dyes [14, 15] e.g. for flow visualisation and material probing and have been shown to produce 3D images [16]. In contrast to imaging in liquids, we employ a dilute thermal gas which provides a very homogeneous medium, allowing direct measurement of the 3D laser profile with negligible distortions arising from interaction with the medium.
2. Summary of technique
Our technique is based on the tomographic reconstruction of images taken via fluorescence imaging in a thermal rubidium vapour. The light that is scattered from an incident laser beam in a fluorescent medium depends on its intensity. In the case of an atomic gas, the two-level photon scattering (fluorescence) rate is [17];
where Γ (2π × 6.06 MHz) is the inverse of the upper state lifetime, I is the laser intensity, Isat (1.6 mw/cm2) is the atom specific saturation intensity and Δ is the laser detuning. As we are using a laser beam well below saturation, the scattering rate becomes directly proportional to the laser intensity. The rubidium vapour is homogeneous and optically dilute such that even at low light power the beam has very limited absorption. This results in an accurate mapping of the laser intensity onto the atomic fluorescence. The recorded fluorescence images then contain the projection of the light profile onto e.g. the yz plane (see Fig. 1(a)) without any information on the x dependence, but the full beam profile can be recovered tomographically.
Fig. 1 a) Illustration of a laser beam and its projections on the xy, xz and yz planes. The beam shown is a superposition of two Laguerre Gaussian beams with l numbers 3 and −3. b) Cut of the same beam at z = 0 illustrating the Radon transform: the two profiles are the projections of the 2D pattern in the x and y planes.
Tomography is the method to reconstruct 2D information from two or more 1D projections. A single column of pixels within the camera image provides the projection of the fluorescence at a particular z-position along the x-direction (or in cylindrical coordinates at a particular angle θ). If we were to rotate the camera around the z axis, we would have access to more projections at more angles. From the combined data we can tomographically reconstruct the 2D xy plane at the chosen z-position. By performing this analysis on all columns in each image, and stacking the retrieved profiles along z we recover the full 3D fluorescence distribution. For practical reasons, instead of rotating the camera around the beam, we rotate the beam itself with a fixed camera position.
3. Tomography
The operation of projecting 2D data along an axis to produce a 1D profile (Fig. 1(b)) is called a Radon transform [18], while the operation to generate 2D data from the 1D profiles is an inverse Radon transform. The most common method to compute the inverse Radon transform is by back projection. This method comprises 4 simple steps: First, each 1D projection is filtered to correct for the oversampling of the central pixels. The ideal filter is simply a ramp filter in frequency space, but it may be optimised to enhance important features in an image. The filtered 1D projections are then converted to a 2D image by copying the value in each column to every row (i.e. ‘smearing’ the values from 1D to 2D). Each 2D image is rotated by the angle at which the projection was taken and finally all of these images are added together.
The reconstruction is performed in Matlab and runs extremely quickly. We test the method by taking 1D projections from a calculated ideal 2D profile, and reconstruct from this the 2D profile (Fig. 2(a) and (b)). Here, the reconstruction is formed in 150 ms from 30 projections at 200 × 200 (x,y) resolution. We have analysed the reconstruction process for different numbers of projections, shown in Fig. 2(c). Increasing the number of projections increases the reconstruction accuracy at the cost of a longer processing time which scales linearly with the number of projections (5ms per projection). A workable balance between speed and accuracy may be at around 30 projections; for the proof of principle experiment reported here, we favour accuracy at the expense of speed, using a large number (116) of projections.
Fig. 2 Sample Reconstruction. a) Calculated transverse profile of the beam shown in Fig. 1. b) Reconstruction of the same profile from 30 1D projections, performed in 0.15 s. c) Analysis of the reconstruction accuracy with the number of projections. The residuals are the absolute difference between the original profile and the reconstruction, summed over all pixels and divided by the total of all pixels in the original image.
4. Experimental results and visualisation
We have tested our beam reconstruction in an experimental set-up shown in Fig. 3(a), composed of a section for beam shaping and one for beam detection.
Fig. 3 a) Experimental setup with beam shaping and detection sections as detailled in the main text. Right: Fluorescence image of the LG superposition of l=3,−3. at 0° (b) and 90° (c). d) Sample reconstruction from 116 projections taken from the same dataset.
