Abstract

Endoscopic imaging through a multicore fiber (MCF) is widely used but is affected by pixelated images, which limits its resolution to a few micrometers. This is due to the spacing between the cores in the MCF, which is large enough to avoid core-to-core light coupling. Wavefront shaping techniques have been shown to focus light to a resolution finer than the inter core spacing, however a long calibration procedure is needed. Moreover the calibration depends on the optical fiber conformation. Here, we show a calibration method using only one digital hologram. The method is based on digital phase conjugation and the memory effect of the MCF to focus and scan a spot. In addition, we show how simple patterns can be projected using the same multicore fiber.

© 2015 Optical Society of America

1. Introduction

Optical fibers are widely used in endoscopic imaging to enable high-resolution optical imaging deep inside the body. Commercially available endoscopes are usually made out of an optical fiber bundle, often referred to as multicore fibers (MCFs) [1,2]. Pixelation artifacts caused by the fiber cores affect the images that can be formed. The individual cores that form the MCF are spaced sufficiently to limit core-to-core light coupling. This separation results in a pixelated image which directly limits the resolution [3]. These limitations have driven research in high-resolution endoscopy by using multimode fibers (MMFs) and wavefront shaping techniques such as optimization algorithms [4,5], digital phase conjugation (DPC) [6,7], and transmission matrix approaches [8,9] to obtain images with diffraction limited resolution. Unfortunately, the scanning ability of the wavefront shaped focus spot through a step index MMF is limited to radial scanning, which prohibits the use of conventional scanning techniques [10] and limits the scanning speed to the refresh rate of a wavefront modulation device. Additionally, bending a MMF changes the relative mode propagation and a new calibration needs to be done.

Recently, the same wavefront shaping approaches used with MMFs have been applied to MCFs to create pixelation-free, diffraction-limited images, adding a new image modality to classical MCFs based endoscopes. Specifically, a transmission matrix approach was used with MCFs to account for mode coupling, as well as for core-to-core crosstalk [11]. Additional work has integrated wavefront shaping for optical focus creation and scanning with a MCF. This effort included the development of a custom-made, single-mode core MCF with 169 cores with spacing large enough to eliminate core-to-core crosstalk and enable focus spot scanning [12,13]. A higher number of cores, which could support many modes, increases the wavefront shaping degrees of freedom, which would contribute to increase the amount of light delivered in the focus and reduce the background.

In this paper, we investigate the characteristics of spot focusing and scanning by DPC through commercially available MCFs with a high number of densely packed multimode cores. We show that the combination of DPC and MCFs is a good compromise between MMFs and transmission matrix approaches with MCFs. We observed that, as in thin scattering media [14,15], phase gradients are conserved within a certain range when coherent light is propagated through MCFs, resulting in a translational “memory effect.” Thus, despite mode coupling and core crosstalk, a focus spot can be steered through a significant scan range thanks to the memory effect present in this kind of optical fiber. This feature enables the use of a single real time feedback mechanism to create a scannable focus spot, which could potentially be steered with high-speed galvomirrors.

The ideal wavefront shaping technique for a single step calibration is DPC, in which only one digital hologram is necessary to create an optical focus. This simplifies the calibration step compared to [12,13], in which a core by core calibration step was required for characterization. DPC has been extensively used for MMF endoscopy, but it also applies to MCF as shown in a pioneering work by Bellanger, et al. [16]. Single step calibration has been shown in MCF in [17], reaching a focus spot with a peak to background ratio of approximately 50. We show here that a ratio of above 1000 can be reached by DPC, since this techniques gives the optimal phase pattern in order to obtain a focused spot.

2. Optical setup

We implemented DPC through MCF using the experimental setup schematically illustrated in Fig. 1. The laser beam from a He-Ne laser is expanded and collimated by the 4f system formed by lenses OBJ1 and L1. A polarizing beam splitter (PBS) splits the beam into two arms: the calibration arm and the reference arm. The calibration arm is directed towards the objective (OBJ2) and is focused at plane S in front of the surface of the distal end of the MCF. The output of the bundle is imaged by the imaging system OBJ3-L3 onto a CMOS camera (MV1-D1312IE-100-G2-12, Photonfocus), where it interferes with the reference arm, generating a digital hologram. A polarizer is placed before the CMOS and is oriented horizontally. The phase of the recorded complex field is digitally calculated, conjugated, and assigned to a spatial light modulator (SLM, Pluto-VIS, Holoeye). For optical phase conjugation the reference beam is shaped by the SLM and sent into the MCF. After propagation the phase conjugated beam recreates a focused spot in the plane S. On the distal end of the MCF the objective (OBJ2), the non-polarizing beam splitter (BS1), the lens (L2), and a CCD camera are all mounted on a translational stage (TS) to control the bending of the MCF and simultaneously check the quality of the phase conjugated spot. For our experiments we use a 3000 core MCF (FIGH-30-215S, Fujikura), the fiber diameter is 190 μm, the core diameters are measured to be on average 2.5 μm (they are actually slightly different to minimize core-to-core coupling [18,19]), and the core and cladding refractive indexes are n1 = 1.497 and n0 = 1.457, yielding a numerical aperture (NA) of 0.34. Each core can support about 4 linearly polarized (LP) modes.

 figure: Fig. 1

Fig. 1 Experimental optical setup. The output of a He-Ne laser is expanded is split by the polarizing beam splitter PBS into the calibration and reference arms. The objective OBJ2 focuses the calibration beam in a plane S in front of the distal end of the multicore fiber (MCF). The proximal end facet of the MCF is imaged on the CMOS sensor through the 4f imaging system OBJ3-L3. Using the non-polarizing beam splitter BS2, the reference arm is combined with the image on the CMOS to form a digital hologram. The phase extracted from the digital hologram is conjugated and projected on the spatial light modulator (SLM), which modulates and reflects the reference beam towards the MCF through BS2. A focus spot forms in the plane S and is imaged using the 4f system OBJ2-L2 on the CCD camera. The translational stage TS bends the fiber in a controllable way. OBJ1 = 10 × , NA = 0.25, Newport, OBJ2 = OBJ3 = 20 × , NA = 0.4, Newport. Focal length lenses: L1 = 150 mm, L2 = 100 mm, L3 = 200 mm.

