Regular modes in a mixed-dynamics-based optical fiber

A multimode optical fiber with a truncated transverse cross section acts as a powerful versatile support to investigate the wave features of complex ray dynamics. In this paper, we concentrate on the case of a geometry inducing mixed dynamics. We highlight that regular modes associated with stable periodic orbits present an enhanced spatial intensity localization. We report the statistics of the inverse participation ratio whose features are analogous to those of Anderson localized modes. Our study is supported by both numerical and experimental results on the spatial localization and spectral regularity of the regular modes.

Figure 1

Presentation of the experimental system. (a) Scheme of the fiber with $\gamma <1$; (b) picture of the cross section of the optical fiber for $\gamma =0.95$ taken with a standard microscope in the transmission mode.

Figure 2

(a) Curvilinear abscissa $s=a\theta$ and angle of reflection $\alpha$ used for the calculation of the PSS and Husimi representations. (b) Three examples of stable periodic orbits of the mixed billiard ($\gamma =0.95$): in continuous pink line the 2-PO, in dashed blue line the 5-PO and in dotted green line the 6-PO. (c–e) PSS representation for three different values of $\gamma$. (c) $\gamma =0.95$, mixed dynamics with a coexistence of regular and chaotic trajectories; the pink star marks the two-bounce stable periodic orbit, the blue squares correspond to the central point of the stability islands of the five-bounce stable periodic orbit, and the green triangles the six-bounce stable periodic orbit. (d) $\gamma =1$, regular dynamics for which one sees the conservation of the angle of reflection, whatever the initial condition is. (e) $\gamma =2/3$, chaotic dynamics, no stability regions, all the periodic orbits are unstable.

Figure 3

Examples of modes of the mixed fiber with $\gamma =0.95$: near-field, far-field, and Husimi representation of (a, b, c) a regular mode of the 2-PO. Superimposed in red is the PSS associated to the 2-PO. (d, e, f) A mode built on the 5-PO. In red, the PSS of the 5-PO. (g, h, i) A mode built on the 6-PO. In red the PSS of the 6-PO. (j, k, l) A chaotic mode.

Figure 4

Probability distribution of the energy level spacing for the symmetric and antisymmetric modes of (a, b) a chaotic cavity ($\gamma =3/2$) and (c, d) a mixed cavity ($\gamma =0.95$).

Figure 5

Spatial distribution of intensity for modes of the mixed fiber ($\gamma =0.95$) associated to different values of $(m,p)$. (a–d) $p=4$ and $m=0,1,4,7$; (e–h) $p=12$ and $m=0,1,4,7$.

Figure 6

Probability distribution of the inverse participation ratio. (a) Chaotic cavity ($\gamma =3/2$). The probability distribution is centered around the value $\text{IPR}=3$. (b) Mixed cavity ($\gamma =0.95$) presenting some larger values of the IPR. The continuous line is a fit based on a nonlinear sigma model [41]. Inset: Same as (b) in logarithmic scale.

Figure 7

IPR as a function of the transverse wave number $\kappa a$ for the chaotic case ($\gamma =3/2$). As expected, the IPR concentrates in the vicinity of $\text{IPR}=3$ except for the modes built on neutral orbits (upper inset, A). Inset B shows a generic ergodic mode.

Figure 8

IPR as a function of the transverse wave number $\kappa a$ for the mixed case ($\gamma =0.95$); Families of modes are distinguishable as shown by the different markers: red circles correspond to 2-PO regular modes, from red to green dotted and dashed lines ( to ) correspond to higher order regular modes denoted from $m=1$ to $m=9$, purple crosses () correspond to modes built on the 6-PO, light and medium plus symbols (, ) and dark blue stars () correspond to whispering gallery modes (insets $A$ to $E$).

Figure 9

IPR as a function of $\kappa a$ for the regular modes and for high values of $\kappa$. The dots and circles correspond to the points presented in Fig. 8, and the continuous lines correspond to the expression (16). The full circles mark the IPR for a fixed value of $p$, while $m$ varies from 0 to 9. Three values of $p$ are marked.

Figure 10

Experimental setup. P: polarizer; M: mirror; $\lambda /2$: half-wavelength wave plate; $\times 10, \times 20, \times 40$: microscope objectives; SLM: spatial light modulator; L: lens needed to collect the FF; CCD: CCD camera.

Figure 11

Initial illumination beams, NF and FF observed at the output of the fiber, corresponding numerical simulations and numerical calculated spectra, for an illumination favorable to the excitation of a family of (a–f) regular modes and (g–l) superposition of modes with no 2-PO regular modes.

Figure 12

Zoom of the spectrum of Fig. 11 between two peaks $p=21$ and $p=22$. Inset: The intensity of the modes corresponding to $u_{4,20}$ and $u_{2,21}$.

Figure 13

Length spectra corresponding to spectra presented in (a) Fig. 11: one can see the regularly distributed peaks corresponding for the first to the length of the 2-PO and for the others to the harmonics; and in (b) Fig. 11 one can see that no specific length exists (see inset) even if a zoom shows some peaks distributed in a disordered way. The arrows mark the 2-PO and 5-PO.

Figure 14

(a, b) Experimental FF measured for two different spatial modulations of the illumination beam. (c) Angular integrated FFs. Blue dashed line corresponds to case (a) and red line to case (b).

Figure 15

(a, b) Numerical FF calculated for two different 2-PO regular modes with $p=12$ and $p=13$, respectively. (c) Angular integrated FFs. Blue dashed line corresponds to case (a) and red line to case (b). The vertical dashed lines correspond to the value of the wave number of the $u_{0,p}$ modes.

Figure 16

Angular integrated FF obtained by numerical simulations. (a) For an illumination favorable to the excitation of regular modes. The vertical dashed lines correspond to the value of the wave number of the $u_{0,p}$ modes; (b) for a modulation along the $y$ direction, so that no regular mode is being excited. The $|c_n{|}^2$ are represented by continuous vertical lines in both cases. Insets show the corresponding FF.