Improved spot formation for flexible multimode fiber endoscope using partial reflector

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1 Impoved spot fomation fo flexible multimode fibe endoscope using patial eflecto RUO YU GU 1, *, ELAINE CHOU 1, CORY REWCASTLE 2, OFER LEVI 2, AND JOSEPH M. KAHN 1 1 E. L. Ginzton Laboatoy and Depatment of Electical Engineeing, Stanfod Univesity, Stanfod, CA Institute of Biomateials and Biomedical Engineeing, Univesity of Toonto, 164 College Steet, Toonto, Ontaio M5S 3G9, Canada *uoyugu@stanfod.edu Abstact: Multi-mode fibe (MMF) endoscopes ae a new type of endoscope that use only a single optical fibe to tansmit images, achieving much highe spatial esolution fo a given diamete than commecial fibe bundle endoscopes. Howeve, a dawback of MMF endoscopes is that imaging quality degades substantially as the MMF is petubed. We popose a method fo impoving spot fomation at the distal end of a petubed MMF, and thus impoving imaging quality, by attaching a patial eflecto to the distal end of the MMF. We expeimentally find that the petubation of the light eflected fom the patial eflecto is highly coelated with the petubation of the light tansmitted though the MMF. We demonstate a simple method exploiting this coelation that enables fomation of spots at the distal fibe end with quality 10-15% highe than if the petubation of the MMF is ignoed. In the futue, moe advanced algoithms exploiting the coelation may enable futhe impovements in spot fomation and imaging though petubed fibes. OCIS codes: ( ) Fibe optics imaging; ( ) Image econstuction techniques; ( ) Noise in imaging systems; ( ) Endoscopic imaging. Refeences and links 1. B. A. Flusbeg, E. D. Cocke, W. Piyawattanametha, J. C. Jung, E. L. M. Cheung, and M. J. Schnitze, "Fibe-optic fluoescence imaging," Nat. Methods 2, (2005). 2. E. R. Andesen, S. Sivankutty, G. Bouwmans, L. Gallais, S. Monneet, and H. Rigneault, "Measuement and compensation of esidual goup delay in a multi-coe fibe fo lensless endoscopy," J. Opt. Soc. Am. B 32, (2015). 3. Y. Kim, S. C. Waen, J. M. Stone, J. C. Knight, M. A. A. Neil, C. Pateson, C. W. Dunsby, and P. M. W. Fench, "Adaptive Multiphoton Endomicoscope Incopoating a Polaization-Maintaining Multicoe Optical Fibe," IEEE J. Sel. Top. Quantum Electon. 22, 1 8 (2016). 4. S. C. Waen, Y. Kim, J. M. Stone, C. Mitchell, J. C. Knight, M. A. A. Neil, C. Pateson, P. M. W. Fench, and C. Dunsby, "Adaptive multiphoton endomicoscopy though a dynamically defomed multicoe optical fibe using poximal detection," Opt. Expess 24, (2016). 5. D. B. Conkey, N. Stasio, E. E. Moales-Delgado, M. Romito, C. Mose, and D. Psaltis, "Lensless twophoton imaging though a multicoe fibe with coheence-gated digital phase conjugation," J. Biomed. Opt. 21, (2016). 6. N. Stasio, D. B. Conkey, C. Mose, and D. Psaltis, "Light contol in a multicoe fibe using the memoy effect," Opt. Expess 23, (2015). 7. S. M. Kolendeska, O. Katz, M. Fink, and S. Gigan, "Scanning-fee imaging though a single fibe by andom spatio-spectal encoding," Opt. Lett. 40, (2015). 8. R. N. Mahalati, R. Y. Gu, and J. M. Kahn, "Resolution limits fo imaging though multi-mode fibe," Opt. Expess 21, (2013). 9. M. Plöschne and T. Ĉiẑmá, "Compact multimode fibe beam-shaping system based on GPU acceleated digital hologaphy," Opt. Lett. 40, (2015). 10. A. M. Caavaca-Aguie, E. Niv, and R. Piestun, "High-Speed Phase Modulation fo Multimode Fibe Endoscope," Imaging Appl. Opt. (2014). 11. M. Plöschne, B. Staka, K. Dholakia, and T. Ĉiẑmá, "GPU acceleated toolbox fo eal-time beamshaping in multimode fibes," Opt. Expess 22, (2014). 12. D. Loteie, S. Faahi, I. Papadopoulos, A. Goy, D. Psaltis, and C. Mose, "Digital confocal micoscopy though a multimode fibe," Opt. Expess 23, (2015).

