Time-of-Flight 3D Single Fibre Endoscopy

. We measure the modal dispersion occurring in a single multimode ﬁbre and account for this using a digital micromirror device to form a raster scanning spot in the far ﬁeld of the distal facet of the ﬁbre. We perform this with a q-switched 700 ps pulsed laser at a 532 nm wavelength. The raster scanning allows us to spatially interrogate the reﬂectivity of a scene while the time of ﬂight of the pulse gives distance information allowing for the generation of a three-dimensional image.


Introduction
Advances in endoscopic technology has enabled imaging in formerly difficult to access areas.Among these, new developments in ultra-thin single fibre imaging has broadened the horizons of endoscopy to imaging in areas which exhibit exceptional sensitivity to probe size [1].Whereas traditional endoscopes may use fibre bundles, where each core in a common cladding works as single pixel, a single multimode fibre (MMF) has the advantage of carrying one to two orders of magnitude more information on the fibre modes for the same probe size [2].Multimode fibres, however, do not act as a simple imaging conduit, rather a light field which is launched into a MMF can be expressed as a superposition of the orthogonal fibre modes.In a typical step-index multimode optical fibre, with a 50um core and a numerical aperture (NA) of 0.22, there are in the order of 1000 modes for visible wavelengths.As these modes propagate down the fibre the input field is transformed due to their differing phase velocities.The resulting output field, therefore, rarely resembles the input, rather a speckle pattern is formed beyond the distal fibre facet.This transformation, however, can be regarded as linear and, after characterising the fibre, the input field can be structured such that a desired output field is achieved.For imaging a common output field is chosen such that the modes constructively interfere at a single point.A scene can be imaged by changing the input field with a spatial light modulator, such as a digital micromirror device (DMD), so that this output point is raster scanned across the entire scene.An image can then be formed by collecting either reflected light, fluorescence emission, non-linear scatter-ing, or a similarly stimulated signal [1,3,4].In this work we extend single fibre imaging endeavours to include 3D time-of-flight measurements by using a picosecond pulsed source.

Imaging in 3D
Calibration is performed by scanning 1961 input plane waves oversampling the fibre modes by almost a factor of two.This is done by the complex addition of a varied grating period, the mode we are measuring, to a static field, the reference mode.The argument of the resulting field is then binarised and placed on a DMD.The resulting first diffraction order is coupled to the fibre which sits in a plane conjugate to that of the DMD.At the distal end of the fibre, images are taken in the far field of the fibre facet as the tested plane wave is phase-stepped through static internal reference.This gives the optimal phase and amplitude for each input plane wave at each pixel in the field of view of the fibre.An internal reference is used to manage the short coherence length of the picosecond pulsed laser, yet this does result in dark spots in the image where singularities existed in the speckled reference field.
For imaging, input DMD masks are calculated to form a single spot in the far field of the fibre's distal facet.The spot is then scanned by changing the input field to interrogate a scene.Reflected light is be collected through a secondary fibre running concurrently with the excitation fibre.These are both held in a custom-made 20-gauge needle as shown in Figure 1(a).In this work, a 50 µm core, 0.22 NA, step index fibre is used as the emission fibre, and a 400 µm core, 0.39 NA, step index fibre is used for collection of reflected light.
, 09022 (2023) Depth information is ascertained by imaging with a 700 ps pulsed, q-switched, 532 nm wavelength laser at a repetition rate of 21 kHz.As the maximum refresh rate of the DMD is 22 kHz, only a single pulse is used for each hologram and, as such, each output spot.A reference photodiode on the excitation arm of the setup is used as a trigger to start recording data from an AC-coupled avalanche photodiode at the end of the collection fibre which is sampled at 2.5 GHz.These data are then interpolated by a factor of 32 for each returning pulse and the interpolated data are temporally localised and converted to a distance to the reflected object.To account for the inverse-square reduction in light intensity, the peak height is scaled by d 2 , where d is the distance to the object and this value is taken as the object's reflectivity for the probed spot.To remove extraneous noise the imaging is temporally gated in post processing limiting the signal only to the distances where objects of interest are observed.
Imaging is performed at 5.0 Hz for a circular field of view with a diameter of 75 scanned spots.The spatial resolution was approximately 1.5 times the diffraction limit for a 50 µm aperture and scaled with distance linearly as expected for a constant angular resolution.Depth accuracy was estimated to within ±5 mm and, as it was limited by pulse duration, was indirectly dependent on distance through the dropping signal to noise ratio at greater distances.An example frame capture is shown in Figure 1(b).
No temporal shaping of the pulse is performed with this system.We calculate a minimum requirement for the coherence length, l co , as, where NA is the fibre numerical aperture, L is the fibre length at 500 mm, and n core is the core refractive index.
The inequality gives a lower bound of 13 mm while we estimate the coherence length of our laser to be of the same order of magnitude of the pulse length at ∼200 mm.We therefore satisfy the inequality.This, however, gives an extremely conservative estimate for the required coherence length, as imaging in the far field spatially separates mode groups meaning that for high contrast scanning spots, only modes with similar propagation velocities must interfere with each other.As such, we expect that, while this method is near a spatial resolution limit, it is not as constrained in axial resolution.Ongoing developments in this work are exploring the effects of bending on image reconstruction without the need for measuring a new transmission matrix [6].

Figure 1 .
Figure 1.Needle endoscope tip and recording snap shot.(a) Needle endoscope tip.Emission fibre (small) is highlighted in blue and collection (large) in yellow.(b) Frame capture from a video recorded with the endoscope at a distance of 1-2.5 m.Adapted from [5].