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3. Digital phase conjugate focus characterization

To understand the DPC focusing characteristics of the MCF, we analyzed the recreated focus spots dependence on distance between the plane S and the fiber facet, which we define as working distance. First, we measured the full width at half maximum (FWHM) of the phase-conjugated spot, with each focus created within plane S aligned to the center of the MCF. This is shown by the blue curve in Fig. 2(a). For working distances less than 250 μm, the spot size remains constant, but beyond this distance, the FWHM increases linearly. Within 250 μm, the focused spot size is determined by the fiber NA. Phase conjugation at larger working distances increases the spot size [7]. In this case, fiber diameter and working distance determine the FWHM. With the same data we also evaluated the enhancement of the focused spot, defined as the ratio between the peak intensity of the focus and the mean intensity of a speckle field generated by a random phase pattern. This quantity depends linearly on the number of degrees of freedom used to form the focused spot [20]. In this case the theoretical maximum enhancement would be approximately the number of fiber cores multiplied by the number of modes per core, if all the modes would be controlled at the same time. If all the modes and/or the cores are not fully excited during the calibration step, the maximum enhancement will be lower than the theoretical one. The enhancement versus the working distance is shown by the red curve in Fig. 2(a). The peak enhancement is reached when all the cores of the MCF are contributing to the focusing, which occurs where the outer cores are excited with an angle equal to arcsin(NA). Beyond this distance the enhancement decreases with the peak intensity (decreases as the FWHM of the focus spot increases) and the number of excited higher order modes (the incidence angle on the outer cores lower than arcsin(NA)).

 figure: Fig. 2

Fig. 2 DPC focus spot characterization. (a) Focus spot size (FWHM, blue points) and enhancement (red points) as a function of the working distance. The lines are drawn only for clarity. (b) Example of DPC focus spot imaged on the CCD camera. (c) The same focus spot of (b) with the CCD exposure time increased 100 times. The six bright speckle areas around the main lobe are a result of the quasi-periodicity of the MCF. Scale bars are 20 μm.

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4. Using the memory effect in a multicore fiber

4.1 Transverse scanning with the MCF

With the DPC generated focus spot, a shift of the focus spot is possible by changing the tilt of the phase conjugated wave incident on the MCF. The memory effect of the MCF allows a transverse shift of the phase-conjugated spot by assigning to the SLM a phase pattern equal to:

ϕSLM(x,y)=arg[ejϕDPC(x,y)ej(kxx+kyy)],
where ϕDPC(x,y) is the optimal phase to recreate a focus spot in a given plane S and the second exponential term represents a plane wave having a carrier frequency, k=(kx,ky). This allows for scanning a focus spot in the plane S with a single digital hologram calibration. Figure 3(a) shows a collection of focus spots scanned in a plane 400 μm from the MCF facet. The spots shown in the figure were created individually with a fixed step size, Δk, and combined into a single image for clarity. The cross-sectional plot in Fig. 3(b) shows the decay in the focus spot peak intensity with increasing|kx|. For a constant wave carrier frequency k, the physical focal spot shift increases linearly with the working distance. The intensity drop can be explained as follows: during phase conjugation, the light sent into the MCF couples into the distinct modes of each core. The mode confinement is the highest for the fundamental mode and decreases as the order of the mode increases [21]. This results in high order modes coupling more easily into neighboring cores. In fact, steering the beam changes the phase relationship between cores, which affects the core-to-core coupling and decreases the phase conjugation efficiency. This aspect is more pronounced when the coupling between cores is more significant, as in the higher order modes case. In other words, core-to-core coupling limits the final scanning range.

 figure: Fig. 3

Fig. 3 Digital scanning of the focus spot using a single calibration hologram. (a) The phase conjugation pattern is combined with a linear phase gradient and projected on the SLM. By changing the direction and the strength of the gradient it is possible to scan the focus spot in a regular grid. The image is a combination of several DPC projections (see Visualization 1). (b) The square markers represent the peak intensities of the DPC spots along the red dashed line in (a). Scale bar 20 μm.

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A proof of this concept is shown using an ultrafast pulsed laser. We replace the He-Ne laser shown in the setup in Fig. 1 with a 140 fs duration pulsed laser (Chameleon, Coherent, wavelength 785 nm). Because of the ultrashort pulse and modal dispersion in the cores, the few modes of the fiber come out of the MCF separated in time. By time gating the acquisition of the digital hologram with the pulsed reference beam, it is possible to select the optical mode to phase conjugate [22]. With this system we were able to analyze the scanning ability of different waveguide modes with a single hologram. The blue and the red curves in Fig. 4 show the scanning range obtained using the fundamental LP01 mode (red) and the LP11 mode (blue) respectively. The spot was scanned at a distance of 225 μm from the MCF facet. The LP01 mode has a wider, more slowly decaying scanning range compared to the LP11 mode. CW focus spot scanning consists of the interference between the LP01, LP11 and possibly even higher order modes depending on the cores and the laser wavelength. Here we see that the decay profile shown in Fig. 3(b) is the result of the LP01 mode and the less resilient higher order modes. The shorter scanning range shown in Fig. 4 compared to Fig. 3(b) is a result of different working distances (225 μm compared to the 400 μm of the previous case).

 figure: Fig. 4

Fig. 4 Using a pulsed laser it is possible to independently phase conjugate distinct modes of the MCF cores. The scanning range of the fundamental LP01 mode is larger than the one of the higher order modes because of the limited core-to-core coupling.

