Two-photon lensless endoscope - OSA Publishing

Two-photon lensless endoscope Esben Ravn Andresen,1,3 Géraud Bouwmans,2 Serge Monneret,1 and Hervé Rigneault1,∗ 1 Aix-Marseille

Université, CNRS, Centrale Marseille, Institut Fresnel, UMR 7249, 13013 Marseille, France 2 PhLAM CNRS, IRCICA, Universit´ e Lille 1, 59658 Villeneuve d’Ascq Cedex, France 3 [email protected] ∗ [email protected]

Abstract: We report a first demonstration of two-photon endoscopic imaging with a lensless endoscope. The endoscope probe is a double-clad bundle of single-mode fibers; point excitation and scanning is achieved by coherent combining of femtosecond light pulses propagating in the single-mode fibers; and back-scattered two-photon signal is collected through the multi-mode inner cladding. We demonstrate the two-photon endoscope on a test sample of rhodamine 6G crystals. © 2013 Optical Society of America OCIS codes: (060.2430) Fibers, single-mode; (170.2150) Endoscopic imaging; (180.2520) Fluorescence microscopy; (180.4315) Nonlinear microscopy; (180.5810) Scanning microscopy; (230.6120) Spatial light modulators.

References and links 1. S. F. Elahi and T. D. Wang, “Future and advances in endoscopy,” J. Biomed. Opt. 4, 471–481 (2011). 2. I. N. Papadopoulos, S. Farahi, C. Moser, and D. Psaltis, “High-resolution, lensless endoscope based on digital scanning through a multimode fiber,” Biomed. Opt. Express 6, 260–270 (2013). 3. T. Cizmar and K. Dholakia, “Exploiting multimode waveguides for pure fibre-based imaging,” Nat. Commun. 3, 1027 (2012). 4. Y. Choi, C. Yoon, M. Kim, T. D. Yang, C. Fang-Yen, R. R. Dasari, K. J. Lee, and W. Choi, “Scanner-Free and Wide-Field Endoscopic Imaging by Using a Single Multimode Optical Fiber,” Phys. Rev. Lett. 109, 203901 (2012). 5. C. LeFort, T. Mansuryan, F. Louradour, and A. Barthelemy, “Pulse compression and fiber delivery of 45 fs Fourier transform limited pulses at 830nm,” Opt. Lett. 36, 292–294 (2011). 6. D. R. Rivera, C. M. Brown, D. G. Ouzounov, I. Pavlova, D. Kobat, W. W. Webb, and C. Xu, “Compact and flexible raster scanning multiphoton endoscope capable of imaging unstained tissue,” Proc. Natl. Acad. Sci. U.S.A. 108, 17598–17603 (2011). 7. C. M. Brown, D. R. Rivera, I. Pavlova, D. G. Ouzonov, W. O. Williams, S. Mohanan, W. W. Webb, and C. Xu, “In vivo imaging of unstained tissues using a compact and flexible multiphoton microendoscope,” J. Biomed. Opt. 17, 040505 (2012). 8. Y. Wu, Y. Leng, J. Xi, and X. Li, “Scanning all-fiber-optic endomicroscopy system for 3D nonlinear optical imaging of biological tissues,” Opt. Express 17, 7907–7915 (2009). 9. S. Tang, W. G. Jung, D. McCormick, T. Q. Xie, J. P. Su, Y. C. Ahn, B. J. Tromberg, and Z. P. Chen, “Design an implementation of fiber-based multiphoton endoscopy with microelectromechanical systems scanning,” J. Biomed. Opt. 14, 34005 (2009). 10. H. Bao, A. Boussioutas, R. Jeremy, S. Russell, and M. Gu, “Second harmonic generation imaging via nonlinear endomicroscopy,” Opt. Express 18, 1255–1260 (2010). 11. H. Bao, J. Allen, R. Pattie, R. Vance, and M. Gu, “Fast handheld two-photon fluorescence microendoscope with a 475 × 475 μ m field of view for in vivo imaging,” Opt. Lett. 33, 1333–1335 (2008). 12. M. R. Myaing, D. J. MacDonald, and X. Li, “Fiber-optic scanning two-photon fluorescence endoscope,” Opt. Lett. 31, 1076–1078 (2006). 13. L. Fu, A. Jain, H. Xie, C. Cranfield, and M. Gu, “Nonlinear optical endoscopy based on a double-clad photonic crystal fiber and a MEMS mirror,” Opt. Express 14, 1027–1032 (2006).

