Two-photon lensless endoscope

Transcription

1 Two-photon lensless endoscope Esben Ravn Andresen, 1,3 Géraud Bouwmans, 2 Serge Monneret, 1 and Hervé Rigneault 1, 1 Aix-Marseille Université, CNRS, Centrale Marseille, Institut Fresnel, UMR 7249, 1313 Marseille, France 2 PhLAM CNRS, IRCICA, Université Lille 1, Villeneuve d Ascq Cedex, France 3 esben.andresen@fresnel.fr herve.rigneault@fresnel.fr 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. 213 Optical Society of America OCIS codes: (6.243) Fibers, single-mode; (17.215) Endoscopic imaging; (18.252) Fluorescence microscopy; ( ) Nonlinear microscopy; (18.581) Scanning microscopy; (23.612) Spatial light modulators. References and links 1. S. F. Elahi and T. D. Wang, Future and advances in endoscopy, J. Biomed. Opt. 4, (211). 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, (213). 3. T. Cizmar and K. Dholakia, Exploiting multimode waveguides for pure fibre-based imaging, Nat. Commun. 3, 127 (212). 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. 19, 2391 (212). 5. C. LeFort, T. Mansuryan, F. Louradour, and A. Barthelemy, Pulse compression and fiber delivery of 45 fs Fourier transform limited pulses at 83nm, Opt. Lett. 36, (211). 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. 18, (211). 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, 455 (212). 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, (29). 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, 345 (29). 1. H. Bao, A. Boussioutas, R. Jeremy, S. Russell, and M. Gu, Second harmonic generation imaging via nonlinear endomicroscopy, Opt. Express 18, (21). 11. H. Bao, J. Allen, R. Pattie, R. Vance, and M. Gu, Fast handheld two-photon fluorescence microendoscope with a μm field of view for in vivo imaging, Opt. Lett. 33, (28). 12. M. R. Myaing, D. J. MacDonald, and X. Li, Fiber-optic scanning two-photon fluorescence endoscope, Opt. Lett. 31, (26). 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, (26). # $15. USD Received 1 Jul 213; revised 1 Aug 213; accepted 4 Aug 213; published 27 Aug 213 (C) 213 OSA 9 September 213 Vol. 21, No. 18 DOI:1.1364/OE OPTICS EXPRESS 2713

2 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, (25). 15. T. Y. Fan, Laser beam combining for high-power, high radiance sources, IEEE J. Sel. Top. Quantum Electron. 11, (25). 16. C. Labaune, D. Hulin, A. Galvanauskas, and G. Mourou, On the feasability of a fiber-based inertial fusion laser driver, Opt. Commun. 281, (28). 17. J. Bourderionnet, C. Bellanger, J. Primot, and A. Brignon, Collective coherent phase combining of 64 fibers, Opt. Express 19, (211). 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, (211). 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, (213). 2. D. J. Richardson, J. M. Fini, and L. E. Nelson, Space-division multiplexing in optical fibres, Nat. Photonics 7, (213). 21. P. S. J. Russell, Photonic-Crystal Fibers, J. Lightwave Technol. 24, (26). 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. 19, (212). 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 [2]. [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. # $15. USD Received 1 Jul 213; revised 1 Aug 213; accepted 4 Aug 213; published 27 Aug 213 (C) 213 OSA 9 September 213 Vol. 21, No. 18 DOI:1.1364/OE OPTICS EXPRESS 2714