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4.2 Axial scanning with MCF

In addition to transverse scanning, by adding a positive or negative parabolic phase pattern to the reference wavefront, ϕDPC(x,y), the phase-conjugated spot can also be shifted in the axial direction to enable 3D digital scanning. Figure 5 shows the axial variation of the focusing PSF. By adding a positive quadratic phase (a positive lens) to ϕDPC(x,y) the focus spot is shifted toward the fiber, and the opposite is true for a negative lens. Defining F as refocusing parameter, ϕSLM(x,y) can be rewritten as:

ϕSLM(x,y)=arg[ejϕDPC(x,y)ejπλF(x2+y2)],
where λ is the laser wavelength. We measured the FWHM of the attainable axial shift to be 60 μm (Fig. 5), where the axial position of the DPC calibration is at the center of the scanning range (400 μm from the MCF facet in this case).

 figure: Fig. 5

Fig. 5 Axial shifting of the DPC focus spot using a single calibration hologram. The central row of images shows a phase-conjugated spot observed at different planes. The calibration hologram was taken focusing at a distance of 400 μm from the fiber facet. By adding a negative lens phase to the phase hologram, the focus shifts towards the fiber. The images above and below the central row show the axial shift of the focus with the increasingly positive or negative lens phases.

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4.3 Single calibration image projection

Moreover, we show that thanks to the memory effect, it is possible to generate patterns with a single calibration hologram and project multiple focused spots within the memory effect range of the MCF. This is possible by combining multiple tilted beams together starting with the phase pattern ϕDPC(x,y). In this case the phase to project on the SLM is given by:

ϕSLM(x,y)=arg[ejϕDPC(x,y)iej(kxix+kyiy)].
Figures 6(a) and 6(b) show the projection of a cross pattern when the two arms are in phase and out of phase, respectively, while Fig. 6(c) shows a nine focus spot projection. The imperfections visible around the cross pattern in Fig. 6(a) (the four blurred spots around the pattern) are artifacts of using phase-only modulation with the SLM. Figures 6(d) and 6(e) show simulated data of the expected pattern assuming phase-only and amplitude and phase modulation, respectively. Figure 6(d) clearly shows the same artifacts as in Fig. 6(a) whereas Fig. 6(e) shows that an amplitude and phase modulation allows the projection of a high quality pattern. The simulated results were obtained by simulating free space propagation with the beam propagation method (BPM [21],). We used the experimental MCF core distribution as the input field, where each core has only the fundamental mode excited.

 figure: Fig. 6

Fig. 6 Pattern projection using a single calibration hologram: (a) cross obtained as a combination of shifted spots; (b) the same pattern as (a) except the spots of one arm are in opposite phase to those of the other arm; (c) multiple spot projection, in this case nine spots are projected simultaneously in a regular grid. Scale bars are 5 μm. (d) Simulation of the cross-pattern projection with phase only modulation; (e) simulation of the cross-pattern projection with amplitude and phase modulation.

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4.4 No coupling scan range

In this section we show that the maximum attainable scanning range depends on the single cores of the MCF, and in particular on their NA Without loss of generality, let us consider an array of identical single-mode cores, where Ea(x) is the mode of each core. The waveguide array is mathematically expressed as:

A(x)=comb(Λxx)Ea(x),
where Λx is the center-to-center distance between the center of two neighboring cores and is the convolution operator.

If a quadratic phase is added to the input of waveguide array to focus at a focal length,f, the optical field in the focusing plane is equivalent to the Fourier transform (FT) of the input field with scaling dependent on f and the wavelength, λ:

A˜(fx)=FT{A(x)}=FT{comb(Λxx)}FT{Ea(x)}.

Thus, in the absence of core-to-core coupling case the scanning range is limited by the maximum spatial extent of the envelope FT{Ea(x)}, which is ultimately determined by the NA of the cores. If Ea(x) is approximated with a Gaussian beam of waist sizew(0), then we can find an expression for the maximum scanning range,wMAX, defined as the waist of the Gaussian beam diffracted after propagating the working distance,f:

wMAX=w(f)=w2(0)+f2NA2.
In intensity this scanning range is decreased by a factor 2.

For imaging purposes, the field of view (FOV) and the maximum scanning range are not necessarily the same. In fact, due to the periodicity of the input the focusing plane contains several focus spots, not just a single focused spot. The spacing between the foci px is given by:

pxfλΛx.

The actual FOV is limited by the presence of these secondary peaks, which bring ambiguity about the position of the optically generated signal. These secondary peaks scan with the focus and have an amplitude which modulates with the envelope, wMAX. In this way, the intensity of the secondary peaks may surpass the primary focus intensity when the secondary peaks are near the primary. To increase the distance between the peaks the distance between cores must be decreased, however this introduces core-to-core coupling.

4.5 Scanning limitations imposed by core-to-core coupling

To understand the effect of core-to-core coupling on the memory effect range, we modeled focusing with a one-dimensional waveguide array with varying core pitch. The model allowed simulation of focusing with a quadratic phase at the waveguide input, or focusing with DPC. The DPC simulation was carried out by performing the following steps: (1) a beam propagation method (BPM [21],) step performs the propagation of a focus spot for a plane in front of the waveguide to the waveguide’s distal facet; (2) the calculated field at the distal facet is coupled to the waveguide array. Coupled mode theory (CMT [23],) is used to calculate the propagation through the array to the proximal facet of the waveguide; (3) to simulate DPC the field at the waveguide proximal end is phase conjugated and a new CMT step is performed; (4) a last BPM step takes the field at the fiber’s output facet and propagates it back to recreate the focus.