2 13. D. Loteie, D. Psaltis, and C. Mose, "Confocal micoscopy via multimode fibes: fluoescence bandwidth," SPIE BIOS 9717, 97171C (2016). 14. E. E. Moales-Delgado, D. Psaltis, and C. Mose, "Focusing and scanning of femtosecond pulses though a multimode fibe: applications in two-photon imaging and polymeization," pesented at the Austalian Confeence on Optical Fibe Technology, Sydney, Austalia, 6 8 Sep E. E. Moales-Delgado, D. Psaltis, and C. Mose, "Two-photon imaging though a multimode fibe," Opt. Expess 23, (2015). 16. E. E. Moales Delgado, D. Psaltis, and C. Mose, "Two-photon excitation endoscopy though a multimode optical fibe," SPIE BIOS 9717, 97171E (2016). 17. S. Sivankutty, E. R. Andesen, R. Cossat, G. Bouwmans, S. Monneet, and H. Rigneault, "Ulta-thin igid endoscope: two-photon imaging though a gaded-index multi-mode fibe," Opt. Expess 24, (2016). 18. S. Ohayon, A. M. Caavaca-Aguie, R. Piestun, and J. J. DiCalo, "Deep bain fluoescence imaging with minimally invasive ulta-thin optical fibes," axiv Pep. axiv (2017). 19. Y. Xu, Y. Lu, Z. Zhu, J. Shi, C. Guan, and L. Yuan, "Multimode fibe focusing lens based on plasmonic stuctues," SPIE/COS Photonics Asia, 10019, C (2016). 20. S. Bianchi, V. P. Rajamanickam, L. Feaa, E. Di Fabizio, C. Libeale, and R. Di Leonado, "Focusing and imaging with inceased numeical apetues though multimode fibes with mico-fabicated optics," Opt. Lett. 38, (2013). 21. Y. Choi, C. Yoon, M. Kim, W. Choi, and W. Choi, "Optical imaging with the use of a scatteing lens," IEEE J. Sel. Top. Quantum Electon. 20, (2014). 22. S. Rosen, D. Gilboa, O. Katz, and Y. Silbebeg, "Focusing and Scanning though Flexible Multimode Fibes without Access to the Distal End," axiv Pep. axiv (2015). 23. T. Čižmá and K. Dholakia, "Optical manipulation, beam-shaping and scanne-fee bight-field and dakfield imaging via multimode optical fibe," SPIE Photonics West, 8637, 86370G (2013). 24. D. Loteie, D. Psaltis, and C. Mose, "Bend tanslation in multimode fibe imaging," Opt. Expess 25, 6263 (2017). 25. W. Xiong, P. Ambichl, Y. Bombeg, B. Redding, S. Rotte, and H. Cao, "Pincipal modes in multimode fibes: exploing the cossove fom weak to stong mode coupling," Opt. Expess 25, (2017). 26. A. M. Caavaca-Aguie and R. Piestun, "Single multimode fibe endoscope," Opt. Expess 25, (2017). 27. R. Y. Gu, R. N. Mahalati, and J. M. Kahn, "Design of flexible multi-mode fibe endoscope," Opt. Expess 23, (2015). 28. D. Macuse, Theoy of Dielectic Optical Waveguides, 2nd ed. (Academic Pess Inc., 1991). 29. R. N. Mahalati, D. Askaov, J. P. Wilde, and J. M. Kahn, "Adaptive contol of input field to achieve desied output intensity pofile in multimode fibe with andom mode coupling.," Opt. Expess 20, (2012). 30. R. Y. Gu, R. N. Mahalati, and J. M. Kahn, "Noise-eduction algoithms fo optimization-based imaging though multi-mode fibe," Opt. Expess 22, (2014). 31. O. Geschke, H. Klank, and P. Telleman, eds., Micosystem Engineeing of Lab-on-a-Chip Devices (John Wiley & Sons, 2004). 32. R. Hoisaki, R. Takagi, and J. Tanida, "Leaning-based imaging though scatteing media," Opt. Expess 24, (2016). 33. M. Lyu, H. Wang, G. Li, and G. Situ, "Exploit imaging though opaque wall via deep leaning," axiv Pep. axiv (2017). 1. Intoduction Endoscopes ae medical instuments that tansmit images though long, thin optical conduits. Small diamete and high esolution ae impotant attibutes of endoscopes, paticulaly when imaging deep tissue egions inaccessible to othe imaging devices. Typical commecial endoscopes ae constucted using a bundle of single-mode fibes (SMFs), and have diametes as small as about 0.5 mm, achieving a spatial esolution of about one thousand pixels [1]. By contast, single-fibe endoscopes ae a new type of endoscope that use only a single optical fibe to tansmit images; typically this single optical fibe is a multi-mode fibe (MMF), though some methods use a multi-coe fibe (MCF) [2 6] o a SMF [7]. Single-fibe endoscopes offe a bette tade-off between diamete and spatial esolution than fibe bundle endoscopes: a single-fibe endoscope with 0.5 mm-diamete using a typical MMF would esolve moe than half a million pixels [8]. Despite these advantages, endoscopes using a

