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Abstract
Focussing light through a multimode fibre (MMF) is the basis of holographic endoscopes, which currently enable detailed imaging of deep tissue. Achieving high fidelity and purity diffraction-limited foci has been shown to be possible, when fully controlling the amplitude, phase, and two orthogonal polarisation states of the input field. Yet, generating more complex field distributions with similar performance remains to be assessed. Here, we demonstrate the generation of Airy beams through an MMF containing in excess of 90 % of the optical power delivered by the fibre. We discuss two distinct methods for generating optical landscapes: the direct field and the Fourier domain synthesis. Moreover, we showcase the flexibility of the Fourier domain synthesis to modify the generated beam.
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1. Introduction
Holographic endoscopes are recent instruments born from the desire to understand and control the physical mechanisms underlying light transport through multimode fibres (MMFs). A hair-thin MMF can guide thousands of spatial modes, each of which propagates with a slightly different propagation constant. Coupling a coherent light source into the MMF leads to a seemingly random, yet deterministic, output speckle pattern [1]. Combining the modes appropriately by means of digital holography can, however, yield to a specific output landscape. This approach requires knowledge of how light propagates in the MMF, being the relationship between input and output of the fibre typically measured empirically and expressed in the form of a transmission matrix (TM) [2–4]. If the TM is measured with a flat-phase reference, and in both polarisation states, it contains all the information necessary to synthesize in practice any light field distribution at the MMF output, given its constraints of numerical aperture (NA) and core dimensions [5,6].
A multitude of microscopy techniques delivered through a MMF were demonstrated in the last decade, including fluorescence microscopy [7,8], wide-field microscopy [9], light-sheet microscopy [10], two-photon excitation [11–14], confocal microscopy [15,16], Raman spectroscopy [17], chemical imaging based on Raman scattering [18], and second-harmonic generation imaging [19]. Furthermore, fluorescence minimally invasive deep-tissue microscopy was also applied in vivo [6,20–22]. Striving for high fidelity of the output field is vital, especially in applications that are photon-hungry or that require high dynamic range imaging. Recently, near perfect focusing through MMF has been demonstrated, in which holographically synthesized diffraction-limited foci were generated containing in excess of 96 % optical power transmitted through the fibre [23]. Such high purity lies in the ability to fully control the amplitude, phase, and two orthogonal polarisation states of the light field coupled at the input of the MMF.
Yet, not all imaging techniques rely on raster scanning a diffraction-limited foci. Some perspective modalities involve the use of complex light fields, such as structured illumination [24], vortex beams [25–27], or even non-diffractive beams [28]. Hence, the capability of realistically synthesizing more complex field distributions through a MMF while achieving high PRs remains to be demonstrated. In this work, we investigate the generation of Airy beams through a step-index MMF. Airy beams were selected as the focus of this case study, given their direct applications in the domain of microscopy [28–31] and their complex nature. Unlike other beams, Airy beams do not follow a circular symmetry and exhibit a non-diffractive quadratic propagation trajectory [32,33]. We evaluate the fidelity of Airy beams in a system, which allows for the control over the distribution of amplitude, phase, and two orthogonal polarisation states of the input fields. These were generated via two distinct methods: direct field and Fourier domain synthesis. Each result was then carefully measured for its distribution of amplitude and phase by means of phase-step interferometry, and compared with the expectation. Finally, we showcase the versatility of Fourier domain synthesis by manipulating the propagation range of the Airy beam and the location of its peak intensity along the propagation trajectory.