In the presence of coupling between the cores, a quadratic phase at the input of the waveguide array is not enough to generate a focus spot, because coupling prohibits the relative phase at the input of the array from being directly transferred to the output. As discussed previously, DPC records the phase pattern at the proximal end of the waveguide array, which generates a focus spot by taking into account the core-to-core coupling. The simulation results in Fig. 7 illustrate two scenarios with two arrays of 21 waveguides: no coupling and core-to-core coupling scenarios. The cores of the array have a diameter d = 1 μm, the cladding refractive index is n0 = 1.47 and the core refractive index is n1 = 1.5, so the waveguides are single-mode for λ = 633 nm. The simulated arrays have core pitch of Λx = 5 μm (no coupling) and Λx = 3.5 μm (coupling) and a length of 30 cm. Figure 7(a) shows that in the no coupling case there is no difference between focusing with a quadratic phase or with DPC. However, in the case with core-to-core coupling (Fig. 7(b)) the focusing ability is lost with quadratic phase focusing, but it is preserved when DPC is performed.

 figure: Fig. 7

Fig. 7 Focusing through a waveguide array in no-coupling and coupling conditions. The propagation in the waveguide array was simulated using coupled mode theory and the free space propagation with the beam propagation method. (a) Focusing in no coupling conditions is possible by assigning a quadratic phase at the input of the waveguide array. The phase relationship between the cores is preserved along the waveguide and it is possible to focus at a given distance for the waveguide facet (400 μm in this case). Focusing using DPC gives equivalent results. (b) In coupling conditions, the phase relationship between cores is not preserved along the waveguide and focusing using a quadratic phase pattern is no longer possible. DPC allows focusing in severe coupling conditions.

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We further investigated the effect of core-to-core coupling on the scan range by varying the coupling conditions and introducing a linear phase gradient. The phase gradient was added to the output of step (2) before performing step (3). In this way we scanned the DPC focus spot and evaluated the effect of coupling in reducing the scanning range. Figure 8 shows the peak intensity of the DPC spot for different tilt angles of the input beam. The curves show the scanning range as a function of the core-to-core distance,Λx. In the absence of coupling, where Λx = 5 μm, the scanning range matches the one described by Eq. (7). When decreasing Λx the increased coupling decreases the scanning range.

 figure: Fig. 8

Fig. 8 Scanning range using DPC in a waveguide array with different coupling conditions by tuning the distance between the cores. Increased coupling decreases the scanning range. The green curve represents the no-coupling conditions, which yields the maximum scanning range.

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5. Focus spot preservation in a bent fiber

The ability to steer the focus spot by adding a linear phase to the phase conjugate field suggests that the MCF should be more resilient to fiber bending. Bending a fiber changes the relative phase relationship between neighboring cores. When bent along a single axis a linear phase gradient, similar to the one used to steer the beam, should be introduced relative to the initial field. To test this, we created a calibration focus 225 μm away from the MCF distal end facet with the objective OBJ2. The complex field recorded by the CMOS when the MCF is not bent has a phase: ϕ1(x,y). After bending the fiber (Fig. 9(a)), by translating the calibration block to position p = i, the recorded phase pattern ϕi(x,y) changes compared to the initial pattern because of the altered conformation of the optical fiber. If the phase change is too dramatic, ϕ1(x,y) cannot be used to recreate a focused spot in the plane S. The 2D maps in Fig. 9(b) show the difference between the phase in the bending position p = i and the reference position p = 1, given by:

 figure: Fig. 9

Fig. 9 Bending of a MCF with DPC. (a) The translational stage where one end of the MCF is mounted is shifted to three different positions to increase fiber bending. The red arrows indicate the MCF. (b) For each bending position, a digital hologram is recorded on the CMOS: the three figures represent the difference between the extracted phase for each hologram and the phase obtained when the fiber is not bent. The insets show the color code used to represent the phase value. Scale bars are 100 μm. (c) The induced linear phase gradient across the fiber leads to a transverse shift of the focus spot in the plane S during DPC. The dashed red line indicates the position of the focus spot of an unbent fiber. Scale bars are 5 μm.

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Δϕ=ϕ1(x,y)ϕi(x,y).

Figure 9(b) clearly shows that a transverse linear phase has been introduced across the MCF compared to the reference wavefront ϕ1(x,y). The phase is a result of the path length differences across the cores induced by the bent fiber. In Fig. 9(c) the focus spot is shown to shift spatially because of the linear phase added to the field across the MCF. This happens in the same way that the focus was steered by adding the linear phase with an SLM.

6. Discussion and conclusion

Here we have demonstrated digital phase conjugation as a method to focus and control light through MCFs. Importantly, the MCFs ability to scan a focus spot could be utilized to enable a bendable, diffraction-limited fluorescent imaging endoscope. The limited core-to-core coupling allows for a wide scanning range with a single calibration hologram. We have shown that a spot can be scanned transversely over several micrometers. This scanning range could be further improved by reducing the coupling events in the MCF which could be done by decreasing the number of modes per core, for example with smaller core diameters. In fact, by reducing the number of high-order modes, we can reduce coupling. Another way to increase the scanning range would be using a shorter fiber to reduce coupling along the MCF length. In this work, we used 30 cm long fibers, but many endoscopic applications are compatible with imaging probes only a few centimeters in length.

A knowledge of the scanning intensity decay could be used to compensate the acquired signal strength during fluorescence imaging depending on the focused spot position. Furthermore, since the scan is made by simply introducing a linear tilt to the calculated phase pattern, this could be implemented by imaging the SLM on a pair of galvomirrors, which in turn would be imaged on the facet of the MCF. In this way, a liquid crystal SLM could be used to project the phase pattern to focus light, while the spot scanning would be achieved by tilting the galvomirrors. In this way, the scan speed could be increased to overcome scan speed limitations placed by the low refresh rate of liquid crystal SLMs.

As with other techniques based on wavefront shaping through optical fibers, this focusing technique depends on the conformation of the optical fiber. A clear advantage of the proposed implementation is that the calibration is extremely fast, since it requires a single hologram. This presents the possibility of calibration of a single, scannable focus spot during imaging. As an example, the light beacon presented in [24] could perform this function. Thus, in this case, while bending the fiber, a light beacon would allow a real time recalculation of the pattern to focus light. Moreover, the multiple spot projection ability could considerably increase the image acquisition speed with the inclusion of a final deconvolution step.

Acknowledgments

This project was conducted partially with the support of the Bertarelli Program in Translational Neuroscience and Neuroengineering (no. 10271) and the Wyss Center for Bio-and Neuro-Engineering (no. 10455). D.B.C. acknowledges support from the ‘EPFL Fellows’ fellowship program co-funded by Marie Curie, FP7 Grant agreement no. 291771.