3 single MMF have weaknesses, including long imaging times, small field of view, and the equiement that the MMF be held igid. Reseach addessing all of these weaknesses has pogessed at a apid pace. To addess long imaging times, new techniques have been developed to incease imaging speed [9 11] and signal-to-noise atio (SNR) [12,13], in paticula, two-photon excitation imaging [14 17]; and impovements in key components, especially spatial-light modulatos, have substantially impoved both speed and accuacy in ecent yeas. In fact, both speed and SNR have impoved to the point that eal-time in vivo biological imaging though a single igid fibe has been demonstated [18]. In paallel, the field of view limitation is being addessed by the design of small passive optical components to be placed the distal end of the optical fibe [19 21]. Finally, many methods have been poposed to addess the igidity equiement to enable imaging though a flexible fibe. Among the most pomising appoaches ae: imaging though a gaded-index MMF by using multi-photon excitation endoscopy, exploiting the coelation popeties of gaded-index MMFs to ceate spots in known locations [22]; imaging though a SMF using boadband specta by encoding spatial infomation in diffeent wavelengths [7]; imaging though a flexible MCF by encoding image infomation in the diffeent coes of a MCF and compensating fo the change in elative phases [4]; and compensation fo mode coupling in MMF using a petubation-invaiant basis by monitoing the physical configuation of the MMF [23]. Recent eseach has also found that mode coupling though a MMF is supisingly weak fo small petubations [24 26]. Each method has stengths and weaknesses; fo example, spatio-spectal encoding equies that the imaged object have known spectal chaacteistics [7]; MCFs ae limited to appoximately 100 coes in ode to emain thin [4]; the method in [23] equies the MMF s physical configuation to be known [23]. In this pape, we extend and expeimentally demonstate a method fist descibed in [27] fo calibating a flexible MMF by placing a stuctued eflecto on the distal end of the MMF, which makes it possible to measue infomation on the mode coupling in the MMF based on measuement of light eflected fom the patial eflecto. A diagam showing how such a flexible endoscope would wok befoe and afte insetion into a sample to be imaged is shown in Fig. 1. The appaatus in Fig. 1 eflects two modifications as compaed to the appaatus descibed in [27]. Fist, a stuctued eflecto is not necessay, and a simple patial eflecto suffices. Second, the imaging modality is fluoescence imaging instead of eflective imaging; because the light eflected fom the patial eflecto and fluoescence fom the sample ae at diffeent wavelengths, the SNR is inceased and a shutte, as descibed in [27], is unnecessay. We show in this pape that the petubation of the modes eflected fom a patial eflecto at the distal end of the MMF is stongly coelated to the petubation of the modes tansmitted fowad though the MMF, as pedicted theoetically. This makes it possible to impove the quality of spots that can be fomed though a petubed MMF. The stong coelation between the petubation of the tansmitted and eflected light signals futhe suggests that a nea one-to-one mapping between the two may be possible, which would allow fo accuate imaging though a flexible MMF.

4 Isolato Befoe Insetion Step-Index MMF SLM Emission Filte Calibation Lase CCD Imaging Appaatus Reflecto CCD Isolato Afte Insetion Step-Index MMF Sample SLM Emission Filte Calibation Imaging Lase CCD Imaging Appaatus Reflecto Reflecto Fig. 1. Schematic showing the concept fo a flexible MMF endoscope. The long-pass emission filte shown in font of the CCD camea can be moved in o out of the beam path. In a calibation pefomed befoe the MMF is inseted into the sample, the tansmission of light though the MMF is measued by a camea at the distal end of the MMF, and the eflection of light fom the patial eflecto at the distal end of the MMF is measued by a camea at the poximal end of the MMF (without the emission filte). Afte this calibation, the MMF is inseted into the sample. Wheneve the MMF is petubed, the eflected light fom the patial eflecto is e-measued (without the emission filte), and this calibation allows imaging though the petubed MMF (with the emission filte). The imaging modality is fluoescence imaging, which means the light emitted fom the sample and collected by the MMF is at a longe wavelength than the light eflected fom the patial eflecto. Since light at the two wavelengths can be sepaated, the SNR is inceased and a shutte at the distal end of the MMF is not needed. The emainde of this pape is as follows. In Section 2, we biefly descibe the theoy descibing the popagation of light in a MMF that we use thoughout the pape. In Section 3, we descibe ou expeimental appaatus and ou pocedue fo coheent measuement of the wavefont of the light. In Section 4, we descibe ou method to compensate fo the petubation of the MMF using eflection measuements. In Section 5, we pesent ou expeimental esults, and in Section 6, we discuss the esults. We conclude in Section 7.