2. Methods
The generation of Airy beams was performed using a system for holographic control of light propagation in multimode fibres, as described in [23]. The system uses a digital micromirror device (DMD) in the off-axis regime to shape the wavefront of a 633 nm laser, in phase and amplitude, before coupling it into the fibre. Although the overall efficiency of these kind of systems is typically as low as 2 %, one can always compensate for the losses with a higher laser power in all forms of microscopy. Moreover, the lost energy in no way interacts with the sample. Two orthogonal polarisation states of the input light were also independently controlled and merged together utilising a beam displacer. A commercially-available step-index MMF (FG050UGA, Thorlabs) with a length of 10.5 cm, a core and cladding diameter of 50 µm and 125 µm, respectively, and NA of 0.22, capable of supporting approximately 745 modes per polarisation at 633 nm, was employed in the experiment. The TM of the fibre was acquired using phase-shift interferometry with an external reference arm using the same methodology as described in [23]. The basis of representation of the measured TM was conveniently selected as a grid of diffraction-limited foci at the proximal MMF facet (inputs) and a grid of camera pixels (outputs), where the distal MMF facet is imaged onto. The TM consists of 3721 input modes (grid of $61\times 61$ diffraction-limited foci) and 20736 output modes (grid of $144\times 144$ camera pixels).
Utilising the information stored in the TM, finite Airy beams were created by two distinct methods: direct field synthesis and Fourier domain synthesis. In the direct field synthesis approach, the input electric field necessary to generate an Airy beam at the MMF output is obtained by direct matrix multiplication between the transpose conjugate of the TM and a matrix containing the desired output field. On the other hand, the Fourier domain synthesis involves projecting the output modes of the TM into the farfield and applying a phase-only cubic mask to them. The input electric field necessary to create an Airy beam is then obtained by directly taking the line of the TM for the respective output position.
2.1 Direct field synthesis
Once one has access to the TM, the input electric field ($E_{in}$) needed to produce a desired light field distribution at the distal end of the MMF ($E_{out}$) can be obtained by direct matrix multiplication in the form of:
where $T\!M^H$ represents the transpose conjugate of the TM. The matrix $E_{out}$ comprising a two-dimensional finite Airy beam can be expressed as:When employing this equation, the Airy beam propagates diagonally with respect to the coordinate system axes. To simplify later analysis, we rotated $E_{out}$ in the coordinate system by $45$ degree to produce an Airy beam accelerating along the horizontal axis. After rotation, the desired field distribution is expressed as:
To emulate the real conditions, the field was restricted, in the Fourier domain, by the fiber NA and further truncated, in the real domain, by a circular mask with the same dimensions as the fiber core. The resultant field ($E_{out}$) is then used in Eq. (1) to obtain the input field ($E_{in}$). At last, the DMD pattern needed to holographically synthesize this input field is calculated based on the Lee hologram approach using complex field modulation with amplitude threshold equal to 2, as described in more detail in [23].
2.2 Fourier domain synthesis
In Fourier domain synthesis, the key aspect involves modifying the phase information stored in the TM. More specifically, incorporating a cubic phase mask is required to obtain an Airy beam. Considering the convolution theorem, this process can be done either by performing a convolution in the real domain or a multiplication in the Fourier domain. The output modes of the TM were projected into the farfield (i.e. into the spatial frequency representation). Here, each output mode in this new representation is multiplied with the same phase mask (M), which for the case of an Airy beam is generally expressed as:
where $b$ represents once more the strength of the cubic phase mask, while $K_{x}$ and $K_{y}$ are the $X$ and $Y$ components of the $K$-vector, which form an orthogonal basis in the Fourier space.We rotated the phase mask by $45$ degree to generate an Airy beam accelerating along the horizontal direction. An example of such mask with the strength ($b$) equal to 5 µm$^{-1}$ is displayed in Fig. 1(a). Moreover, we applied a circular aperture in order to restrict the farfield to the range of available spatial frequencies (NA). The aperture took the shape of a super-Gaussian (SG) distribution to avoid possible effects of hard aperturing. The super-Gaussian aperture is given by:
where $\theta$ is the grid of angular projections of the TM output, and $N$ is a parameter that dictates the strength of the SG aperture. The intensity distribution of the super-Gaussian aperture with a strength of $N = 3$ is shown in Fig. 1(b). After the above-mentioned modifications, the phase mask (M) is described as:
Fig. 1. (a) Rotated cubic phase mask, used for generating an Airy beam via Fourier domain synthesis with $b=5$. (b) Intensity of the super-Gaussian aperture with $N$ = 3. $k$ is the wavenumber, and $k_k$ and $k_y$ represents the x and y component of the propagation constant in the Fourier space.
