1. Introduction

The idea of exploiting multimode fibres in imaging applications dates back to 1970s [1], promising means of high image information delivery through a hair-thin instrument. The first successful control of light transport through MMFs by means of digital holography, resulting in generating optical foci as well as other useful optical landscapes, has been achieved in 2011 [2,3]. To date, the power of such endoscopes has been utilised in various perspective applications, including minimally invasive in-vivo fluorescence endo-microscopy [4–7], scanner-less imaging alternatives [8,9], multiphoton and label-free non-linear chemical imaging approaches [10–13] and other methods of modern photonics including optical trapping [14,15].

The control of light transport through randomising MMFs has been adapted from the discipline of complex media photonics, where focussing through random (scattering) materials has been made possible with the use of holographic modulation of the input light fields [2,16–18]. Altering phases of $N$ input light modes, which despite their spatial scrambling can be delivered through a lossless randomising medium, one can intensify the optical power at a selected point by a factor of $\frac {\pi }{4}N$, when compared to the average level of a randomly distributed signal surrounding the optimised focus. Similarly, attempts of focussing light through MMFs have resulted in high-intensity peak contaminated by certain level of undesired speckle, extending across the whole field of view, while carrying a significant portion of the optical power. The power ratio (PR) describing the fraction of the optical power carried by the desired focus, with respect to the total amount of power transmitted through the fibre, will be used in this paper as a metric for the focussing fidelity. The PR can determined via the enhancement factor ($\eta$), a standard metric in complex photonics which describes the ratio between the intensity of the focus and the mean intensity of the background, here defined as:

There are numerous reasons why it is desirable to strive for the highest purity foci, particularly in imaging applications based on raster-scanning the foci over the field of view [6]. Mainly, the loss of contrast can become debilitating in applications requiring high dynamic range of imaging. For example, in in-vivo neuroscience imaging, which is one of the most perspective application areas of this technology, weak signals from fine processes can quickly become feeble next to large neurone somata giving in excess of four orders of magnitude stronger signals. Other applications requiring high power, such as photoacoustic [19] or two-photon microscopy [20], strongly benefit from high PRs.

Are there any fundamental limits in achieving a perfect focus with no contamination? To answer this, it is essential to realise that MMFs differ in several ways from the idealised model of a random medium. Most importantly, they support a sharply constrained number of modes, which in total may fall well below the amount of light channels controllable by a typical spatial light modulator (e.g. a DMD device). One possible representation of such modes is the basis of propagation invariant modes (PIMs), whose electric field distribution is conserved during propagation [21,22]. Having access to precise control over the amplitude, phase, and polarisation of the input field [23], there should be no fundamental obstacles in generating any optical field, which can be described as a linear combination of PIMs. It can be shown that if a MMF features over 1000 PIMs, their superposition, which matches a diffraction-limited focus best, retains a PR larger than 99% [24]. Additional reductions of PR are, therefore, all to be accounted for as issues originating in the experimental procedures, which we systematically evaluate throughout the paper. As the emphasis of our efforts lies in imaging, we do not follow a single focus, but rather large number of foci distributed in an orthogonal grid across the output facet, utilising the formalism of the transmission matrix (TM) [3,25,26]. Next to optimising the average PR, we also study its uniformity across the whole ensemble. Employing the DMD in an off-axis regime while using the Lee hologram algorithm [27] for the synthesis of binary digital holograms enabled us to investigate the benefits and drawbacks between phase-only and complex modulation, as well as these between controlling one or both polarisation states of the input light. Furthermore, we address multiple important aspects of the system including the influence of ghost order interference, noise in the TM measurement and the bias of the camera. Performing the experimental procedure under the optimal conditions led to an average PR above 96% and intensity enhancements over 20,000.

2. Methods

2.1 Experimental setup

The system, illustrated in Fig. 1, comprises an illumination path, a calibration module, and a wavefront correction (WFC) module. The detailed list of components can be found in appendix A. A 30 mW wavelength-stabilised fibre-coupled diode laser at a wavelength of 633 nm was split into a signal and a reference fibre by means of a 99/1 optical fibre coupler. The optical power of both the signal and reference fibres is controlled via polarisation-maintaining (PM) fibre-controlled electronic variable optical attenuators. The linearly-polarised signal beam from the PM fibre is used to illuminate a temperature-stabilised DMD [28] under an incident angle of approximately $24$°, after being previously expanded and collimated by lens L$1$. A good overfilling of the DMD with the illumination beam is essential to obtain diffraction-limited foci following an Airy disk distribution, as discussed in section 2.3.

 

Fig. 1. Scheme of the experimental setup. During calibration, the fibre is coupled with circularly polarised light, whose wavefront aberrations were previously corrected using the wavefront correction module. At the calibration module, the output signal from the distal fibre facet interferes with a reference signal at the camera. After calibration, a sequence of foci distributed across the fibre facet is generated at the distal fibre facet and measured with the camera (as illustrated in the inset) to later determine their power ratio. Legend: L, lens; HWP, half-wave plate; DMD, digital micromirror device; LP, linear polariser; BD, beam displacer; M, mirror; Obj, objective; QWP, quarter-wave plate; BS, beamsplitter cube; Cam, camera.

