Seeing through chaos in multimode fibres

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1 DOI: 1.138/NPHOTON Seeing through chaos in multimode fibres Martin Plöschner 1, Tomáš Tyc 2 and Tomáš Čižmár 1* 1 School of Engineering, Physics and Mathematics, College of Art, Science & Engineering, University of Dundee, Nethergate, Dundee, DD1 4HN, Scotland, United Kingdom 2 Department of Theoretical Physics and Astrophysics, Masaryk University, Kotlarska 2, Brno, Czech Republic *Corresponding author: t.cizmar@dundee.ac.uk Supplementary Methods 1 Theoretical description of light propagation in multimode fibres The most common description of light propagation in MMF is based on the assumption that the polarisation state of the optical fields propagating through them remains unchanged. Such scalar and paraxial modelling based on the theory of weakly guiding modes leads to the identification of a set of orthogonal linearly polarised (LP) propagation-invariant modes (PIMs) with radial dependence described simply by splicing a Bessel function of the first kind and a modified Bessel function of the second kind defined across the fibre core and the cladding, respectively. The requirement of continuity of the function together with its first derivative leads to a set of discrete solutions (being indexed by letter m in this paper) for every order of the Bessel functions l. The intensities of the fields are azimuthally independent and their phase varies with the azimuthal coordinate ϕ as e ilϕ. This simplified approach serves well to explain several important lighttransport processes, however linearly polarised light in an MMF with parameters such as those used in our experiments will only retain its polarisation state over distances of a few tens of millimetres. A rigorous vectorial modelling also predicts the existence of a set of PIMs; their polarisation is however no longer uniform, with orientation periodically changing across the mode cross-section. Very fortunately, for the majority of the modes the theory predicts degeneracy in the values of propagation constants (axial components of the k vector). A linear combination of such degenerate modes therefore also leads to propagation-invariant optical fields that have an almost identical distribution of the field to the simplified scalar LP PIMs and uniform polarisation across the mode cross-section, but the polarisation state is circular. Due to these qualities, we chose this representation for our experimental study since it is much easier to generate circularly polarised (CP) modes than modes with spatially non-uniform distribution of polarisation. The only exception to this are azimuthally and radially polarised modes sometimes termed as hedgehogs and bagels. While their superposition leads to modes with l = ±1, here such degeneracy does not occur. In our model they are approximated by circularly polarised modes with averaged propagation constants. Intensity of such modes is still azimuthally independent but their energy periodically oscillates between opposite polarisation states, therefore they are no longer propagationinvariant. As shown in Figs. 5 and 6, this insufficiency has a negligible influence on imaging performance. The representation of discrete circularly polarised modes can be conveniently expressed using a quantum formalism describing each mode with the trio of integer quantum numbers l, m and σ. In this formalism, l and σ denote the amount of orbital and spin angular momentum, respectively, in the units of h per photon [25]. This representation is also very convenient to observe the effect of spin-orbit interaction. This effect, known from quantum physics, causes two modes with the same but nonzero l and opposite spin states σ = 1 and σ = 1 to propagate with slightly different phase velocities. The similarity of the vectorial CP modes with the scalar LP modes makes this representation very suitable for implementing corrections for deviations from the ideal step-index profile that can strongly influence the experimental observations. Again, the influence of these aberrations does not considerably affect the distribution of optical fields of the modes. The most pronounced effect is a small variation of propagation constants (relative deviations are in the order of 1 6 ) which can however significantly affect the output phases of the modes over distances of only a few tens of millimetres. As detailed in the following section, these small deviations of propagation constants can be modelled sufficiently precisely and efficiently using scalar LP modes by employing perturbation calculus. Our modelling technique is therefore a hybrid solution employing exact vectorial CP modes with propagation constants being corrected for imperfections in the step-index profile. 2 Calculation of Vector modes CP PIMs constitute the simplest set of modal fields that take into account the vector character of light propagation. In a step-index profile circular fibre the transverse electric field components of CP PIMs can be expressed as: ψ l,m,σ =(ˆx+iσŷ)F lm (R)exp(ilϕ), β = β sc +δβ l,± (S1) where R = r/a is a dimensionless radial parameter with r being the radial coordinate and a the radius of the fibre core. β are the propagation constants of the corresponding modes and we have denoted the polarisation corrections to scalar propagation constants β sc by δβ l,+ for l,σ with equal signs and by δβ l, for l, σ with opposite signs. ˆx and ŷ are unit Cartesian coordinate NATURE PHOTONICS 1