After multiplying every TM output with the same phase mask, in the Fourier domain, an inverse Fourier transform is applied to the result, returning all the TM outputs from the farfield back into the facet plane. The final TM has the same original input modes, but each output mode corresponds now to an Airy beam centred around the respective output position. The input field to generate an Airy beam at a specific output position is obtained by taking the respective output line of the TM. The corresponding DMD pattern for this input field is calculated using the Lee hologram approach, in the same way as described before. The Fourier domain synthesis approach is computationally less intensive than synthesising each output Airy beam, one by one, as in the direct field synthesis approach.
3. Results
To evaluate the produced Airy beams, we used the power ratio (PR) as the quality metrics, here expressed as the overlap integral between the model and experimental Airy beam. We compare two variants of this quantity. The first consists of the overlap integral using the complex fields (amplitude and phase), defined as:
where $E_m$ and $E_e$ correspond to the model and experimental Airy beam fields, respectively, each normalised so that $\iint |E_m|^2=\iint |E_e|^2=1$. In the second approach we considered just the amplitude values of the model and experimental fields, defining the amplitude-only power ratio ($PR_a$) as: once again assuming both fields are normalised.The amplitude and phase of each experimental Airy beam were obtained through 4 phase-step interferometry with the same flat-phase reference used for acquiring the TM. Each Airy beam was then generated with 4 phase offsets (0, $\pi /2$, $\pi$, and $3\pi /2$) and its interference with the reference was recorded with the output camera, followed by a measurement of the reference intensity (without signal). The amplitude and phase of the Airy beam is reconstructed from the 4 phase-steps as $\Sigma _{k=1}^4 I_k.$exp$(i.k.\pi /2)$, where $I_k$ is the captured intensity of the interference between the reference and the Airy beam at a specific phase step. The obtained Airy beam field was divided by the reference intensity, to account for spatial intensity variations (non-uniformity) of the reference. To reduce the measurement noise, which has impact in the amplitude and phase determination, the acquisition of the 4 phase-steps was repeated 500 times in a row. Note that a small angle between the signal and reference during the experimental TM acquisition resulted in a shift of the TM output farfield. Therefore, the appropriate phase ramp was added to all generated Airy beams, centring their farfield with respect to the TM output farfield. After measurement of the amplitude and phase of each Airy beam, the respective phase ramp was removed from the result.
Figure 2 shows both model and experimental Airy beam fields with $b$ set to 5, 10, 15, 20, and 50, respectively, utilising direct field synthesis. In this manner, we were able to create Airy beams with PR rounding 90 %, while the amplitude-only PR is approximately 94 %. The experimental Airy beam intensities are represented in Fig. 2(k-o) in logarithmic scale, showing the uncontrolled speckle background distributed over the field of view.
Fig. 2. Airy beams obtained by direct field synthesis: (a-e) modelled electric field, (f-j) experimentally measured electric field, and (k-o) experimentally measured intensity in logarithmic scale, with $b$ = 5, 10, 15, 20, and 50, respectively. The results in (a-e) were obtained employing Eq. (1) and Eq. (3). The phase of the experimental Airy beam fields was obtained through phase-step interferometry with a flat-phase reference. $PR$ and $PR_a$ in (f-j) were obtained according to Eqs. (7) and (8).
Employing the Fourier domain synthesis, we generated Airy beams using cubic phase masks with the same $b$ parameters as for the direct field case. The model and experimental fields are depicted in Fig. 3. The PR achieved was above 91 %, while the PR in amplitude rounded the 94 %. Figures 3(k-o) display the experimental Airy beam intensities in logarithmic scale. Naturally, the PR for the Airy beam with $b$ = 50 was lower than the rest of the cases, in both approaches, due to the extreme size of the beam, which tends to extend over the core size.