5 2. Mathematical backgound fo light popagation inside MMF Light popagating though a MMF can be concisely descibed in tems of the electic and magnetic fields of a set of othogonal guided modes. Fo convenience we define the poximal and distal ends of the MMF, a fowad diection (fom poximal to distal) along the length of the MMF, and the opposite diection as backwad, all as shown in Fig. 2. At evey point along the MMF we can then oient a local ight-handed Catesian coodinate system x, y, z whose +z axis is a tangent to the cental axis of the MMF and points in the fowad diection. x Poximal m in,p Fowad Backwad MMF m out,d Distal m tan. m out,p m in,d y z Patial Reflecto Fig. 2. Schematic of MMF, showing the Catesian coodinate axes, poximal and distal ends of the MMF, fowad and backwad diections, and the labeling of the mode vectos. A patial eflecto is attached diectly to the distal end of the MMF (with no ai gap). At each x, y, z location, the MMF suppots so-called local nomal modes [28], which ae mathematically equivalent to the ideal TE, TM, HE and EH modes of a staight MMF with the same coodinate system. The pecise E ( x, y, z, t) and H ( x, y, z, t) fields of the fowadpopagating modes ae descibed in [28], and equivalently using phaso notation as complex E ( x, y, z) and H ( x, y, z). Thee ae an equal numbe of backwad-popagating modes, * which have fields E ( x, y, z) * and H ( x, y, z), whee the asteisk epesents the complex conjugate. Each mode is nomalized to have unit powe. We keep tack of popagating modes only and ignoe adiation modes. Let N be the numbe of fowad-popagating modes in the MMF, which is also equal to the numbe of backwad-popagating modes. The amplitude and phase of a mode at one plane in the MMF is descibed by a single complex coefficient; and the amplitude, phase and polaization of the electic and magnetic fields at that plane inside the MMF ae thus epesented by a single complex 2N 1 vecto m. We can make two futhe simplifications. Fist, fo an endoscopic application we ae only concened with the modes at the poximal and distal facets of the MMF, just outside the fibe facet; fo MMFs with low numeical apetue (NA), at these facets it is convenient to use the basis of linealy polaized (LP) modes instead of the moe complicated basis of local nomal modes [28]. Second, since fowad and backwad popagating modes do not couple to each othe inside the MMF, it is convenient to sepaate m at the distal plane into N 1 vectos m in,d and m out,d, and m at the poximal plane into N 1 vectos m in,p and m out,p, as shown in Fig. 2. We ode the coefficients in each of these vectos as follows: LP modes in the x-polaization ae odeed by deceasing popagation constant, followed by LP modes in the y-polaization odeed by deceasing popagation constant. The mathematical elation between the mode coefficients at the poximal and distal ends of the MMF can then be completely descibed in matix notation using matices T and R: m Tmin, p m Rm, (1) out, p in, d

6 whee T is by definition the tansmission matix of the MMF and R is by definition the eflection matix of the MMF; both change if the MMF is physically petubed. Eq. (1) fully descibes the light at both ends of the MMF. Fo example, the (scaled) iadiance I of the fowad-popagating light at the distal end as it would appea on a popely calibated CCD placed at the distal plane can be computed as: 2 k k k 1 N I( x, y) E ( x, y) n ( m ) E ( x, y) n, (2) whee Ek ( xy, ) is the electic field of LP mode k and n is the unit nomal vecto to the CCD suface. 3. Endoscope appaatus and coheent measuement pocedues The appaatus fo measuing petubation of the endoscope is shown in Fig. 3. Fo convenience, the appaatus can be divided into thee sections: lase souce section, endoscope section, and camea section. All lenses and non-polaizing beamsplittes (NPBS) ae antieflection coated at visible wavelengths NPBS SLM 532-nm Lase L1 Shutte L2 Lin. Pol. L3 MMF1 Beam Stop Camea L8 L7 Beam Stop Shutte L4 Mio PBD NPBS L6 Mio (movable) L5 Shutte Beam Stop Petubation MMF2 Fig. 3. Diagam of expeimental setup fo imaging though a petubed multi-mode fibe (MMF2). The lase souce sections, endoscope section and camea sections ae shown. All thee shuttes ae shown open. All beamsplittes ae non-polaizing beamsplittes except the labeled non-polaizing beamsplitte. The lase souce section consists of a lase (single-longitudinal-mode, 200 mw, 532-nm, vetical polaization of beam), two achomatic doublet lenses (L1: f = 19 mm, NA = 0.3 and L2: f = 200 mm, NA = 0.06) fo beam expansion, a NPBS (90% tansmitted, 10% eflected powe), a NPBS, a linea polaize, a phase-only nematic liquid cystal-onsilicon eflective SLM (Meadowlak HSP , pixels, each m 2 in size, with phases contollable fom 0 to 2π with 8-bit esolution), an aspheic lens (L3: f = 4.6 mm, NA = 0.53) to couple light into a fibe, and a MMF labeled MMF1 (Tholabs custom ode, 1 m long, 25 μm coe diamete, 0.1 NA, FC/APC connecto at poximal end, FC/PC connecto at distal end). This aangement allows a complete basis of LP modes to be excited at the distal end of MMF1 by setting diffeent pattens onto the SLM and otating the linea

