Bend translation in multimode fiber imaging

Damien Loterie; Demetri Psaltis; Christophe Moser

doi:10.1364/OE.25.006263

1. Introduction

Multimode fibers have attractive features as imaging devices, such as small thickness, low cost, low loss, and the fact that they can be made of inert and biocompatible materials. Compared to other fiber-based imaging systems (e.g. fiber bundles or MEMS scanners), multimode fibers offer a close to optimum field of view, resolution and collection efficiency for a given cross-sectional diameter. The main challenge is the complex light propagation process inside multimode fibers: light suffers from modal scrambling, meaning that an image injected from one end of a fiber will become scrambled as it propagates through it [1, 2]. In addition, the same fiber can cause a different scrambling effect depending on its spatial configuration, because optical modes propagate differently in a bent fiber [3,4].

Modal scrambling is deterministic and can be undone, as was shown several times in past and recent literature. Early demonstrations used analog holography [5] or nonlinear mixing [6] to perform phase conjugation (also known as time reversal) of the distorted wave. After passing through the fiber in reverse, the image appears undistorted again. Current experiments generally rely instead on wavefront shaping using spatial light modulators. First, modal scrambling is characterized in a prior calibration step. Based on this calibration, the light input of the fiber is modulated to create a spot at the output [7–9]. This spot can be scanned over the target area in front of the fiber tip. An image is then formed from the resulting signal, which travels back from the target via the same fiber to a proximally located detector. A variety of contrast mechanisms have been demonstrated in an endoscopic configuration without distal optics or electronics [10–18]. The imaging speed of many spot-scanning systems is relatively slow, mainly because of limitations in current spatial light modulator technology. A number of solutions have been proposed to address this issue [10, 11, 13, 19, 20] and imaging frame rates of several Hz are now possible in certain implementations.

An important question is how to manage the bend-induced changes in propagation characteristics of multimode fibers, which usually occur after deformations of the order of a few millimeters [10, 21, 22]. Several compensation mechanisms have been proposed. For example, multiple calibrations can be stored for different spatial configurations of the fiber in a semi-rigid endoscope setting. The correct calibration can be loaded based on feedback from a passive holographic beacon at the distal tip [21]. Using a photodetector in the distal end and fast electronic feedback, the system can be recalibrated on a millisecond timescale [23]. Recently, mechanisms were proposed to correct the distortions by exploiting the reflections from the distal facet of the fiber [24, 25]. Finally, a theoretical model describing light propagation in bent fibers was introduced and verified experimentally [4], allowing the propagation characteristics to be predicted based on the curvature at every location and the physical parameters of the fiber.

In this paper, we take a closer look at the following question: are there any ways to significantly alter the geometrical shape of the fiber, while conserving constant propagation characteristics? We show here that this is possible when a bend with constant shape is translated along the length of a multimode fiber, as illustrated in Fig. 1. This effect is observed to different extents on various fibers. Practically, this could be exploited for example to allow a multimode fiber imaging device to translate longitudinally over a certain distance inside a rigid conduit such as a catheter, after a calibration or calculation of the propagation characteristics of the whole device in its initial state.

Fig. 1 Experimental geometry for imaging during translation of a bend in the fiber. Collimated laser light is modulated by a spatial light modulator, and relayed via a lens (L2) and a microscope objective (MO2) to a multimode fiber. The fiber (represented in light blue) passes through a sliding jacket (represented in white) that allows an S-bend with constant shape to be translated between the two ends of the fiber. The output of the fiber is relayed by a microscope objective (MO1) and a lens (L1) to a camera, where a holographic recording can be made by using an off-axis reference beam. The bend has a peak curvature of 40 m⁻¹ and the overall length of the fiber is approximately 250 mm.

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Under the assumption that mode coupling can be neglected, the fiber should maintain approximately the same propagation characteristics independently of the location of the bend [4, 26, 27]. Our goal is to verify if this assumption holds, to compare various types of fibers using a translating bend, and to quantify differences in bend sensitivity between the fibers.