The external cavity diode laser is tuned to 780 nm, resonant with the Doppler broadened 85Rb F=2 to F’=3 transition for maximum fluorescence. The laser is linearly polarised which allows optimal operation of the SLM, however we note that the detection technique itself is polarisation insensitive. The mode is cleaned by a pinhole and then expanded. We refract the laser off an intensity modulated holographic grating [19] displayed on an SLM (Hamamatsu LCOS) to create arbitrary phase and intensity profiles. We select the first order diffracted beam with an aperture placed in the Fourier plane of the SLM. The shaped light beam is then collimated again and sent to the beam detection section. For convenience we rotate the beam by simply turning the pattern on the SLM, however a more general approach would be to use a Dove prism to rotate the beam, thereby allowing detection of arbitrary beams [20].
The beam is directed through a standard rubidium cell with 25 mm diameter and 71.8 mm length (Thorlabs GC25075-RB). The fluorescence emitted from the side of the cell is imaged onto a camera (Prosilica GC660) by a single lens (f = 25.4mm, 25 mm diameter) with a magnification of around 4 and an NA of 0.1. The lens is placed such that the centre of the beam is in focus and has a depth of field such that the outermost parts of the beam have a spread of less than 3 pixels.
The sample beam reported here is the superposition of two Laguerre-Gaussian (LG) modes. A single LG beam is parameterised by its azimuthal and radial numbers l and p respectively. For simplicity we restrict our pattern to have p = 0. A single LG beam has an intensity minimum in the centre, the size of which increases proportional to . A superposition of two beams, with different l numbers produces a ‘petal pattern’ with a number of petals equal to the difference in l numbers. We use a superposition of l = 3 and l = −3, resulting in the pattern shown in Fig. 2(a) with 6 ‘petals’. This pattern is rotated and a video is recorded by the camera. Sample frames, separated by 90 degrees, are shown in Fig. 3(b) and (c).
For the tomographic reconstruction it is necessary to know the beam rotation angle of each frame. This can be achieved either directly by rotating by a known amount from frame to frame, or by extracting the angles from the sinusoidal symmetry of the data. The result of the reconstruction for a single column of pixels is shown in Fig. 3(d). Reconstruction for every column results in a 3 dimensional matrix with values corresponding to the fluorescence of the atoms.
The full 3D intensity profile can be visualised with our Matlab program which renders iso-surfaces and plots these with a colour and transparency which is related to the intensity of the isosurface. An example plot with ten isosurfaces between 45 % and 80 % of the maximum intensity is shown in Fig. 4. The visualisation technique has also been expanded to create fly-by videos as well as images and video capable of being displayed in 3D on compatible displays. We note that in fact the 3D intensity profile can also be observed directly by looking at the scattered light through an IR viewer.
5. Discussion
The resolution of the 3D profiles is generally limted only by the camera. In order to reduce computation time, we have used only 200×200×659 pixels of the 494×494×659 camera resolution. The reconstruction accuracy is limited by the number of projections and the noise. The number of projections increases the accuracy of the reconstruction at the expense of longer measurements and calculation time but is already very high for only 30 projections as discussed in Section 3. Background noise can be reduced by recording a single background measurement at the start of the experiment, which is then subtracted from all further measurements. This is particularly important for improving the final signal to noise ratio, as the back projection method is very sensitive to noise. This noise reduction combined with the homogeneity of the gas results in an extremely accurate representation of the laser beam. We note that the homogeneity of a thermal gas compares favourably with that of flourescent dye, which could provide an alternative medium for similar 3D beam tomography. The fluorescence linewidth in our case is however much narrower, limitting the possible wavelenghts to those that correspond to suitable atomic transitions.
We finally consider high speed measurements or video framerates. Due to the 120 Hz framerate of our camera, a single measurement (at 30 projections) would take 250 ms, however the beam rotation speed using our SLM is capped at 60 Hz, resulting in 30 projections in 500 ms, with possibly slightly faster rotation achievable by rotating Dove prisms. Either method could allow taking 3D measurements at a rate of around 1 Hz, so that the method is limited by reconstruction time. A single 2D reconstruction from 30 projections takes 150 ms, limiting the z resolution to <10 for 1Hz video, or alternatively a lower transverse resolution either by restricting the number of projections or pixels used.
6. Conclusion
We have demonstrated a high resolution, simple, fast experimental method which maps the entire 3D structure of an arbitrary laser beam. As a proof of concept we have generated a superposition of co-propagating LG beams and reconstructed and plotted the results. The method benefits from a far higher resolution in the propagation direction than other invasive methods, without loss of resolution in the transverse dimension. The method is completely automated and capable of producing a single plane reconstruction in only 150 ms, limited by computation time.
References and links
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