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When employed in the off-axis regime, the DMD acts as a spatial light modulator to control the phase of incident light using binary-amplitude gratings based on the Lee hologram approach [27]. Only the first diffraction order of the applied holograms was coupled into the optical fibre. The choice of illumination angle influences the overlap between the maximum diffraction efficiency of the DMD and the location of the desired diffraction order of the DMD grating, and therefore must be chosen accordingly.

After the DMD, the polarisation of the incident beam may become elliptical if not aligned with one of the preferential directions ($s$ or $p$) of the DMD mirrors. Hence, the half waveplate (HWP1) was used to rotate the linear polarisation of the illumination beam to match the preferential direction of the DMD, for which light coupled into the other orthogonal polarisation state is minimum. Then, the residual light converted into the unwanted orthogonal polarisation state was filtered by means of a linear polariser (LP1).

Lenses L$2$ and L$3$ form a telescope to demagnify the beam from the DMD and the HWP2 aligns the beam polarisation at $45$° in relation to the principal axes of the beam displacer (BD). This way, one has easier access to two beams with orthogonal polarisation states at the output of the BD. Moreover, the BD is able to merge two beams with linear polarisations along the BD axes if their physical separation matches the one required by the BD (in this case 2.7 mm). By correctly designing two binary-amplitude gratings, the first diffraction order of each grating can be positioned 2.7 mm apart in the far-field of the DMD (at the focal plane after L$2$), generating two collinear beams along the optical axis with mutually orthogonal linear polarisation states, after propagation through the BD. The BD was aligned under a polarimeter feedback, so each of its principal polarisation components is aligned with the principal polarisation axes of the dielectric mirror M1. After the quarter waveplate (QWP) the two orthogonal linear polarisations are converted into left and right circular polarisations, respectively. As shown in [22], circular polarisation is well-maintained after propagation through step-index multimode fibres.

A $90/10$ beamsplitter cube (BS$1$) splits most of the signal into a $20$x microscope objective (Obj$1$), which focusses the signal beam onto the core of the optical fibre, and a small part into the WFC module, composed of a lens L$5$ and a CMOS camera (Cam$2$). As a result of the DMD curvature, the signal beam becomes strongly aberrated, which brings consequences when focussing onto the fibre core, since a fair amount of uncontrollable light is also coupled into the cladding. Therefore, the WFC module is utilised to measure the wavefront aberration and determine the correction phase-mask to apply to the DMD, producing minimally-aberrated diffraction-limited foci at the proximal fibre facet.

The combination of L$1$, L$2$ and Obj$1$ ensures the necessary demagnification of the DMD holograms to match the numerical aperture (NA) of the MMF. The fibre used is a commercial step-index MMF with a core diameter of 50 µm, outer diameter of 125 µm and $0.22$ NA, supporting approximately 745 modes per polarisation at a wavelength of 633 nm. A total length of 15 cm of straight MMF was used in the experiment.

In the calibration module, a second $20$x microscope objective (Obj$2$) and the lens L$4$ rely the image of the distal fibre facet onto Cam$1$. The external reference beam is expanded and collimated after L$4$. Both the reference beam polarisation state and linear polariser LP$2$ are aligned with one of the principal polarisation axes of the $50/50$ beamsplitter BS$2$, ensuring pure polarisations at Cam$1$. Moreover, such configuration enables the linearly-polarised reference to interfere with the signal from the fibre in both right and left circular polarisation states.

2.2 Optimal far-field positions and wavefront correction

The selected basis of input fields represent an orthogonal grid of diffraction-limited foci distributed at the proximal fibre facet, generated using DMD gratings. The selection of the grating carrier frequency allows to steer the location of the first diffraction order across the far-field plane of the DMD. Similarly, another independent basis of foci can be generated in a distinct position at the DMD far-field by modifying the grating carrier frequency. Morover, both set of bases can also be simultaneously generated across the DMD far-field by complex superposition of the corresponding DMD gratings. This way each basis can be set to one of the orthogonal polarisation states and independently controlled. The BD can combine these two bases together, given that the central carrier frequencies are carefully set to match the 2.7 mm separation defined by the BD. Once aligned with the proximal fibre facet, these two bases provide full control over the amplitude, phase, and polarisation of the input light field coupled into the MMF.

Choosing the proper carrier frequencies, which defines the central position of the diffraction-limited foci grid at the DMD far-field, can impact the maximum achievable PR. More specifically, ghost diffraction orders associated with the spatially truncated gratings can overlap at the desired DMD far-field positions, interfering and degrading the input field. To address this aspect, we have simulated the maximum achievable PR as a function of the central far-field position, taking into consideration the parameters of the experiment. This includes two orthogonal input polarisation states, a generic defocus aberration similar to the real DMD aberration, as shown in Fig. 2(b), and conveniently scaling to the DMD dimensions ($768$x$768$ pixels) (further details in Appendix C). The result is depicted in Fig. 2(a) for the first quadrant of the DMD far-field. For the sake of simplicity, the colourmap was rearranged to display in black the central far-field positions that lead to output foci with PRs lower than 99%, while positions in white lead to PRs above 99.15%. Note that some positions in black (PRs < 99%), especially along the diagonal, can even have PRs less than 70% and are, therefore, highly inefficient solutions. The blue lines in Fig. 2(a) highlight positions symmetrically placed regarding the main diagonal, which matches the required input separation of 2.7 mm for the chosen BD. The best cases correspond, therefore, to positions along these lines, in a region where the expected achievable PR is high (white region). With this in mind, we chose the pairs (76, 246) for one polarisation and the symmetric value (246, 76) for the second polarisation, marked with blue dots in the figure. Although this arrangement of far-field positions was selected for convenience, there are many other possible arrangements that would work similarly.