2. Methods

2.1. Overview

To asses bending behavior, we projected a constant pattern of light into the fiber core, and we monitored the output pattern on the other end of the fiber while translating the bend. The amount of change in the output gives a measure of how sensitive the fiber is to a displacement of the bend. The experimental setup is described in more detail in the next section 2.2.

As our experiments revealed, the results depend on the fiber's specifications but also on the type of pattern being displayed. This is why we included a spatial light modulator (SLM) in the illumination path, as shown in Fig. 1. The SLM allows us to test bending sensitivity under different launch conditions. We tested bending with speckle inputs, and shaped input wavefronts that create focused spots at the output. These will be explained in in section 2.3 and 2.4 respectively.

We validated all our measurements using simulations, which are described in section 2.5.

2.2. Experimental setup

Our experimental setup uses spatially filtered, collimated light from a 532 nm laser (CNI MSL-FN-532-100mW). The light is modulated by a phase-only spatial light modulator (HoloEye Pluto VIS). The modulated light is then injected into the fiber under test via a lens (f = 200 mm) and microscope objective (Newport MV-60X NA 0.85). The light output obtained on the other side of the fiber is expanded by a microscope objective (Newport MV-40X NA 0.65) and a lens (f = 200 mm) and observed on a camera (Photonfocus MV1-D1312-G2). For the speckle experiments, we also superposed an off-axis reference beam to the output pattern, to make an off-axis holographic recording of both the amplitude and the phase of the speckles [28].

All fibers were tested without protective jacket or connectors (bare fibers). The fiber lengths varied between 235 mm and 250 mm (the variations are due to the cleaving process). The list of the fibers we used is shown in Table 1.

Table 1. List of fibers and nominal specifications.

View Table

The fiber under test is held in place on each of its ends by a clamp. In between, the fiber passes through a section of PTFE tubing (inner diameter 550 μm, outer diameter 1.08 mm) which constrains the fiber to a specific geometric shape. The tubing is taped to a motorized translation stage (Thorlabs PT1-Z8), which allows the bend to be smoothly translated along the fiber over a range of 25 mm. The precise shape of the bend is drawn in Fig. 2. The shortest radius of curvature along the bend is approximately 25 mm.

Fig. 2 Shape of the bend used in our experiments.

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2.3. Speckle patterns

Random speckle patterns are the first set of inputs we used to test bending. For each fiber, a speckle pattern was calculated to fill the entire fiber core, and to contain all possible angles of illumination up to the numerical aperture (NA) of the fiber under test. Such a speckle pattern is generated starting from a complex image where the real and imaginary part of every pixel is set by a random number generator following a normal distribution. Then, the image is filtered in the spatial and the Fourier domain by circular masks, limiting the spatial and angular extent of the pattern to the specified fiber core diameter and NA. When this pattern is displayed on the SLM, it excites every mode of the fiber under test randomly.

With a speckle as input, the output pattern on the other side of the fiber is also a speckle pattern. This output speckle is measured holographically for different displacements of the bend. Then, the measured amplitude and phase distribution of the output at each bend position is compared to the initial output using the correlation coefficient [29] as defined in Eq. (1).

ρ = \frac{| 〈 f, g 〉 |}{‖ f ‖ ‖ g ‖} = \frac{| \sum_{k = 1}^{N} f_{k} g_{k}^{*} |}{\sqrt{{\sum_{k = 1}^{N} | f_{k} |}^{2}} \sqrt{{\sum_{k = 1}^{N} | g_{k} |}^{2}}}

where f and g are the initial and displaced measurements respectively, the summations run over all the N pixels of each image, and g_k* denotes the complex conjugate of g_k. In other words, we are calculating the autocorrelation of the output under bend translation, i.e. the correlation of the output with itself at various displacements of the bend. When the autocorrelation decreases, it means that the output speckle has changed and therefore the propagation characteristics of the fiber have changed (since the input is constant).