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References

  • View by:

  1. H. H. Hopkins and N. S. Kapany, “A flexible fibrescope, using static scanning,” Nature 173(4392), 39–41 (1954).
    [Crossref]
  2. B. A. Flusberg, E. D. Cocker, W. Piyawattanametha, J. C. Jung, E. L. M. Cheung, and M. J. Schnitzer, “Fiber-optic fluorescence imaging,” Nat. Methods 2(12), 941–950 (2005).
    [Crossref] [PubMed]
  3. G. Oh, E. Chung, and S. H. Yun, “Optical fibers for high-resolution in vivo microendoscopic fluorescence imaging,” Opt. Fiber Technol. 19(6), 760–771 (2013).
    [Crossref]
  4. S. Bianchi and R. Di Leonardo, “A multi-mode fiber probe for holographic micromanipulation and microscopy,” Lab Chip 12(3), 635–639 (2012).
    [Crossref] [PubMed]
  5. A. M. Caravaca-Aguirre, E. Niv, D. B. Conkey, and R. Piestun, “Real-time resilient focusing through a bending multimode fiber,” Opt. Express 21(10), 12881–12887 (2013).
    [Crossref] [PubMed]
  6. I. N. Papadopoulos, S. Farahi, C. Moser, and D. Psaltis, “Focusing and scanning light through a multimode optical fiber using digital phase conjugation,” Opt. Express 20(10), 10583–10590 (2012).
    [Crossref] [PubMed]
  7. I. N. Papadopoulos, S. Farahi, C. Moser, and D. Psaltis, “High-resolution, lensless endoscope based on digital scanning through a multimode optical fiber,” Biomed. Opt. Express 4(2), 260–270 (2013).
    [Crossref] [PubMed]
  8. T. Čižmár and K. Dholakia, “Exploiting multimode waveguides for pure fibre-based imaging,” Nat. Commun. 3, 1027 (2012).
    [Crossref] [PubMed]
  9. 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(20), 203901 (2012).
    [Crossref] [PubMed]
  10. L. V. Amitonova, A. P. Mosk, and P. W. H. Pinkse, “The rotational memory effect of a multimode fiber,” Opt. Express 23(16), 20569–20575 (2015).
    [Crossref] [PubMed]
  11. D. Kim, J. Moon, M. Kim, T. D. Yang, J. Kim, E. Chung, and W. Choi, “Toward a miniature endomicroscope: pixelation-free and diffraction-limited imaging through a fiber bundle,” Opt. Lett. 39(7), 1921–1924 (2014).
    [Crossref] [PubMed]
  12. E. R. Andresen, G. Bouwmans, S. Monneret, and H. Rigneault, “Toward endoscopes with no distal optics: video-rate scanning microscopy through a fiber bundle,” Opt. Lett. 38(5), 609–611 (2013).
    [Crossref] [PubMed]
  13. E. R. Andresen, G. Bouwmans, S. Monneret, and H. Rigneault, “Two-photon lensless endoscope,” Opt. Express 21(18), 20713–20721 (2013).
    [Crossref] [PubMed]
  14. S. Feng, C. Kane, P. A. Lee, and A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61(7), 834–837 (1988).
    [Crossref] [PubMed]
  15. I. Freund, M. Rosenbluh, and S. Feng, “Memory effects in propagation of optical waves through disordered media,” Phys. Rev. Lett. 61(20), 2328–2331 (1988).
    [Crossref] [PubMed]
  16. C. Bellanger, A. Brignon, J. Colineau, and J. P. Huignard, “Coherent fiber combining by digital holography,” Opt. Lett. 33(24), 2937–2939 (2008).
    [Crossref] [PubMed]
  17. 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(9), 1707–1709 (2011).
    [Crossref] [PubMed]
  18. K. L. Reichenbach and C. Xu, “Numerical analysis of light propagation in image fibers or coherent fiber bundles,” Opt. Express 15(5), 2151–2165 (2007).
    [Crossref] [PubMed]
  19. X. Chen, K. L. Reichenbach, and C. Xu, “Experimental and theoretical analysis of core-to-core coupling on fiber bundle imaging,” Opt. Express 16(26), 21598–21607 (2008).
    [Crossref] [PubMed]
  20. I. M. Vellekoop and A. P. Mosk, “Focusing coherent light through opaque strongly scattering media,” Opt. Lett. 32(16), 2309–2311 (2007).
    [Crossref] [PubMed]
  21. K. Okamoto, Fundamentals of Optical Waveguides (Academic Press, 2006).
  22. E. E. Morales-Delgado, S. Farahi, I. N. Papadopoulos, D. Psaltis, and C. Moser, “Delivery of focused short pulses through a multimode fiber,” Opt. Express 23(7), 9109–9120 (2015).
    [Crossref] [PubMed]
  23. A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications (The Oxford Series in Electrical and Computer Engineering) (Oxford University Press, Inc., 2006).
  24. S. Farahi, D. Ziegler, I. N. Papadopoulos, D. Psaltis, and C. Moser, “Dynamic bending compensation while focusing through a multimode fiber,” Opt. Express 21(19), 22504–22514 (2013).
    [Crossref] [PubMed]

2015 (2)

2014 (1)

2013 (6)

2012 (4)

T. Čižmár and K. Dholakia, “Exploiting multimode waveguides for pure fibre-based imaging,” Nat. Commun. 3, 1027 (2012).
[Crossref] [PubMed]

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(20), 203901 (2012).
[Crossref] [PubMed]

I. N. Papadopoulos, S. Farahi, C. Moser, and D. Psaltis, “Focusing and scanning light through a multimode optical fiber using digital phase conjugation,” Opt. Express 20(10), 10583–10590 (2012).
[Crossref] [PubMed]

S. Bianchi and R. Di Leonardo, “A multi-mode fiber probe for holographic micromanipulation and microscopy,” Lab Chip 12(3), 635–639 (2012).
[Crossref] [PubMed]

2011 (1)

2008 (2)

2007 (2)

2005 (1)

B. A. Flusberg, E. D. Cocker, W. Piyawattanametha, J. C. Jung, E. L. M. Cheung, and M. J. Schnitzer, “Fiber-optic fluorescence imaging,” Nat. Methods 2(12), 941–950 (2005).
[Crossref] [PubMed]

1988 (2)

S. Feng, C. Kane, P. A. Lee, and A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61(7), 834–837 (1988).
[Crossref] [PubMed]

I. Freund, M. Rosenbluh, and S. Feng, “Memory effects in propagation of optical waves through disordered media,” Phys. Rev. Lett. 61(20), 2328–2331 (1988).
[Crossref] [PubMed]

1954 (1)

H. H. Hopkins and N. S. Kapany, “A flexible fibrescope, using static scanning,” Nature 173(4392), 39–41 (1954).
[Crossref]

Amitonova, L. V.

