Wavefront engineering for controlled structuring of far-field intensity and phase patterns from multimodal optical fibers

(a) Optical system for direct far-field shaping of the MMF output: M—mirror, L—lens, MO—objective, MMF—multimode fiber, SLM—spatial light modulator, BS—beam splitter, and CCD—charged coupling device. Dashed lines indicate conjugate planes. Top inset: construction of the phase pattern displayed by the SLM to obtain B_scan and B_ref. Bottom left inset: indexing of the input facet of the MMF in a grid of 25 × 25 pixel elements. (b) Example interferogram $B_{interfere}^{j_{x} j_{y} p}$ ⁠. (c) Intensity variation of $B_{interfere}^{j_{x} j_{y} p}$ at pixel (u, v) over the phase shift. (d) Image on CCD2 of a focused beam. (e) Images on CCD2 of different far-field foci (white circle indicates the theoretical limit of the fiber NA). (f) Focused beam intensity for each targeted far-field point. (g) Scatter plot of focused beam intensity (blue circles) and average focused beam intensity over 1° interval (black squares).

On the scan path, lenses L1 and L2 (f_L2 = 200 mm) formed a 4f relay between the SLM and the back aperture of the microscope objective MO1 (0.65 NA, 40x, AMEP4625, ThermoFisher), while BS2 was used to image the input facet of the MMF onto a camera (CCD1). MO1 focused the beam onto a slack, ∼5 cm-long MMF (0.22 NA, 50 μm core diameter, Thorlabs FG050UGA), which randomized the output pattern resulting in a new beam referred to as B_random. Light transmitted through the MMF was collected by the microscope objective MO2 (10x, 0.3 NA, MPLFLN10x—Olympus, f_MO2 = 18 mm), and lenses L4 (f_L4 = 125 mm) and L5 (f_L5 = 100 mm) formed a 4f optical system conjugating the back focal plane of MO2 on CCD2 (CS505MU—Thorlabs), projecting on it the far field of the fiber facet (referred to as the (u, v) plane).

The distance r of a generic point (u, v) from the center of the far field is given by

r = \frac{f_{L 5}}{f_{L 4}} f_{MO2} \tan θ_{out}

(1)

where θ_out is the angle of the beam emitted by the fiber measured with respect to the optical axis, or in the component form, along the u and v axes, $u = \frac{f_{L 5}}{f_{L 4}} f_{MO2} \tan θ_{u}$ and $v = \frac{f_{L 5}}{f_{L 4}} f_{MO2} \tan θ_{v}$

On the reference path, an adjustable iris was placed at the focal point of L1 blocking the residual scanning beam. B_ref was then expanded by a beam expander formed by lenses L7 and L8 (f_L7 = 30 mm and f_L8 = 100 mm) before being re-joined with the scanning beam by BS3. Thus, B_random and B_ref beams interfere on CCD2 with an angle controlled by BS3 [see the representative interferogram in Fig. 1(b)]. To obtain the B_scan and B_ref beams, blazed sawtooth gratings ϕ_scan and ϕ_ref were generated and summed together to obtain the image $ϕ_{intefere}^{j_{x} j_{y} p} (x y)$ on the SLM, whose screen plane is represented by x and y axes, with

ϕ_{scan}^{j_{x} j_{y} p} (x y) = \mod (a x + b y + p 2 π)

(2)

ϕ_{ref} (x y) = \mod (c x 2 π)

(3)

where a and b define the periodicity of the grating along x and y so that the beam B_scan can scan on a 25 × 25 grid in the input facet of the MMF. Each element of this grid is identified by the indices (j_x, j_y) [see the bottom left inset of Fig. 1(a)]. The parameter c defines the periodicity of ϕ_ref, and it was chosen such that B_ref is aligned with the iris on the reference path and not on the core of the MMF along the scan path. Finally, p is the phase shift applied at each position in the (j_x, j_y) plane as detailed below.