2.4. Spot focusing

The speckle experiment described above is interesting to give a global view of fiber transmission, averaged over many input angles and positions. However, this is not necessarily representative of the spot focusing experiments done in the context of multimode fiber imaging. Whereas a random speckle input excites all the modes of the fiber randomly, a wavefront that is shaped to make spots excites the modes in a specific way [30]. A spot in the center of the core leads predominantly to the excitation of low-order modes (depending on the notation, these are the LP_0n [31], HE_1n [32], or $Ψ_{0, n}$ [4] modes). As the spot is moved closer to the edge of the core, higher order modes are also excited. Because of this, the location of the spot influences the bending characteristics.

In order to obtain results that can directly be related to imaging performance in multimode fiber imaging systems based on spot-scanning, we also performed a series of experiments using shaped inputs. For each fiber, we applied an input wavefront that causes the light to focus into a spot at the output of the fiber. The required input wavefront was found with the transmission matrix method as described in our earlier papers [28, 33]. Then we recorded the intensity of the output spot (non-holographically) as the bend was translated. For each fiber, this was done for two different spot locations: an output spot in the center of the fiber core, and an output spot halfway between the center and the edge of the fiber core.

The spot experiment presented here can be interpreted as an optical correlation experiment as well, similar to the one described in the previous section. In fact, the spot intensity is proportional to the square of the correlation coefficient shown in Eq. (1). The purpose of measuring the spot intensity experimentally in addition to the speckle autocorrelation is that spots excite a different subset of modes, and they are more meaningful measure for spot-scanning imaging applications.

2.5. Simulations

2.5.1 Modes of an optical fiber

For the step-index fibers, we made simulations to verify the results of our experiments. We used a vector mode propagation model [32], which we explain briefly here. Propagation-invariant modes are patterns of light that retain their transverse amplitude and phase distribution as they propagate through a medium. In an optical fiber, only a limited set of such modes can be found that are guided through the fiber. Other modes radiate their energy away from the core of the fiber as they propagate. The guided modes can be found by solving a characteristic equation, which is derived from expressing the boundary conditions of Maxwell's equations at the interface between the core and cladding of the fiber.

By solving the characteristic equation, we determine the propagation constants β_n, the order ν_n and transverse spatial profile of each mode n. Because of the symmetry of the waveguide, the electric field E_n of each mode can be written in a separable way using a cylindrical polarization basis, as shown in Eq. (2). Here, the radial profile of mode n (i.e. E_r,n(r), E_ϕ,n(r) and E_z,n(r)) is composed of Bessel functions [32].

E_{n} (r, ϕ, z, t) = (\begin{matrix} E_{r, n} (r) \\ E_{ϕ, n} (r) \\ E_{z, n} (r) \end{matrix}) e^{i ν_{n} ϕ} e^{i β_{n} z} e^{i ω t}

Each mode E_n can be propagated along a straight segment of fiber by multiplying it with the phase factor $e^{i β_{n} L}$ where L is the length of the fiber segment. Arbitrary input fields must first be decomposed in an orthonormal basis of the fiber modes, yielding a set of modal coefficients. After that, each modal coefficient can be multiplied by the appropriate propagation phase factor $e^{i β_{n} L}$ to simulate propagation over a distance L. The linear combination of the modes, weighted by their modal coefficient at distance L, gives the spatial profile of an arbitrary input after propagating through a straight segment of fiber.