Fig. 3. Airy beams obtained by Fourier domain synthesis: (a-e) modelled electric field, (f-j) experimentally measured electric field, and (k-o) experimentally measured intensity in logarithmic scale, with $b$ = 5, 10, 15, 20, and 50, respectively. The results in (a-e) were obtained by inverse Fourier transform of Eq. (6). The phase of the experimental Airy beam fields was obtained through phase-step interferometry with a flat-phase reference. $PR$ and $PR_a$ in (f-j) were obtained according to Eqs. (7) and (8).
One typical characteristic of Airy beams is their curved trajectory, as illustrated in Fig. 4(a). To analyze the propagation of synthesized Airy beams, we progressively recorded the Airy beam profile from 40 µm in front to 40 µm behind the calibration plane, in steps of 1 µm, using a piezo motor attached to the output objective. For this purpose, we replaced the 20x output objective with an Olympus 40x (RMS40X). A map of horizontal cross-sections through the center of the Airy beam main lobe, along the dashed black line marked in Fig. 4(b), for different planes recorded along the propagation is depicted in Fig. 4(c-g) for direct field synthesis and in Fig. 4(h-l) for the Fourier domain synthesis. With increasing value of the $b$ parameter, the trajectory of produced Airy beams was less curved. Furthermore, with a higher $b$ parameter, Airy beams exhibited stronger side lobes, and therefore the main lobe carries less energy. The intensity distribution at propagation position 40 µm for an Airy beam created by Fourier domain synthesis with $b$ = 10 is shown in Fig. 4(b). Additional videos on the propagation and 3D reconstruction of the same Airy beam can be found in Visualization 1 and Visualization 2.
Fig. 4. (a) Illustration of the curved trajectory of propagating Airy beam. (b) Intensity distribution at $z =$ 40 µm (focal plane position) for an Airy beam generated by Fourier domain synthesis with $b = 10$ (see Visualization 1 and Visualization 2 for propagation cross-section and 3D reconstruction, respectively). The dashed black line indicates the cross-section plane used to construct (c-l). Propagation of Airy beams created by: (c-g) direct field synthesis, and (h-l) Fourier domain synthesis, with $b$ = 5, 10, 15, 20, and 50, respectively.
3.1 Modulation of the propagation characteristics of an Airy beam
The characteristics of an Airy beam, such as the trajectory and peak intensity position, can be further modulated [32], employing both the direct field and the Fourier domain synthesis. The Fourier domain approach provides further flexibility in manipulating the output field over the direct field synthesis. In this particular case, the propagation and peak intensity position of the Airy beam can easily be controlled by acting directly on the cubic phase mask and super-Gaussian aperture, respectively (i.e shifting them with respect to the optical axis).
To showcase this flexibility, we displaced the super-Gaussian aperture or the cubic phase mask ($b$ = 10) horizontally in the plane transversal to the optical axis. The illustrative scheme, together with the chosen Cartesian coordinate system, is displayed in Fig. 5(a). A map of horizontal cross-sections through the main lobe of the Airy beam along the propagation direction obtained when both the super-Gaussian aperture and the phase mask are aligned co-axially is shown in Fig. 5(b). Unlike the previous experiments, here the image acquisition started at the calibration plane, recording only half of the propagation trajectory, yet enabling the observations of the Airy beam behavior. This time the output objective was replaced by an Olympus PLN20X, since the propagation of several modified Airy beams was much longer than before.
Fig. 5. (a) Schematic of the Fourier domain synthesis, with super-Gaussian aperture (SG) and the cubic phase mask (M), with respective coordinate system, used to generate Airy beams with modified propagation characteristics. Measured propagation trajectory: (b) when SG and M are co-axially aligned, (c-d) when shifting M to positive values of the x-axis by 10 and 20 pixels, respectively, and (e-f) after shifting SG to negative values of the x-axis by 10 and 20 pixels, respectively. Data shown in (b-f) are normalized to their maximum. $Z=0$ corresponds to the calibration plane.