To obtain a specific emission angle θ_out from the MMF, a plane wave calibration procedure was applied based on the method proposed in Ref. 3. In the algorithm, the (u, v) point related to a specific θ_out is monitored in intensity on the interference pattern on CCD2 while scanning the beam on the (j_x, j_y) plane using the following equation as an input to the SLM:

ϕ_{interfere}^{j_{x} j_{y} p} (x y) = \arg (\exp (i ϕ_{scan}^{j_{x} j_{y} p} (x y)) + \exp (i ϕ_{ref}))

(4)

For each (j_x, j_y) pair, p was tested from 0 to 2π [the full scan procedure on the input facet is shown in visualization (1) of the supplementary material]. The intensity variation as a function of p on a representative (u, v) point of the far-field plane is displayed in Fig. 1(c), from which the phase step $p_{opt}^{j_{x} j_{y}}$ giving the maximum intensity value at the coordinate (u, v) is extracted and used to generate the phase pattern for the $j_{x}^{t h}$ ⁠, $j_{y}^{t h}$ grid element, $ϕ_{opt}^{j_{x} j_{y}} (x y) = ϕ_{scan}^{j_{x} j_{y} p_{opt}^{j_{x} j_{y}}} (x y)$ ⁠. This input-phase optimization procedure is repeated for all the 625 points on the (j_x, j_y) plane, generating the SLM input image to focus light on the point (u, v) of the far field, as given by the sum of all optimized phase patterns,

Φ (x y) = \arg (\sum_{j_{x} = 1}^{25} \sum_{j_{y} = 1}^{25} \exp (i ϕ_{opt}^{j_{x} j_{y}} (x y))

(5)

III. RESULTS AND DISCUSSION

Typical results of this optimization are shown in Figs. 1(d) and 1(e) for different (u, v) far-field points and in visualization 2 of the supplementary material. Once the fiber is bent or moved, Φ needs to be recomputed. Concerning computational times, for the 30 × 30 square array of focused spots in the (u, v) plane shown in visualization 2 of the supplementary material, the entire process was completed within ∼15 min. The majority of this time is dedicated to the computation of the optimized gratings on the graphics processing unit (GPU—NVIDIA GeForce GTX 960 in our case) and refreshing the SLM. On average, about 9% of the total intensity transmitted through the MMF is contained within the focused spot in the far field. This is comparable to what has been achieved in the near field,²⁰ although by applying polarization control, this could be further increased.³ In the near field, after removing lens L4, the intensity distribution appears to be stochastic and no trace of the far-field focus (shown in 1D) is visible (see Fig. S1 of the supplementary material).

A set of images of the 30 × 30 array of far-field spots [with the optical path in the configuration shown in Fig. 1(a)] were recorded on CCD2. The intensity of each far-field spot for each targeted pixel is shown as a color-map in Fig. 1(f), with related distribution displayed in Fig. 1(g). Although an overall decrease of intensity has been observed for high values of θ, the mean value of the beam waist remains much more stable across the entire fiber NA (see Fig. S2 of the supplementary material) although a small increase is observed close to the limit of the fiber (along the v axis, a 10% increase in average spot size was observed between spots generated at 0° and 12°). It has previously been demonstrated that in the near field, the signal to noise ratio increases at the edge of the fiber facet due to a higher availability of guided modes⁷—this is not the case when the system works in the far field. This can be likely assigned to the fact that the high order modes in the linearly polarized mode formulation (LP_lm) are associated with a higher transversal component of the wavevector k_t and a lower power transfer efficiency.²¹ Indeed, as the number of modes available at a specific k_t increases for high order modes, a given position in the far field does not only foster the system working point at a specific k_t but also a specific pair (k_u, k_v), being $k_{t} = \sqrt{k_{u}^{2} + k_{v}^{2}}$ ⁠. Recently, results showing a similar decrease in the intensity for lower values of θ have been reported for a DMD based calibration used to image distal objects from the fiber.²²

As the fiber length and bending could potentially influence the method, we investigated if far-field structuring is possible also for longer or bent fibers. Figures S3–S5 of the supplementary material show the distribution of beam intensity for a straight 5 cm fiber, a 5 cm-long fiber bent by 90°, and a 15 cm fiber, respectively. In these experiments, due to the changing position of the fiber, an internal reference beam was used to keep the reference signal preserved across different measurements and make them more comparable. In all three cases, more intense beams are generated at the center of the far field; however, beams of lower azimuthal angle generated through the 15 cm fiber appear to be of slightly weaker intensity than the 5 cm fiber. It is important to highlight, however, that the here-applied calibration procedure is sensitive to a dynamic deformation of the fiber. Although the transmission can be calibrated with the bent fiber, once the fiber is moved, a new calibration is required. Methods to compensate for bending have been investigated for near field optimizations,^5,23 which could potentially be applied in our case. The guide star approach has been well established for focusing light through highly scattering media without prior knowledge of the distal side, with some approaches highlighted in Ref. 24, and recently applied to optical fibers.²⁵