2.5.2 Propagation in bent fibers

For propagation in curved segments of fiber, we used the method proposed by Plöschner, Tyc and Čižmár [4]. Instead of multiplying each modal coefficient by a single phase factor, a matrix operation now has to be used since modes can couple to each other. The new modal coefficients are calculated from $c (L) = e^{i B L} c (0)$ , where c(L) represents the vector of modal coefficients at distance L, i.e. the phase and amplitude of each mode E_n at distance L. The matrix exponential $e^{i B L}$ contains in its argument the matrix B defined by Eq. (3),

B_{n m} = β_{n} δ_{n m} - (\frac{n_{c o r e} k_{0}}{ρ / ξ}) 〈 E_{n} | x | E_{m} 〉

where β_n is the propagation constant of mode n, δ_nm is the Kronecker symbol, n_core is the refractive index of the core of the fiber, k₀ is the magnitude of the wavevector of the light in vacuum, ρ is the curvature of the fiber segment and ξ ≈0.77 is a correction factor to account for refractive index changes due to deformation-induced stress [4]. Note that the bending operator

e^{i B L}

is in general not commutative. The factor

〈 E_{n} | x | E_{m} 〉

is given by Eq. (4) and can be interpreted as the overlap between mode n and m, weighted by the position coordinate x along the axis of the bend.

〈 E_{n} | x | E_{m} 〉 = \iint E_{n}^{*} (x, y) x E_{m} (x, y) d x d y

This factor is equal to zero everywhere except between modes of neighboring order (i.e. when the order ν between mode n and m differ by ± 1, or equivalently

a b s (ν_{n} - ν_{m}) = 1

). This can be proven by inserting Eq. (2) in Eq. (4).

Finally, note that the x-axis in Eq. (4) is oriented in the direction of the bend. When the bend changes orientation, a corresponding rotation should be applied to the bending operator [4]. In our case, the curvature changes its direction in the middle of the bend, as is evident from Fig. 2. We accounted for this by letting the curvature ρ in Eq. (3) become negative, which is equivalent to a 180° rotation of the x-axis at that point.

The bending operator describes propagation through a circular segment of fiber. Other shapes of bend must be approximated as a sequence of small circular segments. In our simulations, the segment size was 250 μm for every simulated fiber.

2.5.3 Numerical implementation

The parameters for each fiber simulation were determined based on the corresponding experiments. The shape of the bend, shown previously in Fig. 2, was estimated from a photograph of the experiment at right angle and at approximately 1m distance, with digital correction for the geometric distortion of the camera lens. The fiber's core size was set at the manufacturer's nominal value shown in Table 1. The refractive index of core was assumed to be that of pure silica (1.4607) at the working wavelength (532 nm). The refractive index of the cladding was calculated based on the specified NA in Table 1 and the index of the core.

For efficiency, the spot experiments were implemented in simulation using the time-reversal symmetry of electromagnetic waves: a spot was created as input to the bending simulation, and the outputs for various displacements were correlated with each other. The magnitude squared of the correlation coefficient corresponds to the experimental measurement of the spot intensity as noted in section 2.4.

The bending operator is described above as a multiplication with a matrix exponential. We applied a sparse numerical algorithm [34] for this calculation, and tuned it to exploit the specific block structure of the bending operator. Indeed, as stated before, all overlap factors $〈 E_{n} | x | E_{m} 〉$ are equal to zero except between modes of neighboring order. As an example, a fiber with a 200 μm core and NA 0.39 (fiber S4 in Table 1) has over 100 000 modes at 532 nm, but less than 0.4% of the 10¹⁰ cross-coupling factors are nonzero. Bent propagation is therefore calculated from a sparse matrix, with a diagonal component representing 'normal' mode propagation and small block components representing coupling between modes of order ν and ν ± 1. The blocks are of variable size. Support for this particular matrix structure is not common in sparse algebra libraries, so we implemented C + + routines that handle this type of matrix operations in a multithreaded and cache-efficient manner.

These improvements allowed us to simulate bending with a large number of modes in a reasonable time. For example, propagating light through the bend for fiber S4 with 500 inputs and 500 bend segments takes 48h on our machine (dual Intel Xeon E5-2670 2.6 GHz processors). Note that this problem is intractable with a non-sparse algorithm for this number of modes.