$K l (ur) K l (u) Parameter u is proportional to the transverse wavevector of a considered mode, u = a k 2n2 core β 2, where k = ω/c is the vacuum wavenumber and n core is the refractive index of the$

2 DOI: 1.138/NPHOTON vectors and F lm (R) is the radial profile of the mode defined by Bessel functions of the first (J l ) and second kind (K l ): { Jl (ur) J F lm (R)= l (u),for R < 1 (S2),for R 1. K l (ur) K l (u) Parameter u is proportional to the transverse wavevector of a considered mode, u = a k 2n2 core β 2, where k = ω/c is the vacuum wavenumber and n core is the refractive index of the fibre core. Equation (S1) describes the CP modes. As we have mentioned, they are propagation invariant (PIMs) with the exception of the modes for which l + σ =, i.e., the ones whose total angular momentum is zero. These exceptional modes are not even approximate solutions of the vector wave equation; however, we can construct the correct PIMs by their equal superpositions: ψ hedgehog,m (ψ 1,m, 1 + ψ 1,m,1 )/2 ψ bagel,m (ψ 1,m, 1 ψ 1,m,1 )/2. (S3) Therefore, strictly speaking, in the case of l + σ =, the propagation constant β in Eq. (S1) cannot be defined. For the same reason we cannot define the correction δβ 1, as a correction to the propagation constant of the mode ψ 1,m, 1. We will therefore reserve this symbol for a correction defined with respect to the hedgehog mode, and the correction for the bagel mode will be denoted by δβ 1,. The fact that the modes ψ 1,m, 1 and ψ 1,m,1 themselves are not the PIMs of the fibre shows itself in a very specific way: if the mode ψ 1,m, 1 is injected into the fibre, then after a certain distance (say L m ) it is completely transformed into the mode ψ 1,m,1. Then after propagating through an additional distance of L m, the original mode ψ 1,m, 1 is recovered, and the state continues to oscillate between these two modes along the fibre. This way, the case of l + σ = is the only one where orbital and spin angular momenta are not conserved separately along the fibre, however their sum is still conserved due to the rotational symmetry of the fibre. The bagel and hedgehog modes are in fact members of a complete, although slightly more complicated, representation of the modes of the fibre which we can call generalised bagel and hedgehog (GBH PIMs). This representation of modes in a stepindex profile fibre is constructed as a linear superposition of CP PIMs: HEl+1,m even (l ) 1 ( ) ψl,m,1 + ψ l,m, 1, β = βsc + δβ l,+ 2 EHl 1,m even (l 1) 1 ( ) ψl,m, 1 + ψ l,m,1, β = βsc + δβ l, 2 HEl+1,m odd (l ) 1 ( ) ψl,m,1 ψ l,m, 1, β = βsc + δβ l,+ 2 EHl 1,m odd (l 1) 1 ( ) ψl,m, 1 ψ l,m,1, β = βsc + δβ l, 2 The first index (l + 1 or l 1) of these modes denotes the magnitude of the total angular momentum, and δβ l,± are again corrections of the propagation constants with respect to the scalar theory. However, in the case of l + σ = the corrections split into two values, different for the even and odd modes (hedgehogs and bagels, respectively), which is emphasised by the prime in β l, in the last equation. For l + σ, the corrections β l, and β l, are equal. Specifically, the corrections can be expressed in a simple form: for all l, δβ l,+ = I 1 I 2 ; for l 1, δβ l, = δβ l, = I 1 + I 2, and for l = 1, δβ l, = 2(I 1 + I 2 ) and δβ l, =. Here I 1 and I 2 are simple integrals [1]: I 1 = (2 )3/2 4aV I 2 = l(2 )3/2 4aV RF lm (R)F lm (R) f (R)dR RF 2 lm (R) f (R)dR / / RF 2 lm (R)dR RF 2 lm (R)dR, where V = ak NA is a waveguide parameter with NA being the numerical aperture of the fibre, =(n 2 core n 2 clad )/(2n2 core) is a profile height parameter and f (R) is a dimensionless profile shape function. For the case of step-index fibre f (R) is defined as: { for R < 1 f (R)= (S4) 1 for R 1. This makes the calculation of the above integrals really simple as f (R)=δ(R 1). As a consequence, only the normalisation integrals have to be calculated. Table 1 summarises the corrections for the CP modes as well as their corresponding GBH modes in terms of I 1 and I 2 integrals. In case of l = that is shown separately in the table, I 2 = and therefore the correction is just I 1. The correction I 2 is opposite for the modes ψ l,m,1 and ψ l,m, 1, CP modes corresponding GBH δβ i l values ψ,m,±1 HE1,m even, HE odd 1,m I 1 l = ψ ±l,m,±1 HEl+1,m even, HE odd l+1,m I 1 I 2 l 1 (hedgehog) EH,m even,m 2(I 1 + I 2 ) l = 1 (bagel) EH,m odd,m l = 1 ψ ±l,m, 1 EHl 1,m even l 1,m I 1 + I 2 l > 1 Table 1 Corrections δβ i to the scalar propagation constants β sc. which means that the propagation constants of these modes are slightly different. This is a demonstration of the spin-orbit interaction, which causes rotation of the polarisation vector direction if linearly polarised light with a given l propagates through the fibre, or rotation of the intensity patterns when the superposition ψ l,m,σ + ψ l,m,σ propagates. For completeness, we have also compared the weak guidance corrections δβ i (Supplementary Figs. 1a,b) calculated using the integrals I 1 and I 2 and used them to evaluate higher order vectorial corrections (Supplementary Figs. 1e,f) from: β vec β sc = δβ i +(higher order corrections), (S5) shown in (Supplementary Figs. 1c,d), where β vec is a fully vectorial propagation constant. The higher order corrections are on the order of NATURE PHOTONICS