 

Fig. 2. (a) Simulated far-field position as a function of the maximum achievable power ratio (PR), for the first quadrant of the DMD far-field. The origin corresponds to the DMD zero diffraction order and the k_x,k_y positions are scaled to the DMD pixels. The dashed blue lines mark symmetrical positions in relation to the diagonal, which are physically separated by 2.7 mm (separation required at the BD). The selected far-field positions were (76, 246) and (246, 76), marked with blue dots. (b) Generic aberration mask based on the real DMD aberrations considered in the simulation.

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After determining the far-field positions for both polarisation, their aberrations were measured and corrected before calibration. In light of this, WFC was performed for each polarisation using a subdomain-based approach [29], where the DMD is split into square regions (subdomains) of $16\times 16$ pixels. Each subdomain is modulated by a grating with the central frequency (k_x, k_y) for the corresponding polarisation, as determined before. One central subdomain is selected to provide a static reference beam, while the others are subsequentially displayed and swept over $4$ phase steps, resulting in an interference response recorded by Cam$2$. Such phase-step interferometry allows to find the optimal phase for each subdomain to interfere constructively with the others in a given pixel. The aberration phase-map correction for each pixel is then combined, by removing the specific tip and tilt of the measurements across different pixels at the camera, and averaged to obtain a smoother WFC map [22]. This map provides not only the necessary phase to apply on the DMD for aberration correction, but also information about the illumination at the DMD (see Fig. 3). The latter can be used to correct for the Gaussian illumination profile of the DMD.

 

Fig. 3. Illustration of the electric field composition at the DMD for each polarisation (denominated as $pol1$ and $pol2$, respectively), before binarisation, to generate a specific output focus. The amplitude of each complex field is cropped with a mask and divided by the illumination at the DMD, obtained through WFC. The phase of the complex fields must include the WFC mask and the grating with the respective carrier frequency.

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2.3 Calibration procedures

A total of approximately 1700 focal points (input modes) fit the core of the proximal fibre facet, which compared with the number of guided modes per polarisation supported by the fibre (ca. 745) is highly oversampled. In terms of output fields, the selected basis was a square grid of diffraction-limited foci distributed along the distal fibre facet (grid of $160\times 160$ camera pixels).

The TM was empirically acquired in a so-called calibration procedure, also by means of phase-step interferometry [29]. However, in this case, the utilised phase reference is external and part of the calibration module, as seen in Fig. 1. During calibration, the scanned grid of diffraction-limited foci across the input fibre facet already contains the WFC performed ahead. The TM was measured successively for both polarisations and corrected for phase drifts of the external reference in relation to the signal from the fibre [15].

Since the TM is measured in the basis of focal points at the proximal fibre facet, the inverse 2D Fourier transform of the complex input electric field allows to get the corresponding field at the DMD plane. Here, the amplitude and phase of the field for each polarisation can be binarised and converted to DMD patterns, which result in output foci. In order to aim towards output foci with the highest PR, which implies having practically full control over the propagating light, additional corrections were implemented as explained in the following.

Figure 3 illustrates the different components of the electric field at the DMD for each polarisation, before binarisation, to produce a specific diffraction-limited focus at the distal end of the fibre. The amplitude and phase of the fields are obtained through the inverse 2D Fourier transform of the desired complex input field at the fibre facet, as explained before. The grating with the specific carrier frequency naturally appear by performing the previous inverse Fourier transform in the off-axis regime, centred at the chosen far-field position. The WFC used for each polarisation must also be added to the phase term of the field at the DMD, since the TM does not contain such information. A binary mask was added to the amplitude term, consisting of the average projection of all TM input fields, (thresholded to about 60%), representing the area that is effectively coupled and propagated through the MMF. In other words, the mask represents the NA of the fibre projected at the DMD. Applying such mask avoids light from outside of the masked area with noisy amplitude and phase originated from the inverse Fourier transform to couple into the fibre, especially at the cladding, creating uncontrolled background at the output. Furthermore, the amplitude term is also divided by the DMD illumination obtained during the WFC. This term compensates for the inhomogeneous illumination of the DMD, mainly due to the Gaussian profile of the illumination beam. A Gaussian illumination of the DMD, particularly when it is not considerably overfilling the DMD active area, results in Airy disk-like foci convolved with a Gaussian profile, decreasing the intensity of the Airy disk rings away from the centre and ultimately the number of visible rings.