To measure the azimuthal angle θ_out and the divergence angle α of the beam emitted by the fiber, MO2 was removed and a CCD (CS505MU—Thorlabs) fixed to a single axis translation stage was positioned ∼5 mm from the output of the MMF, as shown in Fig. 2(a). Thus, by measuring the profile of the speckle pattern (emitted directly from the fiber) in two positions (1 mm apart), θ_out and α could be estimated. Images of the beam in both CCD positions are shown in the inset of Fig. 2(a) (profile plots are shown in Fig. S6 of the supplementary material). Uncertainties were calculated by considering the distance between the two positions to be 1.0±0.1 mm account for error in the position of CCD2.

FIG. 2.

(a) Optical setup to measure θ_out and α with CCD2 placed on a translation stage. (b) Measurement of θ_out for a 30 × 30 matrix of far-field foci. (c) Line plot of B—blue dashed lines indicate the error on θ_out. Black dashed lines indicate the theoretical limit from the fiber NA. [(d) and (e)] Measurement of $θ_{o u t_{u}}$ and $θ_{o u t_{u}}$ for a 30 × 30 matrix of far-field foci. (f) Measurement of divergence angle α for a 30 × 30 matrix of far-field foci. Points outside of the NA indicated in white (by setting a threshold on the intensity ratio of the far-field focus). (g) Histogram of α

To calculate θ_out, we considered the coordinate difference in the maximum intensity pixel for the first set (u₁, v₁) and second set (u₂, v₂) of images. For α, we consider the full width at half maximum (FWHM) of the intensity. Two example images of the speckle pattern emitted from the fiber in each position are shown in the inset of Fig. 2(a). The pixel size on the CCD was 3.45 × 3.45 μm, and therefore,

θ_{out} = \tan^{- 1} (\frac{3.45 \sqrt{{(u_{1} - u_{2})}^{2} + {(v_{1} - v_{2})}^{2}}}{1000})

(6)

α = 2 \tan^{- 1} (\frac{3.45 (F W H M_{1} - F W H M_{2})}{1000})

(7)

Figure 2(b) is a color-map showing the measured θ_out for the targeted point of the (u, v) plane. As expected, at the center of the far field, the azimuthal angle is close to zero, whereas at the edge, θ_out approaches the limit of the numerical aperture of the fiber (NA = 0.22 corresponding to 12.7°). The relationship between the far field coordinate and the azimuthal angle is expected to follow (1). The experimental relationship between the far field coordinate and the measured θ_out is shown in the profile plot in Fig. 2(c). The angular components in the u and v directions are shown in Figs. 2(d) and 2(e), calculated as

θ_{o u t_{u}} = \tan^{- 1} (\frac{3.45 (u_{1} - u_{2})}{1000})

(8)

θ_{o u t_{v}} = \tan^{- 1} (\frac{3.45 (v_{1} - v_{2})}{1000})

(9)

The divergence angle α is plotted as a color-map in Fig. 2(f). The histogram of the data in Fig. 2(f) is shown in Fig. 2(g), featuring over 80% of the points between −1° and 1° (an average of 0.70° and a standard deviation of 0.68°). The low divergence angle and uniformity of the profile plots demonstrate that a far-field intensity structuring resulted in complex beam steering at the tip of a MMF.

After demonstrating how far-field beam steering of a laser beam can be performed through a MMF, next, we show how beam splitting may be performed to simultaneously generate n multiple beams $f_{θ_{k}}$ exiting the fiber with azimuthal angle θ_k. If $p_{opt}^{j_{x} j_{y} k}$ is the optimized phase shift for a given k^th far-field focus, with k ranging from 1 to n, the phase image on the SLM is then

Φ (x y) = \arg (\sum_{j_{x} = 1}^{25} \sum_{j_{y} = 1}^{25} \sum_{k = 1}^{n} \exp (i ϕ_{scan}^{j_{x} j_{y} p_{opt}^{j_{x} j_{y} k}}))

(10)

Figure 3 shows the far-field image of three multiple beam arrays with 1, 2, and 3 different angles simultaneously. In this example, between 4% and 6% of the total intensity is contained in each hot-spot array, with a signal to noise ratio reduced as the number of beams is increased: the average beam intensity $(\frac{1}{n} Σ_{(k = 1)}^{n} B_{θ_{k}})$ falls from 1 to 0.5 to 0.39 and the signal to noise ratio falls from 268 to 132 to 108. Therefore, a far-field optimization of light transmission through a MMF can be used to perform complex beam steering of multiple beams, with the above-detailed limitations in terms of divergence angle and signal to noise ratio.