#193195 - $15.00 USD Received 1 Jul 2013; revised 1 Aug 2013; accepted 4 Aug 2013; published 27 Aug 2013 (C) 2013 OSA 9 September 2013 | Vol. 21, No. 18 | DOI:10.1364/OE.21.020713 | OPTICS EXPRESS 20713

14. B. A. Flusberg, E. D. Cocker, W. Piyawattanametha, J. C. Jung, E. L. M. Cheung, and M. J. Schnitzer, “Fiberoptic fluorescence imaging,” Nat. Methods 2, 941–950 (2005). 15. T. Y. Fan, “Laser beam combining for high-power, high radiance sources,” IEEE J. Sel. Top. Quantum Electron. 11, 567–577 (2005). 16. C. Labaune, D. Hulin, A. Galvanauskas, and G. Mourou, “On the feasability of a fiber-based inertial fusion laser driver,” Opt. Commun. 281, 4075–4080 (2008). 17. J. Bourderionnet, C. Bellanger, J. Primot, and A. Brignon, “Collective coherent phase combining of 64 fibers,” Opt. Express 19, 17053–17058 (2011). 18. A. J. Thompson, C. Paterson, M. A. A. Neil, C. Dunsby, and P. M. W. French, “Adaptive phase compensation for ultracompact laser scanning endomicroscopy,” Opt. Lett. 36, 1707–1709 (2011). 19. E. R. Andresen, G. Bouwmans, S. Monneret, and H. Rigneault, “Toward endoscopes with no distal optics: videorate scanning microscopy through a fiber bundle,” Opt. Lett. 38, 609–611 (2013). 20. D. J. Richardson, J. M. Fini, and L. E. Nelson, “Space-division multiplexing in optical fibres,” Nat. Photonics 7, 354–362 (2013). 21. P. S. J. Russell, “Photonic-Crystal Fibers,” J. Lightwave Technol. 24, 4729–4749 (2006). 22. J. Tang, R. N. Germain, and M. Cui, “Superpenetration optical microscopy by iterative multiphoton adaptive compensation technique,” Proc. Natl. Acad. Sci. U.S.A. 109, 8434–8439 (2012).

1.

Introduction

An endoscope consists of a long, narrow, often flexible, endoscope probe. Its task is to bring information about the image of an object located at the far tip (the distal end) of the endoscope probe to the near end (the proximal end). Endoscopes based on linear image contrast e.g. scattering or autofluorescence are in wide spread use as powerful tools that provide quick access to images of hollow organs without the need for surgical tissue removal, aiding e.g. in the localization or early detection of disease. (see e.g. [1] and references therein). Miniaturization of the endoscope probe would widen the range of accessible organs and enlarge the list of potential applications. This has recently fueled a quest for so called lensless endoscopes i.e. endoscopes without optics or other elements attached to the distal tip of the endoscope probe. Notably, several methodologies based on multi-mode fiber (MMF) have recently been proposed [2–4]. While these references achieve impressive spatial control through a complex media like an MMF, they are limited to linear contrast. A lensless endoscope able to achieve nonlinear contrast must support ultrashort optical pulses of fs-duration and hence be based on single-mode fibers (SMF)s where mode dispersion is not a problem and pre-compensation of chromatic dispersion and nonlinearity can ensure that a short, intense pulse is delivered to the sample as was done in e.g. [5] for a single SMF. Researchers are now actively pushing endoscopy towards multi-photon imaging modes, especially two-photon excited fluorescence (TPEF) imaging, using methodologies derived from existing endoscopic methods (with distal optics) [6–14]. As many multi-photon microscopy methods have been invented in recent years to complement standard microscopy, so it is likely that the coming years will see multi-photon endoscopes emerging to complement existing endoscopes. The initial report of lensless endoscopy based on a bundle of SMFs and concepts from coherent laser beam combining [15–17] was published in [18]. We recently demonstrated that the same method, with an optimized bundle, also allows video-rate imaging by point scanning [19]. These recent innovations in lensless endoscopy come at a time where there is also a burgeoning interest in multi-core or multi-mode fibers for large scale applications, in particular telecommunication [20]. [18, 19] both did their demonstration experiments with continuous-wave lasers, linear image contrast and distal collection of signal. In this paper we report the extension of lensless endoscopy based on bundles of SMFs to pulsed lasers and true endoscopic operation (with proximal signal collection) and demonstrate TPEF endoscopic imaging of a test sample of rhodamine 6G (Rh6G) crystals. We thus demonstrate for the first time how nonlinear image contrast can be achieved in a lensless endoscope.