The DMD patterns are obtained by performing the complex superposition of the two fields at the DMD ($E_{pol1}^{DMD}$ and $E_{pol2}^{DMD}$, respectively for each input polarisation):

3. Results

3.1 Impact of polarisation, phase, and amplitude

To evaluate the PR of different foci, a sequence of DMD patterns was generated to display a total of approximately 1100 diffraction-limited foci equally distributed across the distal fibre facet. In each measurement, the foci were successively recorded in a high-dynamic-range (HDR) image with four exposure times (59, 590, 5900, and 59 000 µs, respectively). This way, both the peak intensity of the foci and lower background intensity can be accurately estimated. The protective cover glass from the camera modulates periodically the recorded intensity, introducing artefacts - large interferometric fringes across the field-of-view. Hence, to remove the influence from the camera bias, each HDR image was recorded shifting the camera in relation to the output fibre facet in steps of 2 camera pixels, averaging over a period of the interference modulation. Further considerations and a comparison between the averaged and non-averaged results can be found in Appendix D.

The PR was estimated via Eqs. (1) and 2. Note that no power measurements are needed since the intensity measurements used to determine the enhancement factor are performed with a linear detector (camera). Although the estimate relies on the number of modes, which has an uncertainty associated with the specifications provided by the MMF manufacturer, the error of such estimate is low. As shown in Appendix E, for high PR values as achieved in this work the error associated is less than 0.3 %.

The following compares the PR of foci synthesized with phase-only and complex modulation, while controlling one or both orthogonal input polarisation states. Figure 4(a) shows the radial distribution of the PR for both modulations and polarisations cases. The error bars represent the standard deviation of the averaged foci around the corresponding radius.

 

Fig. 4. (a) Radial distribution of the power ratio for phase-only and complex modulations, as well as considering one or both input polarisations. Power ratio distribution across the fibre facet controlling both input polarisation states for complex modulation (b) and phase-only modulation (c). The decrease in PR around the central region of the fibre results from saturation of the region during calibration. This issue will be addressed in section 3.3.

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The foci with best PRs are consistently accomplished when combining both circular polarisations together with complex modulation. As shown in Fig. 4(a), this situation leads to an average PR of 96.1%, meaning that only about 4% of the input light contributes to the background.

The PR distribution across the fibre facet using both polarisations is shown in Figs. 4(b) and 4(c) for phase-only and complex modulation, respectively. As can be noticed in Fig. 4(a), the drop in PR at the centre of the fibre is caused by saturation of the signal in the region during calibration. Due to the intrinsic higher mode density around the centre of the fibre, more intensity is present in that region, which unavoidably must be considered in order to still have enough visibility and signal across the borders between the core and cladding during calibration. This issue will be addressed later with averaging of TMs.

The complex modulation is implemented by thresholding the amplitude of the electric field at the DMD. A correct selection and implementation of the threshold has a strong impact, not only in terms of the quality of the foci and reduced background, but also negatively on the output intensity, as addressed in the following section.

3.2 Complex modulation

The introduction of amplitude threshold intends to manipulate the DMD mirror so that the most relevant part of the input field is preserved, while regions contributing mainly to background generation are turned off and not coupled through the fibre. Before thresholding, the electric field at the DMD ($E^{DMD}=A \cdot e^{(i\phi )}$, being $A$ the amplitude and $\phi$ the phase) is normalised by the mean value of its modulus restricted to the projection of the NA on the DMD (see Fig. 3 – mask):

This thresholding method preserves a good uniformity of the ensemble output intensity. Another possible normalisation method would be to normalise the field by its maximum value. However, such approach would result in larger standard deviation of the foci intensity, which would be rather inconvenient for real applications. Hence, in the following the first method was taken into consideration.

The amplitude threshold is implemented by setting all amplitude values ($A$) above the threshold equal to the threshold. Mathematically, this procedure is described as:

The PR across the fibre facet was measured for different threshold levels, starting from phase-only modulation (threshold equal to 0) up to a threshold of 4, in steps of 0.5. The result is shown in Fig. 5(a) for one and both circular polarisations, respectively. The insets compare a focus generated using both circular polarisation states with phase-only modulation (threshold equal to 0) and complex modulation with threshold equal to 2. Phase-only modulation by itself already provides high-quality foci for this MMF, here qualitatively seen by the amount of rings of the Airy disk in the inset of Fig. 5(a). The background for this case is still quite visible, nevertheless, it is about 3 orders of magnitude lower than the peak intensity of the focus. On the other hand, for complex modulation with a threshold equal to 2, the background is drastically reduced and a considerable number of rings of the Airy disk are visible. To help visually understand how the background is changing as thresholding is applied, see Visualization 1. The background reduction is visible with an increasing threshold, with no major change after a threshold above 2.

 

Fig. 5. (a) Average power ratio as a function of applied amplitude threshold, while controlling one or two orthogonal circular polarisation states. The maximum occurs for a threshold close to 2. Insets: intensity of a specific focus in the case of phase-only modulation and complex modulation. Introducing complex modulation reduces significantly the background, leading to higher PR. (b) Average foci intensity, normalised to the intensity value for threshold 0, as a function of the applied amplitude threshold. The stronger the complex modulation (higher threshold), the lower is the final intensity of the foci.

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However, introducing complex modulation comes at the cost of intensity: mirrors that mainly contribute to generating background are turned off, decreasing the overall active mirrors sending light into the system. Figure 5(b) represents the average peak intensity of the foci, normalised individually to the case of threshold equal to 0, as a function of the threshold. The error bars correspond to the standard deviation of the peak intensity. As expected, the power efficiency drops when increasing the threshold used for complex modulation. For threshold equal to 2, which provided the best PR values before reaching saturation, the average peak intensity is only about 38.7% of the phase-only modulation case.