3 DOI: 1.138/NPHOTON SUPPLEMENTARY INFORMATION a Deformation twisting 2 c Orientation agreement 1 Supplementary Figure 1 a-b, Vector corrections to β calculated from I 1 and I 2 integrals. c-d, Difference between full vector β vec and scalar β sc. e-f, Higher order vectorial corrections deduced by subtracting (a-b) from (c-d). y [mm] z [mm] b (V) (VI) (IX) (VIII) (VII) (VI) (V) 2 4 x [mm] PIMs exp. original optimised orientation [deg] (V) (VI) (VII) (VIII) (IX) observed orienrtation [deg] 2 PIMs exp. rotated d (V) (VI) amplitude The modal fields above constitute a complete approximate set of solutions of the vector-wave equation and have similar EH/HE nomenclature to that used with the full vectorial approach. Even though the GBH PIMs form a complete basis (including the problematic l + σ = case), their experimental drawback is that their non-uniform polarisation changes orientation periodically across the mode cross-section. As mentioned before, this led us to choose the CP PIMs representation for our experimental study since it is much easier to generate circularly polarised modes than modes with spatially non-uniform distribution of polarisation. This choice of basis has a negligible influence on imaging performance. Due to the very high accuracy of weak guidance approximation in our case, the propagation constants β for the above GBH modes, and also for CP modes, obtained directly from a full vectorial approach, yield virtually identical results to the scalarwave equation with weak guidance polarisation corrections applied. The latter approach, however, has significant advantages over the full vectorial approach. For example, the fibre exhibits small variations of refractive index. These small variations can be readily accounted for by utilising the perturbation theory applied to scalar modes. The same cannot be done easily in the full vectorial description. Due to this clear advantage, we choose the scalar theory with weak guidance and perturbation corrections as a main tool in our calculations. Supplementary Results 1 Deformation operator in 3-D The theoretical model predicts that DO does not change if the plane in which the deformed fibre lies is oriented differently, provided that the output imaging system remains oriented accordingly to the deformation imposed. If this condition is not fulfilled, the DO is accompanied by an orientation operator that rotates all output modes by the differing angle γ (for better understanding of this behaviour see Supplementary Movie 6). To confirm this behaviour we have revolved the deformed fibre (V) together with the imaging apparatus around the axis of the initially straight fibre as shown in Fig. 4a by various angles β. For (VII) (VIII) (IX) l index (VII) (VIII) (IX) l index π π phase Supplementary Figure 2 Influence of deformation orientation. a, Arrangements of deformed fibre used in the experiment with their roman ID number. b, Influence of curvature orientation on PIMs. c, Argument of isolated rotation operator. d, Influence of DO on PIMs after removing rotation operator. this case the model predicts that γ = β. Diagonal components of DO for these cases are presented in Supplementary Fig. 2b. To isolate the orientation operator from these data we can efficiently utilise the fact that for γ =, DO features symmetry of ˆD(+l, m)= ˆD( l, m). Therefore we have searched for the common angle with which output modes have to be rotated in order to maximise this symmetry. The isolated values of γ against the actual orientation β are shown in Supplementary Fig. 2c. The resulting DOs with the orientation operator eliminated are presented in Supplementary Fig. 2d, clearly confirming the orientation invariance of deformation operators. distal imaging a 2 Proximal end imaging proximal imaging b c reference difference Supplementary Figure 3 Demonstration of proximal end imaging. a, Direct distal end imaging. b, Proximal end imaging with USAF target. c, reference image of reflective surface. d, Difference between (c) and (b). During acquisition of images the total transmitted signal was detected using a bucket detector (integrating signal across a CCD d NATURE PHOTONICS 3