FIG. 3.

Multiple far-field foci generated simultaneously and normalized profile plots (averaged over 20 pixels). (a) Single far-field focus. (b) Two far-field foci. (c) Three far-field foci.

Another parameter that can be controlled by the optimization procedure is the phase of the outputted beam. So far, we have optimized the phase of the input beam to $p_{opt}^{j_{x} j_{y}}$ ⁠, which gives the maximum intensity of the interference fringes at the targeted spot (u, v), i.e., giving constructive interference between B_ref and B_out. However, it is also possible for B_ref and B_out to be out of phase by p, where p is in the range of 0–2π. This can be done by applying a shift p to $p_{opt}^{j_{x} j_{y}}$ ⁠, i.e., the phase is shifted by $q = \mod (p_{opt}^{j_{x} j_{y}} + p 2 π)$ ⁠. This gives an input phase image

Φ_{phase} (x y p) = \arg (\sum_{j_{x} = 1}^{25} \sum_{j_{y} = 1}^{25} \exp (i ϕ_{scan}^{j_{x} j_{y} q} (x y))

(11)

generating the beam B_phase(p). To do this, 16 phase steps were applied to a single focused spot in the far-field plane. The intensity distribution is unaffected by the application of phase shift p. To measure the induced phase shift, the phase Ψ of B_phase(p) was reconstructed from the Fourier transform of the interferogram between B_phase(p) and B_ref⁷ The interferogram and the reconstructed phase of B_phase for a representative value of p = π are shown in Figs. 4(a) and 4(b), respectively.

FIG. 4.

https://doi.org/10.1364/ol.32.002309

(a) Interferogram between B_ref and B_phase. (b) Direct phase measurement of B_phase(π)—circle indicates the limit of the NA of the fiber. (c) Top row: measurement of the phase shift between B_phase(0) and B_phase(p). ΔΨ(p) = Ψ(B_phase(p)) − Ψ(B_phase(0)): blue line is the experimental data and black line is a linear fit. (d) Bar graph showing the standard deviation of ΔΨ(p) over the entire core (blue) and in the region corresponding to the far-field focus (orange).

Figure 4(c) shows the calculated phase shift ΔΨ(p) = Ψ(B_phase(p)) − Ψ(B_phase(0)). The linear fit of ΔΨ(p) was ΔΨ(p) = 0.98p + 0.18, indicating that the phase shift applied by the SLM was well preserved at the modulated beam emitted by the MMF. We have observed that this phase modulation takes place within the entire fiber NA, although the system provides a focused spot in a specific point of the far field, where the phase values are less dispersed than in the rest of the core [Fig. 4(d)].

IV. CONCLUSION

In summary, we report evidence that wavefront engineering techniques can be employed to achieve full control on the distribution of intensity and phase of light in the far field of a MMF. We have employed a plane wave based method, which complements other techniques such as the direct measurement of the transmission matrix,⁸ which could potentially be applied to control the far field transmission. Deep learning algorithms have also been applied to structure the near field without the need for any reference beam and could be investigated for the generation of low divergence rays.²⁶ Our results can potentially enable pure fiber-based beam steering of multiple collimated beamlets and open the door to the development of phase-encoded imaging and sensing through a MMF. Potentially, by adding a second fiber for collection, the weak scattering signal from distal objects is decoupled from the background signal from the fiber.

SUPPLEMENTARY MATERIAL

See the supplementary material for Figs. S1–S6.

ACKNOWLEDGMENTS

L.C., M.D.V., and Fe.P acknowledge funding from the European Union’s Horizon 2020 Research and Innovation Program under Grant Agreement No. 828972. Fi.P., A.B., and Fe.P. acknowledge funding from the European Research Council under the European Union’s Horizon 2020 Research and Innovation Program under Grant Agreement No. 677683. Fi.P., M.D.V. and Fe.P. acknowledge that this project has received funding from the European Union's Horizon 2020 Research and Innovation Program under Grant Agreement No 101016787. M.P. and M.D.V. acknowledge funding from the European Research Council under the European Union’s Horizon 2020 Research and Innovation Program under Grant Agreement No. 692943. M.P., Fe.P., and M.D.V. were funded by the U.S. National Institutes of Health (Grant No. 1UF1NS108177-01). M.D.V. was funded by the U.S. National Institutes of Health (Grant No. U01NS094190).