2.

Experimental

a)

λ/2

f1

f2

f3

f4

2D-SLM

b)

Fs-laser f5 ID

f7 f6

LP

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to PC f8

OP f9 Bundle f10

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f8

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APD

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f10

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Multimode inner cladding

d) a)

x5

50 μm Single-mode fibers

b) e)

50 μm

50 μm

Fig. 1. (a) Sketch of the experimental setup. f1 = 100 mm; f2 = 200 mm; f3 = 30 mm; f4 = 200 mm; f5 = 500 mm; f6 = 75 mm; f7 = 100 mm; f8 = 6.24 mm; f9 = 20× objective (LucPlanFLN, Olympus); f10 = 150 mm; Bundle, 30 cm bundle. 2D-SLM, twodimensional spatial light modulator; ID, iris diaphragm; OP, object plane; LP, Long-pass filter (FEL1000, Thorlabs); SP, Short-pass filters (2 FM01, Thorlabs; SP945, Semrock); CMOS, CMOS camera; APD, avalanche photodiode (SPCM-AQRH-13, Perkin-Elmer); PC, personal computer. Dashed line represents back-scattered light. (b) Example of a mask on the 2D-SLM. (c) Scanning electron micrograph of the bundle with inverted contrast, for clarity. (d) Image of OP before phase calibration with a 5× gain, for clarity. (e) Image of OP after phase calibration.

Figure 1(a) shows a sketch of the experimental setup. The pulses from an Yb fs-laser (Amplitude Systèmes t-Pulse, 1035 nm, 150 fs, 50 MHz, 1.3 W, horizontally polarized) pass through magnifying telescopes ( f1 , f2 , f3 , f4 ) so that the collimated beam incident on the two-dimensional spatial light modulator (2D-SLM, X8267, Hamamatsu, 768×768 pixels, 20×20 mm2 ) overfills the aperture. On the 2D-SLM a phase mask Fig. 1(b) is inscribed which consists of N = 169 hexagonal segments of chirped gratings on a triangular grid. The phase of segment i centered on (Xi ,Yi ) on the 2D-SLM is of the form mask Φmask i π proximal (X − Xi )2 + (Y −Yi )2 ) . (1) (X,Y ) = sawtooth φi + 2π fc (X − Xi ) + Φmask i λ fconc Where fc = 1/13 pix−1 is a spatial carrier frequency; and fconc = 300 mm is the focal length of the concave mirror that results in the first order. Each segment i will then diffract a beamlet