7 polaize between configuations of 45, 0 and 45 with espect to the polaization of the beam. Essentially, MMF1 acts as a mode filte. Also, pat of the lase beam is split off by a NPBS to act as a efeence beam fo the camea section. The endoscope section is a MMF (Tholabs custom ode, 1 m long, 25 μm diamete, 0.1 NA, FC/PC both ends) labeled MMF2, cut fom the same fibe as MMF1 and econnectoized; this ensues that both MMFs suppot pecisely the same popagating modes. MMF2 is secued to the optical table, and two matching micoscope objective lenses (L4 and L5: 20 magnification, 1.2 mm woking distance, NA = 0.4) couple light fom MMF1 to MMF2. A NPBS between the objective lenses sends both the light exiting distally fom MMF1 and the light exiting poximally fom MMF2 to the camea section. At the distal end of MMF2, an aspheic lens (L6: f = 10.0 mm, NA = 0.55) is used to collimate light. A 99% eflectivity silve eflecto at the focal plane of the aspheic lens can be used to eflect light back into MMF2 with high coupling efficiency, o moved out of the way to send the collimated light to the camea section. This moveable eflecto acts as the patial eflecto; in a tue endoscopic scenaio a patial eflecto would instead be attached diectly to MMF2. In this way, light descibed by m in,p, m out,p and m out,d (see Fig. 2) can all be independently diected towads the camea section. Duing measuement of the MMF, only one of these beams, designated as the signal beam, is sent to the camea section at a time; the est ae blocked by shuttes. By manually petubing MMF2, the mode coupling of the MMF can be expeimentally petubed. The camea section is compised of a camea (QImaging Retiga 4000R, pixels, each m 2 in size, with 10-bit esolution), a polaizing beam displace (PBD) (2.7 mm displacement), an achomatic doublet lens to image the signal beam (L7: f = 300 mm, NA = 0.04), an achomatic doublet lens to collimate the efeence beam (L8: f = 75 mm, NA = 0.09), and a shutte in the path of the efeence beam. This camea setup allows coheent measuement of both polaizations of any of the thee beams diected into the camea fom the endoscope section. Fom Eq. (1), full chaacteization of the tansmission T and eflection R matices of MMF2 equies measuements of m in,p, m out,p and m out,d of MMF2; this is accomplished as follows. Fist, a set of M SLM pattens ae computed that will excite as closely as possible an othonomal set of modes at the distal end of MMF1. Such a computed set of M pattens achieves a highe SNR than a andom set of pattens. M is at minimum the numbe of modes MMF2 suppots in one polaization, N/2, but fo obustness should be highe by a facto of two to thee. When modulating the SLM, 6 6 pixel binning is used to appoximate amplitude modulation using phase-only modulation [29]. Fo easons that will be explained shotly, in all M pattens, 15% of the SLM is set aside and set to amplitude 0. Patten 1 is then placed onto the SLM, the linea polaize is otated to 45, and E, ( xy, ) is incident upon the camea section while shuttes block all othe beams. E, ( xy, ) is measued coheently (i.e., measuement of amplitude and phase) by obtaining fou measuements: the iadiance of the efeence beam, the iadiance of the signal beam, the iadiance of the coheent intefeence between the signal and efeence beams, and the iadiance of the coheent intefeence between the signal beam, shifted by a phase of π/2 adians with espect to the efeence beam. This pocedue is epeated fo SLM patten 2, etc., up until patten M. Howeve, since each successive coheent measuement is sepaated in time, each E, ( xy, ) measuement is not in phase with the othes due to jitte between the phases of the efeence and signal beams. This is emedied by using the 15% of the SLM set aside ealie to simultaneously excite a second efeence beam that tavels though the MMF along with the signal beam (thus this efeence stays in phase with the signal beam), and taking a set of measuements of the intefeence between this efeence and the signal beam fo evey E, ( xy, ). This ensues that all in p in p in p in p

8 E in, p( xy, ) ae in phase. Afte that, the linea polaize is otated to 45 and the pocedue is epeated fo anothe M measuements, so that complete bases of x- and y-polaized LP modes ae independently excited. Since the coheent measuements of these two sets of modes ae sepaated in time, anothe set of M measuements taken at 0 otation of the linea polaize is used to match the phases of the 45 and 45 measuements. All these E, ( xy, ) ae then adjusted fo x and y position, x and y tilt, and focus, by centeing the aveage of all coheent measuements in both the spatial and spatial fequency domains. Once popely centeed, these measuements ae decomposed in an othonomal basis of modes to obtain m in,p. This othonomal basis is the LP mode basis fo an ideal step-index MMF having the nominal coe adius and NA of MMF2, slightly modified to moe accuately account fo spheical abeation, defocus of the lenses, and small deviations fom nominal coe adius and NA. Finally, this pocess is epeated to measue m out,p and m out,d fo full chaacteization of the MMF. This pocess is epeated fo evey petubation of MMF2. 