Andresen, E. R.

Bellanger, C.

Bianchi, S.

S. Bianchi and R. Di Leonardo, “A multi-mode fiber probe for holographic micromanipulation and microscopy,” Lab Chip 12(3), 635–639 (2012).
[Crossref] [PubMed]

Bouwmans, G.

Brignon, A.

Caravaca-Aguirre, A. M.

Chen, X.

Cheung, E. L. M.

B. A. Flusberg, E. D. Cocker, W. Piyawattanametha, J. C. Jung, E. L. M. Cheung, and M. J. Schnitzer, “Fiber-optic fluorescence imaging,” Nat. Methods 2(12), 941–950 (2005).
[Crossref] [PubMed]

Choi, W.

D. Kim, J. Moon, M. Kim, T. D. Yang, J. Kim, E. Chung, and W. Choi, “Toward a miniature endomicroscope: pixelation-free and diffraction-limited imaging through a fiber bundle,” Opt. Lett. 39(7), 1921–1924 (2014).
[Crossref] [PubMed]

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(20), 203901 (2012).
[Crossref] [PubMed]

Choi, Y.

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(20), 203901 (2012).
[Crossref] [PubMed]

Chung, E.

D. Kim, J. Moon, M. Kim, T. D. Yang, J. Kim, E. Chung, and W. Choi, “Toward a miniature endomicroscope: pixelation-free and diffraction-limited imaging through a fiber bundle,” Opt. Lett. 39(7), 1921–1924 (2014).
[Crossref] [PubMed]

G. Oh, E. Chung, and S. H. Yun, “Optical fibers for high-resolution in vivo microendoscopic fluorescence imaging,” Opt. Fiber Technol. 19(6), 760–771 (2013).
[Crossref]

Cižmár, T.

T. Čižmár and K. Dholakia, “Exploiting multimode waveguides for pure fibre-based imaging,” Nat. Commun. 3, 1027 (2012).
[Crossref] [PubMed]

Cocker, E. D.

B. A. Flusberg, E. D. Cocker, W. Piyawattanametha, J. C. Jung, E. L. M. Cheung, and M. J. Schnitzer, “Fiber-optic fluorescence imaging,” Nat. Methods 2(12), 941–950 (2005).
[Crossref] [PubMed]

Colineau, J.

Conkey, D. B.

Dasari, R. R.

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(20), 203901 (2012).
[Crossref] [PubMed]

Dholakia, K.

T. Čižmár and K. Dholakia, “Exploiting multimode waveguides for pure fibre-based imaging,” Nat. Commun. 3, 1027 (2012).
[Crossref] [PubMed]

Di Leonardo, R.

S. Bianchi and R. Di Leonardo, “A multi-mode fiber probe for holographic micromanipulation and microscopy,” Lab Chip 12(3), 635–639 (2012).
[Crossref] [PubMed]

Dunsby, C.

Fang-Yen, C.

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(20), 203901 (2012).
[Crossref] [PubMed]

Farahi, S.

Feng, S.

S. Feng, C. Kane, P. A. Lee, and A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61(7), 834–837 (1988).
[Crossref] [PubMed]

I. Freund, M. Rosenbluh, and S. Feng, “Memory effects in propagation of optical waves through disordered media,” Phys. Rev. Lett. 61(20), 2328–2331 (1988).
[Crossref] [PubMed]

Flusberg, B. A.

B. A. Flusberg, E. D. Cocker, W. Piyawattanametha, J. C. Jung, E. L. M. Cheung, and M. J. Schnitzer, “Fiber-optic fluorescence imaging,” Nat. Methods 2(12), 941–950 (2005).
[Crossref] [PubMed]

French, P. M. W.

Freund, I.

I. Freund, M. Rosenbluh, and S. Feng, “Memory effects in propagation of optical waves through disordered media,” Phys. Rev. Lett. 61(20), 2328–2331 (1988).
[Crossref] [PubMed]

Hopkins, H. H.

H. H. Hopkins and N. S. Kapany, “A flexible fibrescope, using static scanning,” Nature 173(4392), 39–41 (1954).
[Crossref]

Huignard, J. P.

Jung, J. C.

B. A. Flusberg, E. D. Cocker, W. Piyawattanametha, J. C. Jung, E. L. M. Cheung, and M. J. Schnitzer, “Fiber-optic fluorescence imaging,” Nat. Methods 2(12), 941–950 (2005).
[Crossref] [PubMed]

Kane, C.

S. Feng, C. Kane, P. A. Lee, and A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61(7), 834–837 (1988).
[Crossref] [PubMed]

Kapany, N. S.

H. H. Hopkins and N. S. Kapany, “A flexible fibrescope, using static scanning,” Nature 173(4392), 39–41 (1954).
[Crossref]

Kim, D.

Kim, J.

Kim, M.

D. Kim, J. Moon, M. Kim, T. D. Yang, J. Kim, E. Chung, and W. Choi, “Toward a miniature endomicroscope: pixelation-free and diffraction-limited imaging through a fiber bundle,” Opt. Lett. 39(7), 1921–1924 (2014).
[Crossref] [PubMed]

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(20), 203901 (2012).
[Crossref] [PubMed]

Lee, K. J.

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(20), 203901 (2012).
[Crossref] [PubMed]

Lee, P. A.