with phase φi into the first order (diffraction efficiency around 0.16) and impose on it a convergence. In the first order, the entire 2D-SLM thus acts as an array of concave mirrors producing a hexagonal array of spots at a distance fconc from the 2D-SLM. This spot pattern is imaged ( f5 , f6 , f7 , f8 ) onto the proximal end face of the bundle Fig. 1(c). This approach allows more efficient use of laser light as compared to previous approaches where the SLM itself was imaged on the bundle [18, 19]. An iris diaphragm cuts away all diffracted orders but the first. At the distal end of the bundle, a CMOS camera is conjugated to the object plane OP located a distance zOP = 500 μ m from the bundle distal end face ( f9 , f10 ). This is used for inspection and positioning of the sample as well as phase calibration. Backscattered light (dashed line) is collected at the proximal end by a reflective dielectric filter and the proximal end face of the bundle is re-imaged on an APD ( f8 , f7 , f10 , f8 , effective de-magnification 1.5×). The multimode inner cladding (Ø250 μ m, see below for details) is thus imaged onto a Ø167 μ m spot on the APD (Ø175 μ m). Shortpass filters in front of the APD transmit two-photon fluorescence while blocking any residual laser light. Figure 1(c) shows an electron micrograph of the double-clad silica bundle of N = 169 SMFs with low cross-talk that was designed especially for this experiment. It was fabricated by the stack and draw process [21]. First, Ge-doped rods drawn from a preform (Prysmian Group, parabolic refractive index profile, maximum difference of 31·10−3 compared to silica) were inserted into a stack of 169 capillaries. This “double stacking” step was required in order to achieve very low coupling between the single-mode cores of the final fiber bundle. This multicore preform was then drawn into several all-solid canes. One of these canes was put in a jacketing tube containing the air-clad structure (made by stacking 61 hollow capillaries between two tubes) and drawn to get a bundle with the following parameters. Pitch Λ = 11.8 μ m; Mode field diameter of each SMF 3.6 μ m and corresponding beam divergence 0.12 rad; diameter of multi-mode inner cladding 250 μ m; numerical aperture (NA) of multi-mode inner cladding 0.65; total outer diameter 360 μ m. The measured coupling from one SMF to its nearest neighbor in the piece of bundle used in the experiments was less than -25 dB even when the bundle was coiled into a loop with a 12.5 mm radius of curvature. The presence of the multi-mode inner cladding around the SMFs means that any light that is not coupled into an SMF will instead be coupled into the multi-mode inner cladding where it will propagate to the sample and appear as a wide, background illumination. Efficient coupling into the SMFs is therefore important and we can couple into all 169 SMFs simultaneously with good discrimination. To quantify this, we acquired an image of the distal end face of the proximally illuminated bundle using the CMOS camera. In the image the SMFs appear as bright spots and the multi-mode inner cladding as a fairly flat plateau. From it we measured the relative SMF energy as the integrated intensity over all the SMFs divided by the total integrated intensity. With proper optimization of the 2D-SLM mask Fig. 1(b) we get a relative SMF energy of 62 %. Another good test of the quality of the 2D-SLM mask is to laterally displace it and confirm that in doing so all SMFs are extinguished at the same time. This is indeed what we observe. Furthermore when the mask is displaced by half the pitch the relative SMF energy is only 1 %. Due to the low coupling between SMFs in the bundle [19] there is strong correlation beproximal and the phase of the corresponding beamlet at the distal end φidistal since tween φi φidistal = φiproximal + δ φi . δ φi are static phase offsets stemming e.g. from bundle bends and intrinsic refractive index differences between individual SMFs. As a result, a priori the spot pattern in OP is a speckle pattern, Fig. 1(d). We remind that as a consequence of the low coupling the beamlets propagating in all other cores j = i have no influence on φidistal . With an initial phase calibration it is then possible to compensate for the random δ φi . The phase calibration proceeds as follows. Pairs of SMFs (0, i) are illuminated sequentially. For each pair, the CMOS camera takes an image of the fringe pattern at OP and an algorithm based on spatial fast Fourier proximal