4. Petubation compensation using patial eflecto In this section we descibe the theoy behind compensating fo petubation of the MMF, and hence a petubation of T, by taking a measuement of the petubation of R. The quality of an image obtained though a MMF coesponds diectly to how accuately any desied patten can be excited at the distal end of the MMF [30]; fo simplicity we will assume this patten is a spot, and that the spot foms the point-spead function of the imaging system. The quality of the spot that can be fomed in tun depends diectly on how accuately T is known, since to fom a spot, the tansmission matix must be inveted as shown in Eq. (3): m m Tm in, p T( T m ) 1 in p, (3) whee m ae the desied mode coefficients at the distal end that coespond to a spot, and 1 T is a noisy measuement o appoximation of T. The close TT is to the identity matix, the bette the spot will be; if it is equal to the identity matix, the smallest spot that can be suppoted by the MMF will be fomed. Fo a igid MMF, T can be measued pecisely befoe imaging and does not change. We note that duing imaging it is necessay to fom spots though the patial eflecto, as shown in Fig. 2, but no exta step is equied because its tansmission matix is the (scaled) identity matix. Since T only depends on the cuent physical state of the MMF, imaging though a flexible MMF using Eq. (3) is possible if T is known fo all possible petubations within some allowed ange of motion. To model this, let a MMF have a (known) initial and an (unknown) petubed configuation, with all displacements between the two configuations be consideed the petubation. A patial eflecto is placed at the distal end of the MMF, so that the eflection matix R can be measued. Woking fom Eq. (1), the theoetical elationships between T, R, and the petubed matices ae as follows: m TCm m PTCm in, p t in, p m ROm m P ROm out, p out, p, (4) whee m denotes the petubed mode coefficients, the mode coupling into the MMF is descibed by a mode-coupling matix C, the mode coupling back into the MMF due to the patial eflecto is descibed by a matix O, and Pt and P ae petubation matices. Thee is no loss of geneality in assuming that T is petubed into PtT (i.e., T multiplied by some petubation matix Pt) and R into PR (i.e., R multiplied by some petubation matix P) as

9 long as both T and R ae full-ank. We assume that the initial configuation of the MMF can be measued accuately, but that afte the MMF is petubed thee is no access to the distal end; thus m in, p, m out, p and m can all be measued befoe the MMF is petubed; but only m out, p can be measued while m cannot afte the fibe is petubed. The poblem then becomes how to calculate Pt fom Eq. (4). If we futhe take into account ecipocity (so that R = T T ) [27], the equations become: m TCm m PTCm in, p t in, p m T Om m ( PT) Om T T out, p out, p t. (5) Fom Eq. (5), we descibed a method to calculate Pt in [27]; howeve, that solution made the assumption that coupling into the MMF was pefect (i.e., C = I ), with no mode coupling o mode-dependent loss. Though it is still possible to find a unique solution even with a nonideal C, in a eal system Eq. (5) is futhe complicated by expeimental eo and modedependent loss of T, violating one of the assumptions we made in [27]. This makes exact, closed-fom solutions of Eq. (5) pone to eo and ovefitting. Instead, by matching Eq. (4) and Eq. (5) we can obtain the esult: 1 T Pt T P T. (6) This suggests that when P and T ae close to diagonal, P T should be simila to Pt and thus can be used as an appoximation to Pt. We would expect it to indeed be the case that these matices ae close to diagonal, since modes geneally do not couple to each othe unless thei popagation coefficients ae close and thus inte-goup mode coupling is the stongest effect inside a MMF; it has also been expeimentally obseved that fo small petubations, the tansmission matix changes only a small amount [24 26]. Howeve, looking at Eq. (4), thee is the futhe complication that calculating P equies knowing m ; thus it is necessay to only calculate an appoximate P by using m instead of m. In othe wods, we assume no petubation of the distal modes when calculating the petubation of the eflection matix. Unfotunately, this is a less justified assumption than the assumption that P T is simila to Pt, and we expect this to esult in lowe-quality spots; we show the effect of this assumption in Section 5, and suggest possible altenatives to impoving spot quality in Section 6. The stategy fo compensating fo bending of the MMF can thus be summed up as follows: we know the initial configuation TC and RO using coheent measuements of the unpetubed MMF. When the MMF is petubed, we would like to know the petubation Pt of the tansmission matix in ode to pefom pefect imaging, but we cannot measue it because we do not have access to the distal end. Instead we use the appoximate petubation T P of the eflection matix, which we can measue because of the patial eflecto, as an appoximation of Pt when tying to ceate spots. With efeence to Eq. (3), the spots that we fom will then be descibed by: P ( M' M )( RO) m PT(P T m 1 1 out, p T 1 t ), (7) whee M ' out, p and M ae matices fomed by many independent measuements of the mode vectos m out, p and m, measued using the pocedue descibed in Section 3. This is contasted with the method of tying to fom spots without accounting fo the petubation, as fo example in [24], which esults in the following:

10 m PT T m. (8) 1 t ( ) Both of these can be contasted with ceating spots if Pt is exactly known, which esults in: m PT(PT) m. (9) -1 t t Of couse, it is not possible to measue Pt in an endoscopic application. We would expect that Eq. (8) gives the wost esults, Eq. (7) gives impoved esults, and Eq. (9) gives close to ideal esults. We expeimentally veify these pedictions in the following section. 5. Expeimental esults We affix the MMF to an optical table and displace the cente of the MMF lateally by a distance between 0-50 mm, esulting in the appoximate change to its shape shown in Fig. 3. Fo each petubation, we measue the matices TC, RO, PtTC and PRO, which, as mentioned in Section 4, is sufficient to fully chaacteize the MMF. One set of measuements of these matices fo one such petubation is shown in Fig. 4. x-pol x-pol y-pol Fig. 4. Tansmission (T) and eflection (R) matices, petubed tansmission (P tt) and eflection (P R) matices, and the petubation matices fo tansmission (P t) and eflection (P ). Note that we cannot sepaate T o R fom the input coupling (C) and patial eflecto (O) matices. Matix enties coespond to powe coupling. Modes 1 to 46 ae odeed by deceasing popagation constant in the x polaization; modes 47 to 92 ae odeed by deceasing popagation constant in the y polaization. Scale ba epesents the enegy [0 to 1] in each mode. As can be seen in Fig. 4, thee is substantial mode coupling fo both TC and RO. This could be due to the U-bend in the MMF, mico-bends of the MMF duing manufactue, o mismatched coupling into the MMF epesented by the matices C and O. While we cannot pecisely sepaate these effects, the fact that TC is much less diagonal than RO would suggest that much of the mode coupling in the matices TC and PtTC is due to C (mismatched coupling). This hypothesis is suppoted by the off-diagonal lines that can be seen in TC and PtTC, which we would expect to happen due to mismatched coupling between spatial modes that ae simila to each othe spatially. By contast, bending the MMF esults in petubation matices Pt and P that ae stongly diagonal. We also note that most

11 petubation occus in the lowe-ode modes while highe-ode modes emain unaffected. The loss that can be seen in the highe-ode modes of Pt and P is due to loss fom popagation though the MMF. We then use P to expeimentally ceate a gid of spots at the distal end of MMF2 (as descibed at the end of Section 4), which is shown in Fig. 5. In Fig. 5, column 2 shows PT(P T ) 1 t T and thee epesentative spots we ceate using the matix. Fo compaison puposes, column 1 shows the matix and spots coesponding to no compensation, and column 3 shows the matix and spots coesponding to pefect compensation, as descibed by Eq. (8) and Eq. (9) espectively % 70% 100% 2 51% 54% 100% 3 64% 55% 100% 4 Fig. 5. Row 1: matices coesponding to no compensation, petubation compensation, and ideal compensation. The coelation coefficient between each matix and the identity matix is indicated. Rows 2 to 4: epesentative spots we excite in polaization x using the method coesponding to the matix in the column above. The amount of powe contained in the spot (elative to pefect compensation) fo each excitation is indicated. Row 2 shows a spot that is substantially bette when we use compensation, ow 3 shows a spot that is about the same whethe we use compensation o not, and ow 4 shows a spot that is bette without compensation. Aveaged acoss all spots, petubation compensation esults in 30% moe powe contained inside the spot as compaed to no compensation, almost exactly equal to the

12 Spot quality atio of the matix coelation coefficients between compensation and no compensation. The linea polaize is set to 0 fo all spot excitations. Red indicates the method (compensation o no compensation) yielding highe powe in the spot. As pedicted, using petubation compensation esults in a substantially bette matix and bette spots than no compensation. Both the matix coelation coefficient metic and aveage spot powe metic exhibit appoximately the same impovement, about 30%, as pedicted by theoy. We nomalize the total powe contained in a spot fo both compensation and no compensation to the pecentage of powe contained in a spot fo pefect compensation. The pecentage of total powe contained in a spot fo pefect compensation is limited to aound 50% because, as descibed in Section 3, only 85% of the SLM is available fo the fomation of spots (this is not an inheent limitation, but an implementation choice hee) and we can only launch one polaization at a time into the MMF; this limits the ensemble of modes that we can excite. The fact that PtT (PtT) -1 is not equal to the identity matix (Fig. 5, ow 1) indicates that T has stong loss fo its highest-ode modes. Ou esults also agee with peviously published esults that spots do not completely disappea even fo elatively lage uncompensated petubations [24]. Finally, in Fig. 6, we compae the coelation coefficients of the matices coesponding to no compensation, petubation compensation using P, petubation compensation using P, and pefect compensation fo vaious petubations of the MMF. All matix coelation coefficients ae computed element-wise using the Fobenius nom Compensation using Compensation using Compensation using No compensation Petubation stength 0.65 Fig. 6. Compaison of the quality of spot fomation fo diffeent compensation methods vs. the petubation stength. Each plot includes two points pe petubation, coesponding to exciting spots in the x- and y-polaizations. The y-axis (spot quality) is paameteized by the coelation coefficient between ow 1 of the matix shown in Fig. 5 and the identity matix. The x-axis (petubation stength) is paameteized by the one minus the coelation coefficient between P t and the identity matix. The MMF exhibits non-monotonic behavio, in that lage physical petubations do not necessaily lead to moe substantially diffeent petubation matices. It can be seen that petubation compensation of the matix using P yields vey substantial impovements of the spot quality fo lage petubations and no loss in spot quality fo small petubations. Compensation of the matix using the appoximate eflection matix P is wose, as expected, but still shows impovement. Fo all matix invesions, we use the 1 1 singula-value decomposition with up to 90% of the singula values. Fo P and P we take the exta step of aveaging the matix invese with I; this pocedue makes the matix