3.3 Transmission matrix averaging

The issue of lower PR in the central region of the fibre, visible in Fig. 4, is solved by not saturating the matching region during calibration and performing the average of TMs from multiple calibrations. Additionally, small errors in some calibrations will be averaged out and an increase in the overall PR is expected. Note that other errors can also arise from an unbalanced ratio between the intensity of the signal and reference during calibration, affecting also the obtained PR (more details in Appendix F).

Before averaging, the phase difference between each TM must be determined and compensated, so that all the TMs from different calibrations have the same phase reference. The phase difference ($\theta _{i}$) between the first measured TM (${TM}_{0}$) and one of the following ones (${TM}_{i}$) is given by:

Figure 6(a) presents the average PR for single calibration and multiple calibrations considering the average of 2, 4, 8, and 16 TMs. The calibrations were performed with both circular polarisations and complex modulation with threshold equal to 2. The error bars correspond to the standard deviation of the PR across the fibre facet. The average PR for a single calibration is slightly lower than the one achieved in Fig. 5, since the calibration was executed without saturating the central region of the fibre. This leads to lower signal levels outside the central area, reducing the accuracy in determining the TM complex coefficients. Yet, introducing TM averaging diminishes such inaccuracies, contributing to an increase in the average PR by about 1%. The benefit of TM averaging saturates at around 8 TMs. Considering more than 8 TM only consumes additional time and brings no advantages in further increasing the PR. The distribution of PR across the fibre facet for 8 averaged TMs (marked with a star) is depicted in Fig. 6(c), clearly showing no drop in PR in the central region of the fibre. Figure 6(b) represents the best focus obtained in the same case, achieving a PR of 97.6%.

 

Fig. 6. (a) Power ratio averaged over an ensemble of output foci as a function of the number of TMs consecutively measured for averaging. Error bars correspond to the standard deviation of the power ratio for 1100 foci distributed across the fibre facet. Insets refer to the case of 8 averaged TMs, marked with a star. (b) Intensity of a specific focus while controlling 2 orthogonal circular polarisation states and employing complex modulation with threshold equal to 2. (c) Power ratio distribution across the fibre facet for 8 averaged TMs. Averaging TMs helps to further increase the PR and corrects for the decrease in PR around the central region of the fibre.

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4. Discussion

In this paper, we sought the perfect diffraction-limited focus created after propagation through a MMF and explored its limitations. To the best of our knowledge, we show the highest quality foci with average PRs exceeding 96% by controlling simultaneously the amplitude, phase, and two orthogonal circular polarisation states of the input electric field. Nonetheless, further improvement in PR was also obtained by averaging TMs from multiple calibrations. This helped to eliminate contributions from small errors when determining the TM complex coefficients and resulted in the PR of 97.6% for the best produced spots, standing only 1% away from the best possible focus using this configuration, as shown in Fig. 12 of Appendix G. However, one should note that averaging multiple TMs may not bring improvements when the system suffers from stability. To this extent, the first and last measured TMs may be fundamentally different, decreasing the final PR instead of improving.

Since circularly-polarised light is well-preserved when propagating through step-index multimode fibres, considerably high PRs (above 80%) are already achieved using only single polarisation and phase information (phase-only modulation). For graded-index fibres or other kinds of fibres that do not preserve the input polarisation state so well, using a single polarisation would lead to lower PR values. This particular situation is demonstrated by using single linear input polarisation, as shown in Fig. 4, leading to an average PR of 68%. In those cases, it is crucial to have full control over both orthogonal input polarisation states. Moreover, the PR values for phase-only modulation achievable with a MMF are not comparable to the ones achievable in a typical complex media, since MMFs act as a spatial filter, thus surpassing the normal PR limit of $\pi /4$ (78%).

When introducing complex modulation, by adding the amplitude term together with the mask and DMD illumination suggested in Fig. 3, an average increase of 7.9% was further attained. Even though single polarisation with circularly-polarised input light already provided quite good results, to aim towards perfect focussing one must be in control of all propagation channels in the optical fibre. With both circular polarisations under control, the PR obtained for the case of phase-only modulation could reach values of 90 % and is, in average, 2.5% more than using single circular polarisation.

A qualitative comparison of the imaging performance for the best and worst modulation cases here reported is shown in Fig. 7, where the measured HDR foci images were used to computationally image a 1951 USAF resolution test chart (Fig. 7(a)). Employing phase-only modulation while controlling 1 linearly-polarised input state (Fig. 7(b) and 7(e), PR around 68 %) leads to lower imaging contrasts due to the lower PR values. On the other hand, changing the input polarisation state from linear to circular (Fig. 7(c) and 7(f), PR around 86 %), which is well-preserved during propagation, improves the PR and consequently the imaging contrast. The best imaging performance is obtained when using complex modulation controlling 2 circular orthogonal input states (Fig. 7(d), and7(g), PR around 96 %).