transform extracts phase information at the spatial frequency of the fringe, see App. A for details. The calibration takes about two minutes for N = 169, limited mainly by the 2D-SLM update rate. After the calibration all beamlets interfere constructively in (x,y,z) = (0,0,zOP ), Fig. 1(e), (Note, the bundle was not polarization maintaining, so there was an orthogonally polarized background of about 13 ) and the following phase profile is established at the distal end. 2π 2 φidistal (xi , yi ) = − xi + y2i + z2OP − zOP λ π ≈ − (x2 + y2i ) λ zOP i 1 (2) 2 φ (xi + y2i ). (2) ≡ 2 Which mimicks the presence of a lens with focal length fdistal = zOP at the distal end face of the bundle. At this stage any desired change in φidistal (xi , yi ) can be imposed by imposing the same proximal (xi , yi ) i.e. by inscribing a different mask on the 2D-SLM, and any desired change on φi phase profile can be established. Image formation proceeds by point-scanning of the distal focus. Deflection of the focused (1) (1) beam can be achieved by imposing phase tilts φx and φy in the x- and y directions, i.e. by setting 1 (1) (1) φidistal (xi , yi ) = φx xi + φy yi + φ (2) (xi2 + y2i ). (3) 2 (1)

(1)

And by scanning φx , φy from −π /Λ to π /Λ in a rectangular grid, the distal focus is scanned in the same manner. All 169 cores remain illuminated during the point-scanning, there are no moving parts at the distal end as the point-scanning is achieved by manipulation of the 169 φi only. Here, we used the 2D-SLM itself to realize the point-scanning meaning that the mask Eq. (1) was updated for every pixel and that the image acquisition rate was limited by the 2D-SLM update rate to a few Hz. The image in Fig. 1(e) can be regarded as an approximation of the point-spread function (PSF) of the device. It consists of a bright, central spot surrounded by weaker replicas in a periodic pattern; this is a direct consequence of the periodicity of the bundle. The distance between the main peak and the first ring of satellite peaks is 53 μ m at fdistal = 500 μ m. The peak-satellite distance also corresponds to the field of view. The central spot is 7.4 times more intense than the maximum of the speckle intensity; 102 times more intense than the average speckle; and 1.4 times more intense than the nearest satellites. The average power in the central spot is 2 mW and its typical size 4 μ m which also corresponds to the lateral resolution. 3.

Results

To demonstrate two-photon endoscopic imaging, we place a sample of Rh6G crystals in OP. Figure 2(a) shows a white-light image of the crystal used as target in the subsequent measurements. We recorded the TPEF signal from the Rh6G crystal while using the 2D-SLM and Eq. (2) to scan fdistal . Figure 2(b) shows the result. The obtained curve is centered around fdistal = zOP = 500 μ m, the position of the crystal and has a full-width at half-maximum (FWHM) of 90 μ m. This number represents an estimate of the axial resolution. The curve is slightly assymetric because in the lensless endoscope, the effective numerical aperture for focusing changes reciprocally with fdistal . We then verified that the signal had the expected quadratic dependence on the input power; the curve of TPEF signal versus input power is presented in Fig. 2(c). A fit to the measurement series gave an x2.0 -dependence, proving that a twophoton process is at play. To further confirm TPEF origin of the signal, we examined its polar#193195 - $15.00 USD Received 1 Jul 2013; revised 1 Aug 2013; accepted 4 Aug 2013; published 27 Aug 2013 (C) 2013 OSA 9 September 2013 | Vol. 21, No. 18 | DOI:10.1364/OE.21.020713 | OPTICS EXPRESS 20717

Distal det. Proximal det.