4 DOI: 1.138/NPHOTON ) for each of the output FP generated across the grid of positions (Supplementary Fig. 3a). As some may argue that this is not a real endoscopic method (the detector is not at the proximal side of the instrument), we have also integrated the light returning from the fibre at the proximal end using CCD4. For better understanding see also Supplementary Movie 7. The corresponding image (Supplementary Fig. 3b) clearly exhibits unwanted interference effects present in our imaging pathway. These originate from the light being retro-reflected on many surfaces of our geometry (including fibre facets). This does not pose any obstacles for methods relying on fluorescence emission where the retro-reflected excitation wavelength can be filtered out and only the emission signal is detected. To minimise this effect we have taken a reference image of a uniform reflective surface (Supplementary Fig. 3c). The difference of the two, shown in Supplementary Fig. 3d, gives an image analogous to distal end imaging (Supplementary Fig. 3a) but due to the more elaborate procedure it is more affected by noise. For this reason we have chosen to use the distal end imaging in our demonstrations as it makes observation of subtle effects on imaging quality more evident. 3 Imaging in arbitrary distance from the fibre end Here we show that with highly precise knowledge of the fibre TM (particularly phases of output modes) we can complement the TM with a free-space propagating operator and thus modify the location of imaging plane numerically. As shown in the sequence Supplementary Fig. 4, the further the image plane is set from the fibre, the larger is the accessible field of view but accordingly the resolution obtained is lower. The amount of information imaged remains the same in all the cases shown. z= µm z=15 µm z=3 µm z=45 µm z=6 µm Supplementary Figure 4 Imaging with numerically updated TM to image at various distances behind the fibre end. The white scale bar corresponds to the size of fibre core, 5 µm. 4 NATURE PHOTONICS

assist latex not to break page a b c d Supplementary Figure 5

the representation of linearly polarised FPs.

5 DOI: 1.138/NPHOTON SUPPLEMENTARY INFORMATION Supplementary Figures text to assist latex not to break page a b c d Supplementary Figure 5 Experimental transformation matrix data after processing introduced in Methods. This example represents TM for 1mm long segment of fibre in the representation of linearly polarised FPs. a, Polarization component S S. b, Polarization component S P. c, Polarization component P S. d, Polarization component P P. 5 NATURE PHOTONICS

DOI: 1.138/NPHOTON.215.112 Supplementary Figure 6 Conversion matrix.