M.D.V. and F.P. are founders and hold private equity in Optogenix, a company that develops, produces, and sells technologies to deliver light into the brain.

DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request. The code used in this study will be made available at https://www.projectnanobright.eu

REFERENCES

1.

I. M.

Vellekoop

and

A. P.

Mosk

, “

Focusing coherent light through opaque strongly scattering media

,”

Opt. Lett.

32

,

2309

–

2311

(

2007

).

https://doi.org/10.1038/ncomms2024

2.

T.

Čižmár

and

K.

Dholakia

, “

Exploiting multimode waveguides for pure fibre-based imaging

,”

Nat. Commun.

3

,

1027

(

2012

).

https://doi.org/10.1364/oe.19.018871

3.

T.

Čižmár

and

K.

Dholakia

, “

Shaping the light transmission through a multimode optical fibre: Complex transformation analysis and applications in biophotonics

,”

Opt. Express

19

,

18871

–

18884

(

2011

).

https://doi.org/10.1364/oe.22.002933

4.

M.

Plöschner

,

B.

Straka

,

K.

Dholakia

, and

T.

Čižmár

, “

GPU accelerated toolbox for real-time beam-shaping in multimode fibres

,”

Opt. Express

22

,

2933

–

2947

(

2014

).

https://doi.org/10.1038/nphoton.2015.112

5.

M.

Plöschner

,

T.

Tyc

, and

T.

Čižmár

, “

Seeing through chaos in multimode fibres

,”

Nat. Photonics

9

,

529

–

535

(

2015

).

https://doi.org/10.1364/oe.21.012881

6.

A. M.

Caravaca-Aguirre

,

E.

Niv

,

D. B.

Conkey

, and

R.

Piestun

, “

Real-time resilient focusing through a bending multimode fiber

,”

Opt. Express

21

,

12881

–

12887

(

2013

).

https://doi.org/10.1364/oe.20.010583

7.

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

,

10583

–

10590

(

2012

).

https://doi.org/10.1364/oe.23.023845

8.

D.

Loterie

,

S.

Farahi

,

I.

Papadopoulos

,

A.

Goy

,

D.

Psaltis

, and

C.

Moser

, “

Digital confocal microscopy through a multimode fiber

,”

Opt. Express

23

,

23845

–

23858

(

2015

).

https://doi.org/10.1038/s41377-018-0111-0

9.

S. A.

Vasquez-Lopez

,

R.

Turcotte

,

V.

Koren

,

M.

Plöschner

,

Z.

Padamsey

,

M. J.

Booth

,

T.

Čižmár

, and

N. J.

Emptage

, “

Subcellular spatial resolution achieved for deep-brain imaging in vivo using a minimally invasive multimode fiber

,”

Light: Sci. Appl.

7

,

110

(

2018

).

https://doi.org/10.1364/oe.25.013782

10.

I.

Gusachenko

,

M.

Chen

, and

K.

Dholakia

, “

Raman imaging through a single multimode fibre

,”

Opt. Express

25

,

13782

–

13798

(

2017

).

https://doi.org/10.1038/s41377-018-0003-3

11.

A.

Lombardini

,

V.

Mytskaniuk

,

S.

Sivankutty

,

E. R.

Andresen

,

X.

Chen

,

J.

Wenger

,

M.

Fabert

,

N.

Joly

,

F.

Louradour

,

A.

Kudlinski

, and

H.

Rigneault

, “

High-resolution multimodal flexible coherent Raman endoscope

,”

Light: Sci. Appl.

7

,

10

(

2018

).

https://doi.org/10.1364/oe.27.030055

12.

J.

Trägårdh

,

T.

Pikálek

,

M.

Šerý

,

T.

Meyer

,

J.

Popp

, and

T.

Čižmár

, “

Label-free CARS microscopy through a multimode fiber endoscope

,”

Opt. Express

27

,

30055

–

30066

(

2019

).

https://doi.org/10.1039/c1lc20719a

13.

S.

Bianchi

and

R.

Di Leonardo

, “

A multi-mode fiber probe for holographic micromanipulation and microscopy

,”

Lab Chip

12

,

635

–

639

(

2012

).

https://doi.org/10.1364/optica.6.001185

14.

V.

Tsvirkun

,

S.

Sivankutty

,

K.

Baudelle

,

R.

Habert

,

G.