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Fig. 2. TPEF endoscopic images of a Rh6G crystal. (a) White-light image of the crystal. (b) Rh6G TPEF signal (proximal detection) versus the focal length fdistal of the lensless endoscope. (c) Power dependence of the Rh6G TPEF signal. Note that the scales are logarithmic. (d) Endoscopic (proximal detection) TPEF images of the Rh6G crystal for different focal lengths. (e) Same as (d) but with distal detection, for comparison. Scale bar 20 μ m.

ization dependence (not shown) and found that it was unpolarized. A spectrum of the signal (not shown) showed that the wavelengths of emitted light lay in the range 600 nm–720 nm (edge of filter) further supporting that the origin is two-photon fluorescence. No second-harmonic generation at half the laser wavelength was observed. We then acquired a series of endoscopic TPEF images for seven different fdistal in the range 394 μ m to 685 μ m. These images are displayed in Fig. 2(d). When fdistal = zOP = 500 μ m we get the highest image intensity and we observe a structure resembling the white-light image Fig. 2(a), as expected. And as the focal length is increased or decreased, image intensity drops and blur sets in. The images presented are raw data. Note that there is virtually zero background. For the purpose of comparison, we acquire images of the same sample, under the same circumstances (though not simultaneously) with the detector placed at the distal end. These images are presented in Fig. 2(e). Qualitatively similar results appear. A point to note is that the image intensity is not dramatically different from Fig. 2(d) to Fig. 2(e). This is a testimony to the effectiveness of the double-clad bundle at collecting back-scattered signal photons. We remind that in our bundle, virtually 100 % of the bundle surface area contributes to the collection. A second point to note is the presence of a weak background in the images with distal detection. This background had a quadratic dependence upon the input power, and we therefore assign it to two-photon autofluorescence in the bundle. Note, that our approach reduces this background linearly in the number of fibers N of the bundle, compared to endoscopes sending all excitation light through only one fiber at a time. This autofluorescence background is greatly reduced in the proximally detected images for reasons of (i) low amount of scatterers in the sample and (ii) divergence in conjunction with collection efficiency. This conclusion applies to all other possible nonlinearities in the SMFs; no spectral broadening due to self-phase modulation is


apparent in the spectra after the bundle (not shown). 4.

Conclusion and outlook

We have reported the first experimental demonstration of a two-photon lensless endoscope. We stress the fact that this is a true lensless endoscope—all optics, mechanics, and detectors are situated before the bundle that acts as flexible endoscopic probe. The only necessary operation to be done at the distal end was the phase calibration. But there is no conceptual obstacles to a phase calibration with proximal optics only, e.g. by iterative optimization of sample twophoton fluorescence [22] which, incidentally, would also be beneficial for compensating sample scattering. At the moment the performance of the presented two-photon lensless endoscope does not equal that of other recently demonstrated two-photon endoscopes e.g. the fiber-scanning twophoton endoscope [7] where frames were acquired in vivo at 4.1 Hz with 75 mW in a 1 μ m spot. We obtain only 2 mW in a 4 μ m spot and thus do not have high enough intensity to achieve high framerate in two-photon imaging, although we previously demonstrated how to achieve video-rate imaging in one-photon lensless endoscopy [19]. Our endoscope is limited by the equipment employed, notably the 2D-SLM where 84 % of light is lost into other diffracted orders, and the laser itself whose center wavelength and pulse duration (150 fs) are not adapted to effective excitation of commonly used two-photon fluorophores. We applied no dispersion pre-compensation, although the 150 fs pulse duration increases by a factor 1.25 in the 30 cm long bundle. We are confident that addressing those issues will bring our endoscope much closer to useful applications. We note that in the case where the bundle is free to flex, the use of ultrashort pulses can become a problem because of curvature-induced differences in the group velocity or optical path length which can lead to loss of temporal overlap at the distal end. At the moment, we are not compensating for this and instead keep the bundle fairly straight meaning that there was less than one half-loop on the bundle. Many other improvements are possible and desirable, especially on the subject of the bundle—a bundle with smaller pitch but without cross-talk, polarization-maintaining SMFs would lead to immediate improvements. As would higher NA of the SMFs because the size of the distal focus is given by an effective NA for focusing that is approximately equal to the NA of a single SMF. Our proof of principle two-photon lensless endoscope opens the door to many other nonlinear image contrasts in lensless endoscopes. Any kind of nonlinear microscopy that can proceed by point-scanning (e.g. three-photon fluorescence, second-harmonic generation, and thirdharmonic generation microscopy) could straightforwardly be implemented in our endoscope. We reckon that this kind of lensless endoscope—owing to its flexibility, small diameter, and wavefront shaping capability—could one day find potential in demanding deep imaging applications where space is constrained and scattering high, such as the brain of living animals. Acknowledgments Financial support from the CNRS interdisciplinary mission, the Weizmann NaBi European Associated Laboratory, ANR grant France Bio Imaging (ANR-10-INSB-04-01) and France Life Imaging (ANR-11-INSB-0006) Infrastructure networks, and the Région PACA is gratefully aknowledged. GB acknowledges financial support from Région Nord-Pas de Calais and FEDER (CPER 2007–2013).