Supplementary Figure 7 Converted transformation matrix for 1mm long

6 DOI: 1.138/NPHOTON Supplementary Figure 6 Conversion matrix. λ = 164nm, NA=.22 and core diameter d = 5µm. Supplementary Figure 7 Converted transformation matrix for 1mm long segment of fibre. TM is expressed in the representation of LP PIMs 6 NATURE PHOTONICS

7 DOI: 1.138/NPHOTON SUPPLEMENTARY INFORMATION Supplementary Figure 8 Converted transformation matrix for 1mm long segment of fibre. TM is expressed in the representation of LP PIMs Supplementary Figure 9 Converted transformation matrix for 1mm long segment of fibre. TM is expressed in the representation of CP PIMs NATURE PHOTONICS 7

$43 1.45 n 1.43 1e 5 n n c a profile of refractive index d g ideal step index dopant diffusion n fine modulation 2e 5 radius [µm] 2 3 n c.1 b phase agreement c difference φ 1.$

8 DOI: 1.138/NPHOTON Supplementary Figure 1 Converted transformation matrix for 3mm long segment of fibre. TM is expressed in the representation of CP PIMs 1.45 n n e 5 n n c a profile of refractive index d g ideal step index dopant diffusion n fine modulation 2e 5 radius [µm] 2 3 n c.1 b phase agreement c difference φ 1.2 e 1 h m 1 m length [mm] l Supplementary Figure 11 Optical phases of PIMs, data for the 3mm long fibre. a, b and c, Assumed profile of refractive index, phase agreement and difference between experimentally obtained and theoretically predicted phases of PIMs respectively for the case of ideal step-index fibre. d, e and f, corresponds to a model with included dopant diffusion. g, h and i, represents correction for diffusion as well as fine index modulation (three parameters) across the fibre core. 1 m f i π π π π π π 8 NATURE PHOTONICS

deformation operator corresponding to fibre deformation (V) from Figure 4.

9 DOI: 1.138/NPHOTON SUPPLEMENTARY INFORMATION Supplementary Figure 12 Experimentally measured deformation operator corresponding to fibre deformation (V) from Figure 4. Supplementary Figure 13 Theoretically predicted deformation operator corresponding to fibre deformation (V) from Figure 4. NATURE PHOTONICS 9

DOI: 1.138/NPHOTON.215.112 L1 L2 HWP1 M1 L3 L7 HWP2 SMF M2 QWP1 L4 PBD2 MO1 PBD1 HWP3 MMF MO2 L6 L8 HWP4 Portable unit PBS M3 NPBS1 L5 QWP2 NPBS2 Supplementary Figure 14 The experimental geometry.

10 DOI: 1.138/NPHOTON L1 L2 HWP1 M1 L3 L7 HWP2 SMF M2 QWP1 L4 PBD2 MO1 PBD1 HWP3 MMF MO2 L6 L8 HWP4 Portable unit PBS M3 NPBS1 L5 QWP2 NPBS2 Supplementary Figure 14 The experimental geometry. LASER: CrystaLaser CL164-5-S with Faraday isolator Thorlabs IO-3D-164-VLP, L1, L7 and L8: Thorlabs A24TM-C, L2: Thorlabs AC254-4-C-ML, SLM: Boulder Nonlinear Systems / Meadowlark Optics HSPDM512 DVI, L3: Thorlabs AC254-3-C-ML, HWP1-HWP4: Thorlabs WPH1M-164, PBD1: Thorlabs BD4, PBD2: Thorlabs BD27, M1-3: Thorlabs BB1-E3, L4: Thorlabs AC254-2-C-ML, NPBS1-2: Thorlabs CM1-BS14, MO1-2: Olympus Plan N 2x, MMF: Thorlabs FG5UGA, L5-6: Thorlabs AC C-ML, PBS: Thorlabs CM1-PBS253 CCD1-4: Basler pia64-21gm, SMF: Thorlabs P1-98A-FC-1, QWP1-2: Thorlabs WPQ1M-164. Optomechanical components, beam-blocks and attenuation filters are not shown in the simplified scheme. 1 NATURE PHOTONICS

Supplementary Figure 15 Enhancement of wavefront correction.

b, Enhanced wavefront correction obtained averaging 1 single pixel corrections.

corrected Supplementary Figure 16 Coarse misalignment compensation.