S. Feng, C. Kane, P. A. Lee, and A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61(7), 834–837 (1988).
[Crossref] [PubMed]

Monneret, S.

Moon, J.

Morales-Delgado, E. E.

Moser, C.

Mosk, A. P.

Neil, M. A. A.

Niv, E.

Oh, G.

G. Oh, E. Chung, and S. H. Yun, “Optical fibers for high-resolution in vivo microendoscopic fluorescence imaging,” Opt. Fiber Technol. 19(6), 760–771 (2013).
[Crossref]

Papadopoulos, I. N.

Paterson, C.

Piestun, R.

Pinkse, P. W. H.

Piyawattanametha, W.

B. A. Flusberg, E. D. Cocker, W. Piyawattanametha, J. C. Jung, E. L. M. Cheung, and M. J. Schnitzer, “Fiber-optic fluorescence imaging,” Nat. Methods 2(12), 941–950 (2005).
[Crossref] [PubMed]

Psaltis, D.

Reichenbach, K. L.

Rigneault, H.

Rosenbluh, M.

I. Freund, M. Rosenbluh, and S. Feng, “Memory effects in propagation of optical waves through disordered media,” Phys. Rev. Lett. 61(20), 2328–2331 (1988).
[Crossref] [PubMed]

Schnitzer, M. J.

B. A. Flusberg, E. D. Cocker, W. Piyawattanametha, J. C. Jung, E. L. M. Cheung, and M. J. Schnitzer, “Fiber-optic fluorescence imaging,” Nat. Methods 2(12), 941–950 (2005).
[Crossref] [PubMed]

Stone, A. D.

S. Feng, C. Kane, P. A. Lee, and A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61(7), 834–837 (1988).
[Crossref] [PubMed]

Thompson, A. J.

Vellekoop, I. M.

Xu, C.

Yang, T. D.

D. Kim, J. Moon, M. Kim, T. D. Yang, J. Kim, E. Chung, and W. Choi, “Toward a miniature endomicroscope: pixelation-free and diffraction-limited imaging through a fiber bundle,” Opt. Lett. 39(7), 1921–1924 (2014).
[Crossref] [PubMed]

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(20), 203901 (2012).
[Crossref] [PubMed]

Yoon, C.

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(20), 203901 (2012).
[Crossref] [PubMed]

Yun, S. H.

G. Oh, E. Chung, and S. H. Yun, “Optical fibers for high-resolution in vivo microendoscopic fluorescence imaging,” Opt. Fiber Technol. 19(6), 760–771 (2013).
[Crossref]

Ziegler, D.

Biomed. Opt. Express (1)

Lab Chip (1)

S. Bianchi and R. Di Leonardo, “A multi-mode fiber probe for holographic micromanipulation and microscopy,” Lab Chip 12(3), 635–639 (2012).
[Crossref] [PubMed]

Nat. Commun. (1)

T. Čižmár and K. Dholakia, “Exploiting multimode waveguides for pure fibre-based imaging,” Nat. Commun. 3, 1027 (2012).
[Crossref] [PubMed]

Nat. Methods (1)

B. A. Flusberg, E. D. Cocker, W. Piyawattanametha, J. C. Jung, E. L. M. Cheung, and M. J. Schnitzer, “Fiber-optic fluorescence imaging,” Nat. Methods 2(12), 941–950 (2005).
[Crossref] [PubMed]

Nature (1)

H. H. Hopkins and N. S. Kapany, “A flexible fibrescope, using static scanning,” Nature 173(4392), 39–41 (1954).
[Crossref]

Opt. Express (8)

S. Farahi, D. Ziegler, I. N. Papadopoulos, D. Psaltis, and C. Moser, “Dynamic bending compensation while focusing through a multimode fiber,” Opt. Express 21(19), 22504–22514 (2013).
[Crossref] [PubMed]

E. E. Morales-Delgado, S. Farahi, I. N. Papadopoulos, D. Psaltis, and C. Moser, “Delivery of focused short pulses through a multimode fiber,” Opt. Express 23(7), 9109–9120 (2015).
[Crossref] [PubMed]

A. M. Caravaca-Aguirre, E. Niv, D. B. Conkey, and R. Piestun, “Real-time resilient focusing through a bending multimode fiber,” Opt. Express 21(10), 12881–12887 (2013).
[Crossref] [PubMed]

I. N. Papadopoulos, S. Farahi, C. Moser, and D. Psaltis, “Focusing and scanning light through a multimode optical fiber using digital phase conjugation,” Opt. Express 20(10), 10583–10590 (2012).
[Crossref] [PubMed]

L. V. Amitonova, A. P. Mosk, and P. W. H. Pinkse, “The rotational memory effect of a multimode fiber,” Opt. Express 23(16), 20569–20575 (2015).
[Crossref] [PubMed]

E. R. Andresen, G. Bouwmans, S. Monneret, and H. Rigneault, “Two-photon lensless endoscope,” Opt. Express 21(18), 20713–20721 (2013).
[Crossref] [PubMed]

K. L. Reichenbach and C. Xu, “Numerical analysis of light propagation in image fibers or coherent fiber bundles,” Opt. Express 15(5), 2151–2165 (2007).
[Crossref] [PubMed]

X. Chen, K. L. Reichenbach, and C. Xu, “Experimental and theoretical analysis of core-to-core coupling on fiber bundle imaging,” Opt. Express 16(26), 21598–21607 (2008).
[Crossref] [PubMed]

Opt. Fiber Technol. (1)

G. Oh, E. Chung, and S. H. Yun, “Optical fibers for high-resolution in vivo microendoscopic fluorescence imaging,” Opt. Fiber Technol. 19(6), 760–771 (2013).
[Crossref]

Opt. Lett. (5)

Phys. Rev. Lett. (3)

S. Feng, C. Kane, P. A. Lee, and A. D. Stone, “Correlations and fluctuations of coherent wave transmission through disordered media,” Phys. Rev. Lett. 61(7), 834–837 (1988).
[Crossref] [PubMed]

I. Freund, M. Rosenbluh, and S. Feng, “Memory effects in propagation of optical waves through disordered media,” Phys. Rev. Lett. 61(20), 2328–2331 (1988).
[Crossref] [PubMed]

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(20), 203901 (2012).
[Crossref] [PubMed]

Other (2)

K. Okamoto, Fundamentals of Optical Waveguides (Academic Press, 2006).

A. Yariv and P. Yeh, Photonics: Optical Electronics in Modern Communications (The Oxford Series in Electrical and Computer Engineering) (Oxford University Press, Inc., 2006).