13 invese slightly wose fo lage petubations, but moe stable fo small petubations, and so ensues that petubation compensation gives bette esults fo all petubations. 6. Discussion The expeimental esults of Section 4 demonstate substantial impovement of the ability to ceate spots at the distal end of a petubed fibe by using infomation fom the eflection matix, paticulaly fo lage petubations. This may be supising, given the numbe of appoximations made in Section 3. Futhe, the wost-case scenaio of substantial inte-goup mode coupling and fully andom, non-diagonal petubation matices descibed in [27] is found to be pessimistic; in the expeiments pesented hee, inte-goup mode-coupling is found to be weak, and petubation mainly affects the coupling between lowe-ode modes. Thus it is sufficient hee to use a simple patial eflecto instead of a stuctued eflecto as descibed in [27]. We note that it is feasible to use eithe a patial eflecto o a stuctued eflecto. Fo example, a stuctued eflecto can be fabicated by coating chome on a glass o fused silica cove slip, then using electon-beam lithogaphy and a lift-off standad pattening pocess to ceate two-dimensional pattens with m featue size and tansmission coefficient in the desied ange. Such a stuctued eflecto can be easily diced and attached to the end of the MMF [31]. Altenatively, the end of the MMF can be coated with a thin laye of chome, which can be pattened using a focused ion beam to ealize a compact stuctued eflecto. While the method we descibe fo petubation compensation is useful in inceasing the SNR fo the ceation of spots as compaed to no compensation, it falls shot of a fully ealized method to completely compensate fo the petubations of a MMF and so enable fully flexible MMF imaging. Notably, the use of the appoximate eflection matix petubation P instead of the exact matix P leads to much educed impovement in quality of spot fomation, which is not unexpected given the majo assumptions made, as descibed in Section 3. Howeve, ou esults suggest moe pomising methods to achieve accuate spots beyond simply assuming that the petubation of the eflection matix is equal to the petubation of the tansmission matix. Fo example, a staightfowad method would be to measue, while the endoscope is not in use, the tansmission and eflection matices fo a lage numbe of petubations; then, assuming that the space of petubations is sampled well enough, when the endoscope is in use, one could intepolate the expected tansmission matix by measuing the eflection matix. This method only equies that the tansmission and eflection matices map to each othe one-to-one, which we would expect to be the case fom the small space of petubations we obseve, and that the esults ae epeatable. Of couse, this method is computationally inefficient; it might also be possible to use moe complex and obust algoithms to analyze the tansmission and eflection matices and find a mathematical elationship between them, simila to ecent wok that uses machine leaning to tansmit images though andom media [32,33]. A pessing question is how these esults scale up as the numbe of modes N of the MMF inceases. A useful imaging system is likely to equie at least N = 1000 modes, and ideally would employ many moe. As N scales up, the imaging time will incease by O(N), which is acceptable given ecent developments in imaging speed mentioned peviously. Image pocessing time will incease by O(N 2 ), which is acceptable given that the algoithm is highly paallelizable. The key question, then, is how much P and T will change fo a MMF with many moe modes. Fo example, it is possible that mainly the lowest-ode modes will undego significant petubation in mode coupling, as we obseve hee? In that case, we would expect petubation compensation to wok as well as it does hee. Convesely, MMFs with moe modes could be much moe sensitive to petubations, in which case the method demonstated hee would not wok. Extapolating fom ecent eseach, the moe optimistic

14 scenaio seems moe likely; it is possible to substantially incease the NA of the MMF and thus the numbe of modes N without affecting the stength of mode coupling [24 26]. The esults pesented hee could be impoved upon futhe by educing expeimental eo, souces of which include impefect coupling of light into the MMF, phase jitte (a vibation at a fequency of 11 Hz is obseved), dift of the lase powe, spheical abeation of the lenses, and technical limitations of the SLM and camea. 7. Conclusion We have poposed and expeimentally veified a method to obtain infomation about the tansmission matix of a petubed step-index MMF using a patial eflecto attached to the end of the MMF. We expeimentally find that the petubation of the eflection matix is highly coelated with the petubation of the tansmission matix. Exploiting this obsevation, we demonstate a simple method that allows the tansmission matix to be calculated and spots to be ceated with 10-15% highe quality compaed to methods that assume no change in the MMF. Futue methods could bette exploit this coelation by using moe advanced algoithms to bette pedict the tansmission matix fom the eflection matix, which could enable imaging though fully flexible MMFs. Acknowledgements We would like to thank Iliya Sigal of the Univesity of Toonto fo his help with the camea and optical setup.