11 DOI: 1.138/NPHOTON SUPPLEMENTARY INFORMATION a single pixel b 1k pixels 1 amplitude π π phase Supplementary Figure 15 Enhancement of wavefront correction. a, Single pixel correction. b, Enhanced wavefront correction obtained averaging 1 single pixel corrections. a input facet b input farfield c output facet d output farfield original e f g h corrected Supplementary Figure 16 Coarse misalignment compensation. a, Transmission profile of the input facet area, scanned during calibration. b, Transmission profile of input angular spectrum. c, Transmission profile of output facet. d, Transmission profile of output angular spectrum. Both (a) and (b) were up-sampled to match the resolution of output transmission profiles. The large shift of the transmission spectrum towards the top left corner in (d) was introduced into the system deliberately by adjusting the angle of incidence of the reference signal onto the CCDs. This was done in order to eliminate strong interference effects caused by multiple reflections of the reference signal within our system. e-h, The equivalent of (a-d) after applying coarse correction for misalignment. NATURE PHOTONICS 11

12 DOI: 1.138/NPHOTON Supplementary Media Supplementary Movie 1 Record of progression of optimisation procedure introduced in Methods. 12 NATURE PHOTONICS

DOI: 1.138/NPHOTON.215.112 SUPPLEMENTARY INFORMATION Supplementary Movie 2 Experimentally generated full set of PIMs as introduced in Methods.

The left column shows images of the modes prior they enter the optical fibre (recorded by CCD1). The right column shows PIMs at the output, recorded by CCD1 and CCD2.

13 DOI: 1.138/NPHOTON SUPPLEMENTARY INFORMATION Supplementary Movie 2 Experimentally generated full set of PIMs as introduced in Methods. PIMs are generated in circular polarisation and propagate through 1mm long segment of fibre. Colour corresponds to individual polarisation components. The left column shows images of the modes prior they enter the optical fibre (recorded by CCD1). The right column shows PIMs at the output, recorded by CCD1 and CCD2. Supplementary Movie 3 Experimentally generated interfering pairs of PIMs with opposite l indices. PIMs are generated in circular polarisation and propagate through 1mm long segment of fibre. NATURE PHOTONICS 13

DOI: 1.138/NPHOTON.215.112 Supplementary Movie 4 Experimentally generated full set of PIMs as introduced in Methods.

14 DOI: 1.138/NPHOTON Supplementary Movie 4 Experimentally generated full set of PIMs as introduced in Methods. PIMs are generated in circular polarisation and propagate through 1mm long segment of fibre. Supplementary Movie 5 Experimentally generated interfering pairs of PIMs with opposite l indices. PIMs are generated in circular polarisation and propagate through 1mm long segment of fibre. 14 NATURE PHOTONICS

DOI: 1.138/NPHOTON.215.112 SUPPLEMENTARY INFORMATION Supplementary Movie 6 Invariance of deformation orientation.

We deliberately image the FPs onto the CCD chip out of focus. This is to spread the total power over a larger area thus allowing larger power to be detected and reducing overall noise.

15 DOI: 1.138/NPHOTON SUPPLEMENTARY INFORMATION Supplementary Movie 6 Invariance of deformation orientation. Supplementary Movie 7 Imaging on the distal and the proximal side of the multimode fibre. a, Record of CCD3 (at the distal end of the fibre) during imaging. We deliberately image the FPs onto the CCD chip out of focus. This is to spread the total power over a larger area thus allowing larger power to be detected and reducing overall noise. b, Progress of the image acquisition from data obtained by CCD3. c, Record of CCD4 (at the proximal side of the fibre) during imaging. Here we collect the light returning from the fibre after reflecting off the object. Due to its propagation through the fibre the signal appears scrambled. d, Progress of the image acquisition from data obtained by CCD4. Due to space constraints both (a) and (c) represent only a fraction of the whole CCD chips areas. NATURE PHOTONICS 15

FIBER OPTICS. Prof. R.K. Shevgaonkar. Department of Electrical Engineering. Indian Institute of Technology, Bombay. Lecture: 07

FIBER OPTICS. Prof. R.K. Shevgaonkar. Department of Electrical Engineering. Indian Institute of Technology, Bombay. Lecture: 07 FIBER OPTICS Prof. R.K. Shevgaonkar Department of Electrical Engineering Indian Institute of Technology, Bombay Lecture: 07 Analysis of Wave-Model of Light Fiber Optics, Prof. R.K. Shevgaonkar, Dept. of