Figure 5(c,d) depicts the measured propagation curve obtained after displacement of the cubic phase mask to positive values of the horizontal axis by 10 and 20 pixels, respectively. Compared with the original trajectory, shown in Fig. 5(b), one can easily see that the horizontal displacement of the phase mask enables control over the trajectory starting position and propagation range, while the peak intensity always remains at the maximum of the curvature.
Secondly, we moved the super-Gaussian aperture along the horizontal axis, to its negative side, by the same amount of pixels as in the previous experiment, resulting in the Airy beams depicted in Fig. 5(e,f). Clearly, the beam trajectory remains the same regardless of the decentering magnitude. However, with a more significant aperture displacement, the intensity peak position shifts away from the maximum of the curvature.
Moreover, misaligning both the mask and the super-Gaussian aperture in opposite directions of the horizontal axis allows to control the range of propagation as well as the position of the intensity peak [32]. These results, together with results obtained while shifting the mask and aperture together in the positive direction of the horizontal axis, can be seen in the Appendix. The fundamental background behind these phenomena can be found in [32].
4. Discussion
In this paper, we have demonstrated the capability of the holographically generating Airy beams after propagation through a MMF with high fidelity and power ratio. We have assessed two distinct methods to produce such complex light fields: the direct field and Fourier domain synthesis. Both methods provided Airy beams with good fidelity of the amplitude and phase profiles. The obtained power ratios for both approaches is very similar, in the range of 90 to 92 %, considering both amplitude and phase of the experimental Airy beams. If the phase is ignored, the amplitude-only power ratio ranges from 93 to 95 %.
If the mathematical expression of the desired output field is known, the direct field synthesis provides a straightforward solution for its generation, involving solely a single matrix multiplication. Alternatively, the Fourier domain synthesis provides a different approach to generate the desired beams, by multiplying a single phase mask with the TM, where each output mode is projected into the Fourier domain (spatial frequency representation).
Moreover, the Fourier domain synthesis has more flexibility over the direct field synthesis in modifying the generated landscapes. Operations such as beam defocussing, adding a vortex phase, or trimming the output numerical aperture can easily be performed with the Fourier domain synthesis. We have showcased this flexibility by manipulating the Airy beam propagation trajectory and peak intensity position.
The methods addressed in this work can be directly applied to generate other complex light fields with high fidelity and power ratio for applications in structured illumination and light-sheet microscopy (e.g. bessel beams), or even in stimulated emission depletion (STED) microscopy (e.g. vortex beams and bottle beams).
5. Appendix
5.1 More on control of the propagation characteristics of an Airy beam
Figure 6(b-c) display the measured propagation trajectory of an Airy beam generated after shifting the cubic phase mask (M) to positive values of the horizontal axis, while the super-Gaussiann (SG) aperture was shifted to the opposite side of the horizontal axis. The magnitude of the shift was the same for both M and SG (5 and 10 pixels, for Fig. 6(b) and (c) respectively). Shifting both, M and SG, in the same direction of the horizontal axis leads to the propagation trajectory shown in Fig. 6(d-e), for a shift of 10 and 20 pixels in the positive direction, respectively.
Fig. 6. Measured propagation trajectory: (a) when the super-Gaussian aperture (SG) and cubic phase mask (M) are co-axially aligned, (b-c) when displacing M and SG in opposite directions along the horizontal axis by 5 and 10 pixels, respectively, and (d-e) when displacing M and SG in the same direction to positive values of the horizontal axis by 10 and 20 pixels, respectively. Data displayed in (a-e) are normalized to their maximum. $Z=0$ corresponds to the calibration plane.
Funding
Freistaat Thüringen (2018-FGI-0022, 2020-FGI-0032); Horizon 2020 Framework Programme (101016787); European Regional Development Fund (LM2018129); Ministerstvo Školství, Mládeže a Tělovýchovy (CZ.02.1.01/0.0/0.0/15_003/0000476); European Research Council (101069245, 101082088, 724530).
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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