Appendix A.

Phase calibration

From SMF i a beamlet emerges which we describe as 2π ( x−xi x+ y−yi y)+iφ i z z

Ei (x, y, z) = Ei (x − xi , y − yi , z)ei λ

(4)

where we are assuming that z xi and Ei is a real Gaussian-like function. When SMFs 0 and i are illuminated simultaneously, Fig. 3(a), the following fringe pattern is observerved in the plane z = zOP (setting (x0 ,y0 ) = (0,0) and φ0 = 0) I0i (x, y, zOP ) = |E0 (x, y, zOP ) + Ei (x, y, zOP )|2 = |E0 (x, y, zOP )|2 + |Ei (x − xi , y − yi , zOP )|2 2π +2E0 (x, y, zOP )Ei (x − xi , y − yi , zOP )cos (xi x + yi y − φi ) . λ zOP

(5)

Figure 3(b) shows a calculated example of the fringe pattern. Note that in this notation the bright fringe is at (0,0) when φi = 0. Therefore, we wish to determine φi and subsequently apply a corrective phase with the same magnitude but opposite sign. Doing so for all i will set all bright fringes in the center and establish the desired focus. I0i (x, y, zOP ) is 2-D Fourier transformed to yield (6) I˜0i (kx , ky , zOP ) = F I0i (x, y, zOP ) . Figure 3(c) shows a calculated example of |I˜0i | which contains peaks pertaining from the fringe pattern at the spatial frequencies (kx , ky ) = ±( λ2πz xi , λ2πz yi ) with phase ∓φi . For each pair (0,i) OP OP φi is then found as Arg[I˜0i (− 2π xi , − 2π yi , zOP )]. Subsequently, we illuminate all N SMFs and λ zOP

λ zOP proximal

= −φi and as a result all beamlets interfere conin the 2D-SLM mask Eq. (1) we set φi structively in (x,y,z) = (0,0,zOP ), Fig. 1(e) and Eq. (2). In Figs. 4(a)-4(c) are shown examples equivalent to Figs. 3(a)-3(c), but on experimentally acquired data.

I0i (x, y, 0)

I˜0i (kx , ky , zOP )

I0i (x, y, zOP )

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b)

c)

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ky [a.u.]

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y [a.u.]

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0 kx [a.u.]

Fig. 3. Calculations illustrating of the phase calibration procedure. (a) Intensity distribution in the distal end plane of the bundle. (b) Intensity distribution in the plane OP. (c) Absolute value of the 2D Fourier transform of (b), with labels marking the peaks where the phase φi can be read off from the phase of the Fourier transform.


I0i (x, y, 0) a)

c)

0

0 x [a.u.]

ky [a.u.]

b) y [a.u.]

y [a.u.]

I˜0i (kx , ky , zOP )

I0i (x, y, zOP )

0

0 x [a.u.]

0

0 kx [a.u.]

Fig. 4. Illustration of the phase calibration procedure on experimental data. (a) Intensity distribution in the distal end plane of the bundle. (b) Intensity distribution in the plane OP. (c) Absolute value of the 2D Fourier transform of (b).