Bouwmans

,

O.

Vanvincq

,

E. R.

Andresen

, and

H.

Rigneault

, “

Flexible lensless endoscope with a conformationally invariant multi-core fiber

,”

Optica

6

,

1185

–

1189

(

2019

).

https://doi.org/10.1038/s41592-020-01024-z

15.

W.

Zong

,

R.

Wu

,

S.

Chen

,

J.

Wu

,

H.

Wang

,

Z.

Zhao

,

G.

Chen

,

R.

Tu

,

D.

Wu

,

Y.

Hu

,

Y.

Xu

,

Y.

Wang

,

Z.

Duan

,

H.

Wu

,

Y.

Zhang

,

J.

Zhang

,

A.

Wang

,

L.

Chen

, and

H.

Cheng

, “

Miniature two-photon microscopy for enlarged field-of-view, multi-plane and long-term brain imaging

,”

Nat. Methods

18

,

46

–

49

(

2021

).

https://doi.org/10.1088/2515-7647/ab3f0e

16.

V.

de Miguel Soto

and

M.

Lopez-Amo

, “

Truly remote fiber optic sensor networks

,”

J. Phys.: Photonics

1

,

042002

(

2019

).

https://doi.org/10.1038/s41467-017-02434-y

17.

A.

Xomalis

,

I.

Demirtzioglou

,

E.

Plum

,

Y.

Jung

,

V.

Nalla

,

C.

Lacava

,

K. F.

MacDonald

,

P.

Petropoulos

,

D. J.

Richardson

, and

N. I.

Zheludev

, “

Fibre-optic metadevice for all-optical signal modulation based on coherent absorption

,”

Nat. Commun.

9

,

182

(

2018

).

https://doi.org/10.1109/jlt.2015.2404860

18.

N.

Yu

and

F.

Capasso

, “

Optical metasurfaces and prospect of their applications including fiber optics

,”

J. Lightwave Technol.

33

,

2344

–

2358

(

2015

).

https://doi.org/10.1364/oe.27.032638

19.

R.

Florentin

,

V.

Kermene

,

A.

Desfarges-Berthelemot

, and

A.

Barthelemy

, “

Shaping of amplified beam from a highly multimode Yb-doped fiber using transmission matrix

,”

Opt. Express

27

,

32638

–

32648

(

2019

).

https://doi.org/10.1364/oe.19.000247

20.

R.

di Leonardo

and

S.

Bianchi

, “

Hologram transmission through multi-mode optical fibers

,”

Opt. Express

19

,

247

–

254

(

2011

).

https://doi.org/10.1364/boe.6.004014

21.

M.

Pisanello

,

A.

Della Patria

,

L.

Sileo

,

B. L.

Sabatini

,

M.

de Vittorio

, and

F.

Pisanello

, “

Modal demultiplexing properties of tapered and nanostructured optical fibers for in vivo optogenetic control of neural activity

,”

Biomed. Opt. Express

6

,

4014

–

4026

(

2015

).

https://doi.org/10.1063/5.0038367

22.

I. T.

Leite

,

S.

Turtaev

,

D. E.

Boonzajer Flaes

, and

T.

Čižmár

, “

Observing distant objects with a multimode fiber-based holographic endoscope

,”

APL Photonics

6

,

036112

(

2021

).

https://doi.org/10.1364/oe.25.006263

23.

D.

Loterie

,

D.

Psaltis

, and

C.

Moser

, “

Bend translation in multimode fiber imaging

,”

Opt. Express

25

,

6263

–

6273

(

2017

).

https://doi.org/10.1038/nphoton.2015.140

24.

R.

Horstmeyer

,

H.

Ruan

, and

C.

Yang

, “

Guidestar-assisted wavefront-shaping methods for focusing light into biological tissue

,”

Nat. Photonics

9

,

563

–

571

(

2015

).

25.

S.

Li

,

S. A. R.

Horsley

,

T.

Tyc

,

T.

Cizmar

, and

D. B.

Phillips

, “

Guide-star assisted imaging through multimode optical fibres

,” arXiv:2005.06445 [physics.optics] (

2020

).

https://doi.org/10.1038/s41377-018-0074-1

26.

B.

Rahmani

,

D.

Loterie

,

G.

Konstantinou

,

D.

Psaltis

, and

C.

Moser

, “

Multimode optical fiber transmission with a deep learning network

,”

Light: Sci. Appl.

7

,

69

(

2018

).