Robustness of Light-Transport Processes to Bending Deformations in Graded-Index Multimode Waveguides

Light transport through a multimode optical waveguide undergoes changes when subjected to bending deformations. We show that optical waveguides with a perfectly parabolic refractive index profile are almost immune to bending, conserving the structure of propagation-invariant modes. Moreover, we show that changes to the transmission matrix of parabolic-index fibers due to bending can be expressed with only two free parameters, regardless of how complex a particular deformation is. We provide detailed analysis of experimentally measured transmission matrices of a commercially available graded-index fiber as well as a gradient-index rod lens featuring a very faithful parabolic refractive index profile. Although parabolic-index fibers with a sufficiently precise refractive index profile are not within our reach, we show that imaging performance with standard commercially available graded-index fibers is significantly less influenced by bending deformations than step-index types under the same conditions. Our work thus predicts that the availability of ultraprecise parabolic-index fibers will make endoscopic applications with flexible probes feasible and free from extremely elaborate computational challenges.

Figure 1

Top: Diagram of transmission matrix measurement. A SLM is used to project a series of focal spots across a grid on the proximal end of the rod. The distal end output fields are measured and stored in the columns of a TM. (a) TM of the PI rod before numerical alignment, in a propagation-invariant mode (PIM) basis, ordered by $l$ and then $m$ in Eq. (2), as per Fig. 1(c) of Ref. [18]. Amplitude and phase color scale indicated in lower left. For clarity, only every third mode is displayed. (b) TM after numerical alignment. Inset shows enlarged area around the diagonal, demonstrating that only the diagonal carries significant power. (c) Measured phases and amplitudes of PIMs.

Figure 2

(a) Modal phase difference between expected and measured PIMs after optimizing PI rod length, assuming a parabolic refractive index profile. PA is given in the figure. (b) Phase difference between expected modal phases and measured modal phases after fitting fine scale modulations to the refractive index profile. (c) PA as a function of rod length before (black) and after (red) optimizing refractive index profile. (d) Refractive index profile (blue), the detected perturbation (red), with 95% confidence interval indicated in gray; see Sec. S3 of Ref. [22]. All color scales are the same as in Fig. 1.

Figure 3

Initial TM of the graded-index fiber in a PIM basis before (a) and after (b) the optimization procedure. (c) Measured phase retardations. Lower right: Predicted responses for the $(l,m)=(-6,0)$, $(-3,2)$, (0,0), and (2,0) modes at the output facet of the fiber. All color scales are the same as in Fig. 1.

Figure 4

Bending sensitivity of imaging performance for a step-index (SI) MMF and two different graded-index (GRIN) MMFs. The TM is measured for a straight fiber and used to image a sample object with three different fiber layouts.