 

Fig. 7. Illustration of the impact of foci quality in imaging. (a) 1951 USAF resolution test chart: groups 8 and 9. Raster-scan imaging simulation using the real measured foci for phase-only modulation controlling (b) 1 linear polarisation state, (c) 1 circular polarisation state, and (c) complex modulation with threshold equal to 2 controlling 2 circular orthogonal polarisation states. The same figures are represented in (e-g) using logarithmic scale.

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However, this result does not represent the real scenario of fluorescence imaging of a 3-dimensional structure (ex. neurones), where uncontrolled background light together with out-of-focus light further degrades the imaging contrast. In particular, the impact of PR when imaging fine structures in the vicinity of larger ones is very different from the 2-dimensional case. To demonstrate such impact the imaging simulation of a 3-dimensional neurone model is presented in Fig. 8. Here, we considered two distinct mechanisms of PR reduction. The first arises from employing different modulation and polarisation cases (Fig. 8(b)–8(d)), leading to uncontrolled background light with some correlation from point to point. The second type of PR reduction results from noise in the TM measurement (Fig. 8(e)–8(h)), where the uncontrollable background from pixel to pixel is uncorrelated. Small structures, such as dendritic spines, clearly visible in Fig. 8(d) start to be feeble even for PRs of 80% as shown in Fig. 8(h).

 

Fig. 8. Impact of foci quality in imaging of a 3-dimensional sample. (a) Model of a neurone, scaled to the real dimensions of the system. Focal plane marked in red ($z=0$). Result for phase-only modulation while controlling (b) 1 linear input polarisation and (c) 1 circular polarisation. (d) Complex modulation with 2 circular polarisations (ideal PR of 100 %). (e-h) Impact of PR reduction due to noise in the TM measurement. Images are represented in logarithmic scale. The width of the images corresponds to 65 µm and the focal plane is located 32.5 µm away from the distal fibre facet.

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Complex modulation effectively reduces the uncontrolled signal spread through the background, but has a negative impact on the power efficiency. On the one hand, the intensity loss of employing complex modulation can be a limiting factor when the available power is restricted. On the other hand, the effective reduction of background leads to higher imaging contrast, and therefore the use of lower excitation intensities might now be sufficient since the contaminating background is no longer a major issue.

We removed the bias of the camera by averaging the recorded foci images over a period of the camera interference modulation. Such artefact comes exclusively from the camera protective cover glass, and the produced foci are, in fact, not contaminated by the camera-induced modulation. The successful averaging of the modulation shows that the camera bias does not influence the calibration procedure and determination of the TM, since it is a static modulation and, therefore, simply an intensity offset.

Beyond full control of amplitude, phase, and both polarisations of the input electric field, we also addressed other important aspects of the system that contribute towards reaching the limits. The ghost order interference, that naturally appears from the binarisation of the DMD grating and placement of the first diffraction order in a specific DMD far-field positions, affects the input field’s fidelity at the fibre facet and decreases the achievable PR. Through simulations, we determined and selected the best far-field position for each polarisation. We additionally implemented WFC of the input beam for each polarisation state, plus masking the amplitude of the electric field at the DMD by the corresponding NA of the fibre. Both measures aim to minimize the amount of light coupling into the cladding, which can contribute to the uncontrolled background at the output.

The optimized system and methods here presented, capable of forming high-purity foci, represent an important step for microscopy techniques that rely on single-photon excitation, where the presence of background influences significantly the contrast of the image and causes out-of-focus photobleaching. Moreover, gaining practically full control over the light propagation mechanisms opens doors for applications relying on high-fidelity generation of advanced optical beams, such as vortex beams, Airy beams, or even structured-illumination beams.

Appendix

A. Equipment and components

The components used in the experimental setup (see Fig. 1) are detailed as follows. Laser, QPhotonics: QFBGLD-633-30; 99/1 optical fibre coupler, Thorlabs: PN635R1F1; PM fibre-controlled electronic variable optical attenuator, Thorlabs: V600PA; DMD, ViALUX: V-7001; lenses: L1, Thorlabs: AC254-250-A-ML; L2, Thorlabs: AC254-200-A-ML; L3,L4, and L5, Thorlabs: AC254-100-A-ML; Objectives Obj1 and Obj2, Olympus: 20x Plan Achromat; Linear polarisers L1 and L2, Thorlabs: LPVISC100-MP2; half waveplates HWP1 and HWP2, Newport: 10RP52-1B; quarter waveplate, Newport: 10RP54-1B; beam displacer BD, Thorlabs: BD27; 90/10 beamsplitter cube BS1, Thorlabs: BS070; 50/50 beamsplitter cube BS2, Thorlabs: BS010; mirror M1 and M2, Thorlabs: BB05-E02; cameras Cam1 and Cam2, Basler: acA640-750um; multimode fibre, Thorlabs: FG050UGA.

B. Example of Lee holograms

Figure 9 illustrates Lee holograms with phase-only modulation and complex modulation with threshold equal to 2, which respectively produce the focus displayed in the insets of Fig. 5.

 

Fig. 9. Example of Lee holograms for phase-only modulation and complex modulation with threshold equal to 2.