Supplementary Material (1)

NameDescription
Visualization 1: AVI (527 KB)      Focusing and scanning using digital phase conjugation and memory effect in a multicore fiber.

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Figures (9)

Fig. 1
Fig. 1 Experimental optical setup. The output of a He-Ne laser is expanded is split by the polarizing beam splitter PBS into the calibration and reference arms. The objective OBJ2 focuses the calibration beam in a plane S in front of the distal end of the multicore fiber (MCF). The proximal end facet of the MCF is imaged on the CMOS sensor through the 4f imaging system OBJ3-L3. Using the non-polarizing beam splitter BS2, the reference arm is combined with the image on the CMOS to form a digital hologram. The phase extracted from the digital hologram is conjugated and projected on the spatial light modulator (SLM), which modulates and reflects the reference beam towards the MCF through BS2. A focus spot forms in the plane S and is imaged using the 4f system OBJ2-L2 on the CCD camera. The translational stage TS bends the fiber in a controllable way. OBJ1 = 10 × , NA = 0.25, Newport, OBJ2 = OBJ3 = 20 × , NA = 0.4, Newport. Focal length lenses: L1 = 150 mm, L2 = 100 mm, L3 = 200 mm.
Fig. 2
Fig. 2 DPC focus spot characterization. (a) Focus spot size (FWHM, blue points) and enhancement (red points) as a function of the working distance. The lines are drawn only for clarity. (b) Example of DPC focus spot imaged on the CCD camera. (c) The same focus spot of (b) with the CCD exposure time increased 100 times. The six bright speckle areas around the main lobe are a result of the quasi-periodicity of the MCF. Scale bars are 20 μm.
Fig. 3
Fig. 3 Digital scanning of the focus spot using a single calibration hologram. (a) The phase conjugation pattern is combined with a linear phase gradient and projected on the SLM. By changing the direction and the strength of the gradient it is possible to scan the focus spot in a regular grid. The image is a combination of several DPC projections (see Visualization 1). (b) The square markers represent the peak intensities of the DPC spots along the red dashed line in (a). Scale bar 20 μm.
Fig. 4
Fig. 4 Using a pulsed laser it is possible to independently phase conjugate distinct modes of the MCF cores. The scanning range of the fundamental LP01 mode is larger than the one of the higher order modes because of the limited core-to-core coupling.
Fig. 5
Fig. 5 Axial shifting of the DPC focus spot using a single calibration hologram. The central row of images shows a phase-conjugated spot observed at different planes. The calibration hologram was taken focusing at a distance of 400 μm from the fiber facet. By adding a negative lens phase to the phase hologram, the focus shifts towards the fiber. The images above and below the central row show the axial shift of the focus with the increasingly positive or negative lens phases.
Fig. 6
Fig. 6 Pattern projection using a single calibration hologram: (a) cross obtained as a combination of shifted spots; (b) the same pattern as (a) except the spots of one arm are in opposite phase to those of the other arm; (c) multiple spot projection, in this case nine spots are projected simultaneously in a regular grid. Scale bars are 5 μm. (d) Simulation of the cross-pattern projection with phase only modulation; (e) simulation of the cross-pattern projection with amplitude and phase modulation.
Fig. 7
Fig. 7 Focusing through a waveguide array in no-coupling and coupling conditions. The propagation in the waveguide array was simulated using coupled mode theory and the free space propagation with the beam propagation method. (a) Focusing in no coupling conditions is possible by assigning a quadratic phase at the input of the waveguide array. The phase relationship between the cores is preserved along the waveguide and it is possible to focus at a given distance for the waveguide facet (400 μm in this case). Focusing using DPC gives equivalent results. (b) In coupling conditions, the phase relationship between cores is not preserved along the waveguide and focusing using a quadratic phase pattern is no longer possible. DPC allows focusing in severe coupling conditions.
Fig. 8
Fig. 8 Scanning range using DPC in a waveguide array with different coupling conditions by tuning the distance between the cores. Increased coupling decreases the scanning range. The green curve represents the no-coupling conditions, which yields the maximum scanning range.
Fig. 9
Fig. 9 Bending of a MCF with DPC. (a) The translational stage where one end of the MCF is mounted is shifted to three different positions to increase fiber bending. The red arrows indicate the MCF. (b) For each bending position, a digital hologram is recorded on the CMOS: the three figures represent the difference between the extracted phase for each hologram and the phase obtained when the fiber is not bent. The insets show the color code used to represent the phase value. Scale bars are 100 μm. (c) The induced linear phase gradient across the fiber leads to a transverse shift of the focus spot in the plane S during DPC. The dashed red line indicates the position of the focus spot of an unbent fiber. Scale bars are 5 μm.

Equations (8)

Equations on this page are rendered with MathJax. Learn more.

ϕ SLM ( x,y )=arg[ e j ϕ DPC ( x,y ) e j( k x x+ k y y ) ],
ϕ SLM ( x,y )=arg[ e j ϕ DPC ( x,y ) e j π λF ( x 2 + y 2 ) ],
ϕ SLM ( x,y )=arg[ e j ϕ DPC ( x,y ) i e j( k xi x+ k yi y ) ].
A( x )=comb( Λ x x ) E a ( x ),
A ˜ ( f x )=FT{ A( x ) }=FT{ comb( Λ x x ) }FT{ E a ( x ) }.
w MAX =w( f )= w 2 ( 0 )+ f 2 N A 2 .
p x f λ Λ x .
Δϕ= ϕ 1 ( x,y ) ϕ i ( x,y ).

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