speckle patters arise whenever lights from a variety of directions hit a scree, When the number of different directions is large (≈>50) the one sees the "normal" speckle pattern (BTW each fo the black spec is actually a phase singularly, or optical vortex). A multimode fibre typical supports >1000 fibre modes so their addition/interference is what creates the speckle.

within a ray optical picture, the rays associated with different fibre modes zig-zag down the fibre at different angles from each other 8every mode having ≈ its own zig zag). the higher order modes zig zag more tightly. remember that the wavevector has x, y, and z components (kx^2+ky^2 +kz^2 = k0^2). if the ray has non zero kx and ky then kz must be reduced below k0. the more tight the zig zag the more kz is reduced. phase velocity in z-direction is omega/kz, smaller kz gives larger Vphase.

speckle patters arise whenever lights from a variety of directions hit a scree, When the number of different directions is large (≈>50) the one sees the "normal" speckle pattern (BTW each fo the black spec is actually a phase singularly, or optical vortex). A multimode fibre typical supports >1000 fibre modes so their addition/interference is what creates the speckle.

within a ray optical picture, the rays associated with different fibre modes zig-zag down the fibre at different angles from each other 8every mode having ≈ its own zig zag). the higher order modes zig zag more tightly. remember that the wavevector has x, y, and z components (kx^2+ky^2 +kz^2 = k0^2). if the ray has non zero kx and ky then kz must be reduced below k0. the more tight the zig zag the more kz is reduced. phase velocity in z-direction is omega/kz, smaller kz gives larger Vphase.

Thank you so much for your helpful answer.

Let's consider a two-mode (only) fiber with a step index. The fundamental mode has tails that extend into the lower index cladding that surrounds the higher index core. The higher order mode also has tails that extend into the low index cladding but these tails extend further than for the fundamental mode. We say that the confinement factor of a fundamental mode is higher in the core than higher order modes. You can see this if you calculate the first two modes of a step index fiber or even if you visualize the far-field pattern of these modes when they exit the fiber.

The effective index is essentially a confinement weighted average of the material indices of the mode. So let's take our two-mode example. Fundamental mode: core confinement = 0.8 in material with core material index 1.6 (I'm making up these numbers) and 0.2 with cladding with index 1.5. Second mode: core confinement = 0.6 and cladding confinement 0.4. So the effective index of the fundamental mode is about 0.8 x 1.6 + 0.2 x 1.5 = 1.58 while the effective index of the next higher order mode is 0.6 x 1.6 + 0.4 x 1.5 = 1.56. The second order mode has lower effective index and therefore travels faster. You can think of it crudely as 'sticking out more into faster material' than the fundamental mode, which is dragging along with more of itself in the slower (higher index) core material. Now so far, we've been talking about two mode fiber.

Let's now talk about multimode fiber in which lots of modes can propagate at once. In this case, modes 1, 2, ... N all travel down the fiber and, as they do, they interfere with one another creating local maxima and minima, i.e., speckles. But the modes are not affecting with one another because the fiber is linear (in this approximation). So each of the modes just travel along as in our original two mode fiber. As the modes travel at their own speeds, each having their own effective index (and remember that speed = c / neff), the modes beat against one another. But, your question asks why each mode travels at a different speed -- and that is because each mode has a different spatial cross-section in the step index fiber and therefore a different effective index. The beating (interference) of the multiple modes creates the speckle. As you bend such a fiber, you change the speeds of the modes and the speckle changes accordingly. It's a very complex system made up of very simple things, much like ocean waves are simple but there are so many of them that the resulting surface pattern is complicated. Hope this helps.

Thank you so much. Your answer was helpful. Now I can imagine it better.

By the way, Prof. Padgett's clear and concise answer above is another view of the many modes within your step-index, multimode fiber. By assigning each eigenmode a k-vector, as Prof. Padgett has done, the far-field of the light exiting the fiber can be reconstructed as it strikes a distant screen, assuming random phases between modes. Sort of a Fraunhofer picture of the Fourier transform of the fiber's modal structure. Sort of. I didn't know that the black spots in the speckle pattern were optical vortices. Very cool.