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C. Considerations on the far-field positions

The simulated results presented in Fig. 2(a) were obtained following the steps below:

• 1. Calculate the TM for each polarisation, considering the parameters of the MMF used (i.e. core and cladding dimensions, NA, and length).
• 2. Starting from a desired diffraction-limited focus at the output, the result is backpropagated through the calculated TMs to obtain the input electric field at the proximal fibre facet for each polarisation.
• 3. Perform cycle through all the far-field positions:

  • 3.1. Place the input field for each polarisation, "off-axis", centered at the respective far-field position, in a $768$x$768$ grid corresponding to the area-of-interest of the DMD.
  • 3.2. Calculate the input field at the DMD plane through the 2D inverse Fourier transform.
  • 3.3. To consider the aberrations caused by the DMD itself, a generic aberration mask shown in Fig. 2(b), based on the real aberrations (see Fig. 3 – WFC) was added to the calculated input field at the DMD.
  • 3.4. Introduce complex modulation, by thresholding the amplitude of the input field at the DMD using Eq. (5)).
  • 3.5. Binarise the input field at the DMD plane using the Lee hologram approach according to Eq. (6).
  • 3.6. Perform the 2D Fourier transform of the binarised input field at the DMD plane to obtain the new input field at the proximal fibre facet.
  • 3.7. Trim the input field at the proximal fibre facet by the core size.
  • 3.8. Propagate the trimmed field using the initial calculated TMs and determine the PR comparing the result with the initial input field (point 2), also propagated using the TMs.

D. Averaging of camera fringes: removing bias of the camera

A vast number of cameras, including the ones used in the context of this experiment, contain a protective glass covering their sensor. When working with coherent light sources, such protective glass cover acts as a Fabry-Perot interferometer, causing a small percent of light to be successively transmitted and reflected at the glass interfaces. Hence, the measured signal at the sensor is modulated by interference fringes caused by a local change of the signal’s original intensity.

In the particular case of this experiment, even though the camera fringes are static and have no impact in the calibration procedure, they will affect the measurement of the generated foci. The intensity distribution of each foci is modulated by the camera fringes and, as a consequence, the PR distribution across the fibre facet follows the same trend. This artefact ends up masking the real results, since in reality the intensity and PR of the foci are quite uniform and far from being modulated by any similar periodic interference. Performing camera averaging, by recording each foci while moving the camera stepwise in relation to the output fibre facet over approximately a period of the interference modulation, allows to average out this effect. Ultimately, the average value of PR across the fibre facet with and without camera fringes averaging remains mostly unaffected, while the standard deviation of the PR reduces.

Figure 10(a)–10(f) compares the results from Figs. 4(b)–4(c) and 6 with and without camera averaging. The average PR without camera fringes averaging is 88.9%, 95.9%, and 96.5%, respectively for phase-only modulation, complex modulation without and with 8 TM averaging. The same values for the case of camera averaging are 89.1%, 95.7%, and 96.5%, respectively, presenting deviations in the order of 0.2% from the previous results. However, in terms of fluctuations in the PR values, the standard deviation reduced by 11%, 8%, and 37%, respectively for each case, when applying camera averaging.

 

Fig. 10. Power ratio distribution across the fibre facet without camera averaging (a-c) and with camera averaging (d-f). Figures (d-f) correspond to the results presented in Figs. 4(b-c) and 6. (g) Average power ratio as a function of the ratio between the signal and reference intensities, integrated over the camera region-of-interest.

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E. Error associated with PR estimation via enhancement factor and number of modes

The PR was estimated using Eq. (2), which relies on having knowledge about the number of propagating modes ($N$) in the MMF. This value is conditioned by the specifications of the MMF provided by the manufacturer, which naturally have an uncertainty associated with it. In this appendix we aim to clarify the impact on the PR estimation caused by uncertainties of the MMF parameters, specifically the numerical aperture (${{NA}}$) and the fibre core radius ($a$).

The number of modes per polarisation supported by the MMF is determined via the normalized frequency parameter (V-number) as:

(9)$$N = \frac{V^2}{4} = \frac{\pi^2}{\lambda^2}a^2{{NA}}^2,$$

where $\lambda$ is the operation wavelength. The error associated to the number of modes (${\Delta N}$) given the uncertainties of the fibre core radius and the ${NA}$ is expressed as:

(10)$$\Delta N = \sqrt{\left(\frac{\delta N}{\delta a}\Delta a\right)^2+\left(\frac{\delta N}{\delta {{NA}}}\Delta {{NA}}\right)^2},$$

where $\delta N/\delta a$ and $\delta N/\delta {{NA}}$ are the derivative of $N$ in relation to the core radius and ${{NA}}$, respectively. For the specific MMF used in this work, the manufacturing uncertainties are $0.22 \pm 0.02$ for the ${{NA}}$ and 50 µm $\pm$ 5 % for the fibre core diameter. The number of propagating modes per polarisation has then an uncertainty of $\pm 15$ modes. The error associated with PR estimation ($\Delta PR$) based on Eq. (2) is given by:

(11)$$\Delta PR = \frac{\delta PR}{\delta N}\Delta N,$$

being $\delta PR/\delta N$ the derivative of Eq. (2) in relation to the number of modes ($N$). The result of Eq. (11) is depicted in Fig. 11. In this work we are dealing with high PR values, ultimately above 90%, whose PR estimate has less than 0.2% of associated error. Therefore, this result validates the use of Eq. (2) as the metrics for this work.