3 2. Experimental a) Fs-laser λ/2 f 1 f 2 f 3 f 4 2D-SLM b) f 5 ID f 7 f 8 OP f9 to PC f 6 LP f 1 Bundle PC f 8 SP f 8 SP f 1 f 1 APD CMOS c) Multimode inner cladding d) a) x5 e) b) 5 μ m 5 μm Single-mode fibers 5 μm Fig. 1. (a) Sketch of the experimental setup. f 1 = 1 mm; f 2 = 2 mm; f 3 =3mm; f 4 = 2 mm; f 5 = 5 mm; f 6 = 75 mm; f 7 = 1 mm; f 8 = 6.24 mm; f 9 =2 objective (LucPlanFLN, Olympus); f 1 = 15 mm; Bundle, 3 cm bundle. 2D-SLM, twodimensional spatial light modulator; ID, iris diaphragm; OP, object plane; LP, Long-pass filter (FEL1, Thorlabs); SP, Short-pass filters (2 FM1, 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, 135 nm, 15 fs, 5 MHz, 1.3 W, horizontally polarized) pass through magnifying telescopes ( f 1, f 2, f 3, f 4 ) so that the collimated beam incident on the two-dimensional spatial light modulator (2D-SLM, X8267, Hamamatsu, pixels, 2 2 mm 2 ) 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 mask Φ mask i of segment i centered on (X i,y i ) on the 2D-SLM is of the form Φ mask i (X,Y )=sawtooth [ φ proximal i + 2π f c (X X i )+ π ( )] (X X i ) 2 +(Y Y i ) 2 ). (1) λ f conc Where f c = 1/13 pix 1 is a spatial carrier frequency; and f conc = 3 mm is the focal length of the concave mirror that results in the first order. Each segment i will then diffract a beamlet # $15. USD Received 1 Jul 213; revised 1 Aug 213; accepted 4 Aug 213; published 27 Aug 213 (C) 213 OSA 9 September 213 Vol. 21, No. 18 DOI:1.1364/OE OPTICS EXPRESS 2715

4 with phase φ proximal i into the first order (diffraction efficiency around.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 f conc from the 2D-SLM. This spot pattern is imaged ( f 5, f 6, f 7, f 8 ) 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 z OP = 5 μm from the bundle distal end face ( f 9, f 1 ). 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 ( f 8, f 7, f 1, f 8, effective de-magnification 1.5 ). The multimode inner cladding (Ø25 μm, see below for details) is thus imaged onto a Ø167 μm spot on the APD (Ø175 μm). 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 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 μm; Mode field diameter of each SMF 3.6 μm and corresponding beam divergence.12 rad; diameter of multi-mode inner cladding 25 μm; numerical aperture (NA) of multi-mode inner cladding.65; total outer diameter 36 μm. 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 between φ proximal i and the phase of the corresponding beamlet at the distal end φi distal since φi distal = φ proximal i + δφ i. δφ i 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 φi distal. With an initial phase calibration it is then possible to compensate for the random δφ i. The phase calibration proceeds as follows. Pairs of SMFs (, 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 # $15. USD Received 1 Jul 213; revised 1 Aug 213; accepted 4 Aug 213; published 27 Aug 213 (C) 213 OSA 9 September 213 Vol. 21, No. 18 DOI:1.1364/OE OPTICS EXPRESS 2716

5 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) = (,,z OP ), Fig. 1(e), (Note, the bundle was not polarization maintaining, so there was an orthogonally polarized background of about 1 3 ) and the following phase profile is established at the distal end. φi distal (x i,y i ) = 2π ( ) xi 2 λ + y2 i + z2 OP z OP π (xi 2 + y 2 i ) λz OP 1 2 φ (2) (x 2 i + y 2 i ). (2) Which mimicks the presence of a lens with focal length f distal = z OP at the distal end face of the bundle. At this stage any desired change in φi distal (x i,y i ) can be imposed by imposing the same change on φ proximal i (x i,y i ) i.e. by inscribing a different mask on the 2D-SLM, and any desired phase profile can be established. Image formation proceeds by point-scanning of the distal focus. Deflection of the focused beam can be achieved by imposing phase tilts φ x (1) and φ y (1) in the x- and y directions, i.e. by setting φi distal (x i,y i )=φ x (1) x i + φ y (1) y i φ (2) (xi 2 + y 2 i ). (3) And by scanning φ x (1), φ y (1) 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 φ i 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 μm at f distal = 5 μm. 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; 12 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 μm 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 f distal. Figure 2(b) shows the result. The obtained curve is centered around f distal = z OP = 5 μm, the position of the crystal and has a full-width at half-maximum (FWHM) of 9 μm. 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 f distal. 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 x 2. -dependence, proving that a twophoton process is at play. To further confirm TPEF origin of the signal, we examined its polar- # $15. USD Received 1 Jul 213; revised 1 Aug 213; accepted 4 Aug 213; published 27 Aug 213 (C) 213 OSA 9 September 213 Vol. 21, No. 18 DOI:1.1364/OE OPTICS EXPRESS 2717