Graded-index fibers were not simulated. Their modes cannot be calculated as accurately due to the lack of an analytical solution and the requirement for the precise knowledge of the refractive index profile.

3. Results

In our first set of experiments, we applied a static random pattern of light to the input of the fiber and we recorded the output speckle holographically as the bend was translated. From the holographic recordings we calculated the autocorrelation coefficient, which describes the similarity of the output between various states of translation of the bend. The experimental data is shown in Fig. 3(a) and the corresponding simulations in Fig. 3(b).

Fig. 3 Autocorrelation of the output as the bend is translated, for a random input. (a) Experimental autocorrelation of the speckle pattern at the output of the fiber as the bend is translated. (b) Simulated autocorrelation for the step index fibers. Refer to Table 1 for the fiber specifications.

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In a second set of experiment, we measured the intensity of focused spots at the output of the fiber, as shown experimentally in Fig. 4(a) for a spot in the center of the core and Fig. 4(c) for an off-center spot. The corresponding simulations are Fig. 4(b) and 4(d) respectively. We also projected a line and a grid of spots at the output of the fiber (i.e. patterns where spots at many different positions are displayed simultaneously). The grid is shown in Fig. 5(a) before deformation and in Fig. 5(b) after deformation. The line is shown in Fig. 5(c) before deformation and in Fig. 5(d) after deformation.

Fig. 4 Change in intensity of a focused spot due to the translation of the bend. (a) Experiments and (b) simulations for the intensity of a spot created in the middle of the fiber's core. (c) Experiments and (d) simulations for a spot created half-way between the center and the edge of the fiber core. Refer to Table 1 for the specifications of the various fibers.

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Fig. 5 Patterns at the tip of a fiber. (a) Spots before translating the bend and (b) after 25 mm translation. The average full width at half maximum of the spots is 480 nm. The inset is a zoom on the center of the spot grid. (c) Line before translating the bend and (d) after 25 mm translation, showing the extent of the central region of the core where bending resilience is lower. The patterns shown here were made via fiber S5 (70 μm core, NA 0.64). The boundary of the core is indicated with a dashed circle. The scale bars are 10 μm.

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4. Discussion

Several interesting conclusions can be drawn from the data presented in Fig. 3 and Fig. 4. First, the data shows that with some fibers a bend can be translated over a few centimeters while still preserving a nearly constant output. However, the assumption of limited mode coupling is not universally applicable. For example, a fiber with a 200 μm core and NA 0.22 tolerates very little translation before the output is lost. In contrast, a fiber with the same NA but a smaller core of 50 μm has a much more stable output during the experiment: after 25 mm translation, the output remains constant to within 89% as measured by the correlation coefficient. In comparison with the diameter of the fiber, this is a significant distance.

The output patterns change more rapidly in fibers with larger core sizes, as can be seen by comparing the traces of fibers S1, S2 and S3 in Fig. 3 and Fig. 4. This behavior is expected from Eq. (3), because the overlap factors $〈 E_{n} | x | E_{m} 〉$ increase in magnitude with increasing core diameter, so that the overall effect of bending is larger. On the contrary, the output patterns are more stable in fibers with larger NA, as can be seen by comparing fibers S1 and S5, or S3 and S4. Both core size and NA affect the number of modes in the same way (i.e. doubling either the core size or the NA quadruples the number of modes). However, an increase in core diameter increases the density of modes over the same range of propagation constants, whereas an increase in NA spreads the modes over a larger span of propagation constants.

The step-index fibers show a sharp change in the output as the bend is moved from its initial position, but after this point there is a plateau phase where the output remains constant over a large range. This behavior is consistent across the experiments in Fig. 4(a) and 4(c) and the corresponding simulations in Fig. 4(b) and 4(d). In our simulations, it was possible to verify that this plateau extends to much larger distances. For example, the intensity of an off-center spot in fiber S5 remains the same to within 1% whether the bend is translated by 25 mm or 1 m (assuming a fiber that long). Graded-index fibers show a different trend. While the range over which they maintain the output is rather large, graded-index fibers do not exhibit the plateau behavior and the integrity of the output keeps decreasing as we move away from the starting position.