 

Fig. 11. Percentage of error associated to the power ratio estimate as a function of the enhancement factor. For PR values of 0.9 or higher (high enhancement factors), the error associated is less than 0.2 %.

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F. Impact of the ratio between the intensity of the signal and reference during calibration

The intensity ratio between the signal and reference during calibration has an influence on the PR. An unbalanced intensity ratio between these two leads to a weaker interference visibility, and therefore to errors in determining the complex coefficients of the TM. We performed a parametric study varying the relative intensity between the signal and the reference, while keeping the maximum intensity in the camera region-of-interest below saturation (maximum intensity approximately 90% below the limit of saturation). Using a camera region-of-interest of 160x160 pixels, the intensity ratio was determined dividing the signal intensity by the reference intensity, integrated over the region-of-interest of the camera. The average PR of foci distributed across the fibre facet, generated using 2 polarisations and complex modulation with threshold equal to 2, was measured for each ratio. The result is depicted in Fig. 10(g). The error bars correspond to the standard deviation of the PR across the fibre facet. For this specific setup, considering the region-of-interest used and the magnification of the output fibre facet on the camera, the maximum average PR was obtained for a ratio of $0.07$. This study highlights the importance of finding the correct ratio between the intensity of the signal and reference, nevertheless the best ratio is system specific and should be determined for each system independently.

G. Simulations

To further support the overall empiric observations of this work, we include a simulation of forming a focus at the distal end of a MMF. The parameters of the simulation closely follow the experimental conditions, particularly the properties of the MMF, the used wavelength and the magnification of the relay optics, the spatial resolution of the DMD, and the placement of the polarisation contributions in the Fourier plane of the DMD. The model considers a DMD featuring the same aberrations and illumination uniformity as detected in Fig. 3. The model further assumes straight and perfectly cylindrically symmetric waveguide, which supports a series of PIMs that maintain the circular polarisation state. It has been shown to reflect the experimental reality very well, except for a few modes in which the sum of spin and orbital angular momentum equals zero [22]. Within the model, a perfect diffraction-limited focus with NA corresponding to that of the MMF is send backwards through the fibre and the randomised field at the proximal end is projected to the plane of the DMD by two-dimensional Fourier transform mimicking the relay optics. Here, a phase conjugate is synthesised by the means of the Lee hologram, as in Eq. (6), with a variable threshold, while considering the illumination non-uniformity and phase aberrations. The result is sent forward through the relay optics and the fibre, leading to a simulated field of the focus. Having access to the field distribution, we can evaluate the resulting PR according to its definition, i.e. from the inner product of the resulting field with the optimum focus. The optimum focus can be understood as the ideal diffraction-limited focus, as well as the one representing the best superposition of PIMs, therefore we provide our study for both. Finally, we also evaluate the PR estimate from the enhancement factor, as it was done in our experimental data, thereby verifying its suitability. The results are summarised in Fig. 12.

 

Fig. 12. Average power ratio as a function of the applied threshold for (a) 2 polarisations and (b) 1 polarisation. For each case, the PR was determined via three distinct approaches. (c) Modulation efficiency as a function of the applied threshold. For a single polarisation state we model the desired focus having a circular polarisation state, while for the 2-polarisation case we demand linear polarisation (with equal contributions propagating through each polarisation arm of the system).

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The imaging simulations presented in Fig. 7 are based on the real measured HDR images of foci with phase-only modulation controlling 1 linearly- or circularly-polarised input state, and with complex modulation of threshold equal to 2 controlling 2 circularly-polarised orthogonal states. The 1951 USAF resolution test chart (groups 8 and 9) was properly sampled and the HDR foci images were upscaled to match the respective size of the USAF chart. Each foci image was successively multiplied with the target image and integrated over the field-of-view, building computationally the raster-scanned image of the USAF chart.

In case of Fig. 8, the foci used for each modulation and polarisation case (Fig. 8(b)–8(d)) were generated based on the previous numerical simulation adapted for each particular case, including free-space propagation of the respective foci. Hence, the 3-dimensional neurone model, respectively scaled to the dimensions of the fibre and system, was computationally raster-scanned with 3-dimensional foci, with focal plane at $z=0$. To simulate the impact of uncorrelated noise arising from the TM measurement (Fig. 8(e)–8(h)), random noise was generated and computationally propagated through the MMF model, respectively normalised and added to the case of complex modulation (PR=100 %) accordingly to obtain the desired PR. The particular case of phase-only modulation controlling 1 linear polarisation state was simulated for a fibre length of 10 cm, since the corresponding data in Figs. 4 and 7 was taken with such fibre length.

Funding

Bundesministerium für Bildung und Forschung; Thüringer Aufbaubank; Thüringer Ministerium für Wirtschaft, Wissenschaft und Digitale Gesellschaft; Freistaat Thüringen (2018-FGI-0022, 2020-FGI-0032); Ministerstvo Školství, Mládeže a Tělovýchovy (CZ.02.1.01/0.0/0.0/15_003/0000476); Horizon 2020 Framework Programme (101016787); European Research Council (724530).

Acknowledgments

Benjamin Rudolf is gratefully acknowledged for its valuable criticism while preparing the manuscript.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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