6 a) TPEF intensity [kcps] 1 5 b) FWHM = 9 μm 5 1 f distal [μm] TPEF intensity [a.u.] c) y x P in [mw] 1 3 Proximal det. Z = 685 μm 61 μm 55 μm 5 μm 459 μm 424 μm 394 μm d) [kcps] 2 Distal det. e) 2 μm 1 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 f distal 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 2 μm. 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 6 nm 72 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 f distal in the range 394 μm to 685 μm. These images are displayed in Fig. 2(d). When f distal = z OP = 5 μm 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 1 % 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 # $15. USD Received 1 Jul 213; revised 1 Aug 213; accepted 4 Aug 213; published 27 Aug 213 (C) 213 OSA 9 September 213 Vol. 21, No. 18 DOI:1.1364/OE OPTICS EXPRESS 2718

7 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 μm spot. We obtain only 2 mw in a 4 μm 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 (15 fs) are not adapted to effective excitation of commonly used two-photon fluorophores. We applied no dispersion pre-compensation, although the 15 fs pulse duration increases by a factor 1.25 in the 3 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-1-INSB-4-1) and France Life Imaging (ANR-11-INSB-6) Infrastructure networks, and the Région PACA is gratefully aknowledged. GB acknowledges financial support from Région Nord-Pas de Calais and FEDER (CPER ). # $15. USD Received 1 Jul 213; revised 1 Aug 213; accepted 4 Aug 213; published 27 Aug 213 (C) 213 OSA 9 September 213 Vol. 21, No. 18 DOI:1.1364/OE OPTICS EXPRESS 2719

8 Appendix A. Phase calibration From SMF i a beamlet emerges which we describe as E i (x,y,z)=e i (x x i,y y i,z)e i 2π λ ( x x i z x+ y y i z y)+iφ i (4) where we are assuming that z x i and E i is a real Gaussian-like function. When SMFs and i are illuminated simultaneously, Fig. 3(a), the following fringe pattern is observerved in the plane z = z OP (setting (x,y ) = (,) and φ =) I i (x,y,z OP ) = E (x,y,z OP )+E i (x,y,z OP ) 2 = E (x,y,z OP ) 2 + E i (x x i,y y i,z OP ) 2 [ 2π ] +2E (x,y,z OP )E i (x x i,y y i,z OP )cos (x i x + y i y φ i ). (5) λz OP Figure 3(b) shows a calculated example of the fringe pattern. Note that in this notation the bright fringe is at (,) when φ i =. Therefore, we wish to determine φ i 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. I i (x,y,z OP ) is 2-D Fourier transformed to yield [ ] Ĩ i (k x,k y,z OP )=F I i (x,y,z OP ). (6) Figure 3(c) shows a calculated example of Ĩ i which contains peaks pertaining from the fringe pattern at the spatial frequencies (k x,k y )=±( 2πx i λz, 2πy i OP λz ) with phase φ OP i. For each pair (,i) φ i is then found as Arg[Ĩ i ( 2πx i λz, 2πy i OP λz,z OP OP )]. Subsequently, we illuminate all N SMFs and in the 2D-SLM mask Eq. (1) we set φ proximal i = φ i and as a result all beamlets interfere constructively in (x,y,z) = (,,z OP ), 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. I i (x, y, ) I i (x, y, z OP ) Ĩ i (k x,k y,z OP ) a) SMF i SMF b) c) φ i y [a.u.] y [a.u.] k y [a.u.] φ i x [a.u.] x [a.u.] k x [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 φ i can be read off from the phase of the Fourier transform. # $15. USD Received 1 Jul 213; revised 1 Aug 213; accepted 4 Aug 213; published 27 Aug 213 (C) 213 OSA 9 September 213 Vol. 21, No. 18 DOI:1.1364/OE OPTICS EXPRESS 272

9 I i (x, y, ) I i (x, y, z OP ) Ĩ i (k x,k y,z OP ) a) b) c) y [a.u.] y [a.u.] k y [a.u.] x [a.u.] x [a.u.] k x [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). # $15. USD Received 1 Jul 213; revised 1 Aug 213; accepted 4 Aug 213; published 27 Aug 213 (C) 213 OSA 9 September 213 Vol. 21, No. 18 DOI:1.1364/OE OPTICS EXPRESS 2721