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In the previous sections, we have seen that, depending on its refractive index profile and wavelength, a fiber can guide a different number of modes. If the numerical aperture and refractive index contrast are small, it is probably only a single guided mode (LP01 mode). In this case, the fibers are called single-mode fibers LP 11, LP 20 etc. Higher order modes do not exist – only the cladding modes, which are not confined around the fiber core.

Note that in most cases light with different polarization states can be guided. The term “single-mode” ignores the fact that often (for radially symmetric refractive index profiles and no birefringence) one actually has two distinct modes, with identical intensity profiles but orthogonal linear polarization directions. Any other polarization state can be considered as a linear superposition of these two. (See also Section 6 on polarization issues.)

**single-mode guidance condition**

For step-index fiber designs, there is a simple criterion for single-mode guidance: the V number must be below ≈2.405. The V number is defined as:

where * λ *is the vacuum wavelength, a is the core radius, and NA is the numerical aperture. For other radial dependencies of the refractive index, even for refractive index profiles that are not radially symmetric, the single-mode condition usually has to be calculated numerically. It is incorrect to use the criterion V < 2.405, for example, to calculate V from the largest exponent difference

**Effect of core size**

In general, one might think that a smaller core means smaller fiber modes. This is true for a constant V number; the exponential contrast gets smaller and smaller for larger cores. However, if we keep the numerical aperture constant, the V number varies with the core radius, and the mode radius shows a non-monotonic dependence on the core radius, as shown in Figure 2, with an NA of 0.1:

Figure 1: Dependence of mode radius on core radius for a constant numerical aperture of 0.12. The mode radius is defined by the second moment (D4σ value) of the intensity distribution. The unimode state is to the left of the gray vertical line.

It can be seen that for core radii below ≈4.8 μm, the V-number becomes considerably smaller and the mode radius increases. At small values of V, the mode extends far beyond the core and deviates considerably from a Gaussian shape.

Figure 3 shows the case for a higher NA of 0.3:

Figure 2: Dependence of mode radius on core radius for a constant numerical aperture of 0.12.

**Example: Typical Single Mode Fiber**

A typical single-mode fiber at a wavelength of 1.5 μm might have a step-index profile with a core radius of 4 μm and a numerical aperture of 0.12. The guided mode has a mode radius of 5.1 μm and an effective mode area of 75 μm2. That’s not far off the figure for Corning’s commonly used SMF-28 telecom fiber.

Figure 3: The dashed curve for single-mode fiber LP 01 shows a very similar Gaussian distribution. Gray vertical lines show the location of the core/cladding boundary.

As with single-mode fiber, the field distribution extends significantly beyond the core; only 54.4% of the power propagates in the core. (It may seem more based on Figure 3, but note the factor r in the area integral, which makes the outer part of the profile more important.) However, the intensity drops off rapidly with increasing radial coordinates. The intensity distribution is close to a Gaussian distribution.

When we decrease the wavelength, we find that the fiber is no longer single-mode below the cut-off wavelength of 1254 nm: in addition to the LP 01 mode, it also supports the LP 11 mode (in fact two of them have an orthogonal orientation). Below 787 nm, an additional LP 02 mode is added.

Optical In principle, the fiber remains single-mode for any wavelength above the cut-off wavelength (ie 1254nm). However, for longer wavelengths, the mode becomes larger and it becomes more sensitive to bending losses, caused by macroscopic bending and microscopic defects. For the designs discussed here, another problem is actually more serious: beyond ≈2 μm, the base material (silica) starts to absorb. Therefore, in practice, the wavelength spacing over which single-mode fiber can be used is limited.

**Launch light into a single-mode fiber**

Efficient launching of light into single-fiber modes requires a high degree of overlap between the complex amplitude distribution of the incident light (assumed to be monochromatic) and the corresponding mode amplitude distribution. Fortunately, the fundamental mode of a single-mode fiber has a profile close to that of a Gaussian beam in most cases (with sufficiently large values of V for robust guidance), and a Gaussian beam can be well approximated by Output mode laser. So the remaining tasks are:

- Correctly focus the laser beam so that the beam radius is close to the radius of the fiber mode,
- Position the fiber end at the beam focus (beam waist), and
- Align the fiber so that the beam focus hits the fiber core in the correct direction.

Clearly, the position error of the incident beam should be small compared to the mode radius. The formula below illustrates how the emission efficiency (disregarding possible reflections from the interface) depends on the position error * Δx *and the possible deviation between the input beam radius w1 and the mode radius w2, if we can assume a Gaussian mode distribution:

We see that for a perfect beam size, an offset of one beam radius already reduces the coupling efficiency to 1/e ≈ 37%, while a 5x smaller error achieves a coupling efficiency of 90%. Note that this equation only works for Gaussian profiles, but in most cases this is a good approximation.

The beam direction must also be correct. However, this is not so sensitive for typical single-mode fiber. Angle error should be much lower than beam divergence, but relatively large for smaller more areas.

**Effects of Imperfect Launch Conditions**

For example, what happens if we slightly misalign the input beam? Figure 2 shows an example simulation where the input laser beam is displaced by 1/10 of the beam radius. After a certain propagation length, only the light in the guided mode remains. All other light is lost in the cladding. (Losses at the cladding/coating interface are usually large.) For example, at the end of a 10 cm long fiber, we would find that there is only light in the core, whose distribution is determined only by the mode distribution. The emission conditions only affect the emission power, but not the output beam profile.

Figure 4: Light propagation at 1.5 μm wavelength in a single-mode fiber showing the input beam. Numerical simulations were performed with RP Fiber Power software.

**get large pattern area**

For some applications, it is desirable to have a relatively large pattern area while still having single pattern guidance. For example, one may wish to minimize nonlinear effects or maximize the energy stored in a pulsed fiber amplifier while maintaining high beam quality. In principle, it is easy to obtain single-mode guidance even for fairly large cores: just reduce the index contrast (and thus the numerical aperture). For example, the core radius in the example above can be increased fivefold to 20 μm. If we reduce the numerical aperture by the same factor to 0.024, we still get single-mode guidance – now the effective area is 1869 μm2, which is 5 2 = 25 times larger than before. Unfortunately, however, this created some troubles:

- Since the index contrast is now very weak (0.0002), the fiber is extremely sensitive to small random index changes due to incomplete control of fabrication conditions. An NA of 0.024 is considered impractically small for currently available techniques.
- Even if a perfect fiber were made based on this design, it would be very sensitive to bending. Figure 2 shows a numerical simulation where the reverse bend radius grows linearly along the fiber. On the right, the bending radius reaches 1 m. Before that, light undergoes severe bending losses. This means that fiber optics should only be used if they remain essentially straight and any significant microbending can be avoided.

Figure 5: Amplitude distribution in a large mode area fiber bent increasingly to the right (bending radius up to 1 m). On the right, severe bending losses occur. Assume that the fiber cladding has a radius of 125 μm and that light is completely reflected at this outer interface.

More advanced fiber designs have been developed in which mode areas well above 1000 μm can be obtained as well as better bend performance and lower susceptibility to fabrication errors. However, the described problems can be completely avoided. The fundamental problem is that a very large pattern is necessarily only weakly guided and thus sensitive to various additive effects.

A single-mode fiber that efficiently launches light into a large mode region is simpler in terms of positional alignment than a small region. Note, however, that angular alignment becomes more sensitive as the beam divergence becomes smaller.

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