The type of excitation has a great influence on the bending stability. For speckle (Fig. 3) or off-center spots (Fig. 4(c) and 4(d)), the output can be maintained over a significant range for certain fibers. However, for spots created in the center of the core (Fig. 4(a) and 4(b)), the stability of the output is much lower and the spots disappear monotonically as the bend is translated in all cases. This shows the large difference in bending sensitivity between the low order modes and the high order modes. For fiber imaging, it means that the regions of the core that are not in the center have a much greater tolerance to this type of deformation. This is illustrated visually Fig. 5 using a grid and a line pattern. The central region of the core is visibly attenuated after deformation, but the other spots maintain high intensities.

An important remark concerns the symmetry of the bend used in these experiments. For experimental simplicity, we used an S-shaped bend which allows us to change the shape of the fiber while both ends remain fixed. Since the first half of this curve has roughly the same shape as the second half except for an opposite curvature, one could hypothesise that some of the distortions incurred in the first half are somehow compensated when the light travels through the second half because of the symmetry. We have verified in simulation, however, that the effects discussed above also exist for a translating U-shaped bend. For a bend with the same absolute curvature as the S-bend in Fig. 2, but with everywhere positive curvature value, we obtained similar intensity and correlation curves as shown in Fig. 6.

Fig. 6 Simulations for a U-shaped bend. (a) U-shaped bend having the same absolute curvature at every point as the bend in Fig. 2, except the curvature is everywhere positive. (b) Autocorrelation trace versus horizontal translation of the bend. (c) Intensity of a spot created in the center of the fiber core and (d) intensity of an off-center spot versus translation of the bend.

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5. Conclusion

We compared various multimode fibers with each other in terms of their ability to maintain a constant output while a bend is translated along their length. The results give an indication of the mode coupling effects in each fiber.

There are large differences in the stability of the propagation characteristics of fibers undergoing bending. Fibers with limited core sizes and large numerical apertures (e.g. 70 μm and NA 0.64) are resilient and maintain a nearly constant output over significant ranges. Fibers with large cores and small numerical apertures (e.g. 200 μm and NA 0.22) are very sensitive to any translation of the bend.

Our findings may help select fibers with a greater tolerance to certain deformations, and motivate the search for other classes of deformations that preserve propagation characteristics. Finally, we remark that a sliding U-shaped bend could allow linear motion of the tip of a fiber while the other end remains fixed to the measurement apparatus. In conjunction with bending compensation techniques, this could improve the flexibility of imaging systems based on multimode fibers.

Funding

Swiss National Science Foundation (200021_160113/1, project MuxWave).

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Label	Model	Supplier	Index	NA	Diameter
S1	FG050LGA	Thorlabs	Step	0.22	50 μm
S2	FG105LCA	Thorlabs	Step	0.22	105 μm
S3	FG200LEA	Thorlabs	Step	0.22	200 μm ^(*)
S4	FT200EMT	Thorlabs	Step	0.39	200 μm ^(*)
S5	GOF85	Schott	Step	0.64	70 μm
G1	GIF625	Thorlabs	Graded	0.29	62.5 μm
G2	F-MLD	Newport	Graded	0.29	100 μm

Bend translation in multimode fiber imaging

Abstract

1. Introduction

2. Methods

2.1. Overview

2.2. Experimental setup

2.3. Speckle patterns

2.4. Spot focusing

2.5. Simulations

2.5.1 Modes of an optical fiber

2.5.2 Propagation in bent fibers

2.5.3 Numerical implementation

3. Results

4. Discussion

5. Conclusion

Funding

References and links

Cited By

Figures (6)

Tables (1)

Equations (4)

Optics Express