定义：
在光纤或者其它波导中，由于弯曲带来的传输损耗。

弯曲损耗是光纤光学中常见的问题：当光纤被弯曲的时候会有附加的传输损耗。通常情况下，一旦光纤达到某一个临界曲率半径后，损耗便会迅速上升。这个临界曲率半径对于不同的光纤差别很大：对于具有较好导波特性的光纤（即具有高数值孔径的光纤），这个临界值就很小（几毫米）；但是对于普通模式面积很大的单模光纤，这个临界值通常很大（几十厘米）。 

通常情况下，尽管由于包层和涂覆层或者更外面的涂覆层表面的反射光干涉效应随波长是振荡变化的，但是弯曲损耗对于波长越长的分量的影响越大。在长波段高的弯曲损耗通常都会限制单模光纤的可传输光的波长范围。例如工作在1mm波段的单模光纤，其单模截止波长为800nm，但是它不能用于波长1500nm的光的传输，因为在该波长处具有很高的弯曲损耗。需要注意的是即便光纤不存在宏观维度的弯曲，它依然可能会有明显的弯折损耗，这部分弯折损耗很可能是由制造过程中的一些缺陷导致了微观维度的弯曲。 

图1：大模式面积光纤中的振幅分布，沿着右侧弯曲越来越大。光学模式变得越来越小并且损耗越来越大，光耦合出去变成包层模式。数值模拟采用了RP 光纤功率软件。 
光子晶体光纤的弯曲损耗可以非常低，即便在大于单模截止波长的情况下。因此在某种程度上可以称其为“无截止单模”，也就是说在一个很大的范围内都保持单模特性。 
在多模光纤中，相对于基模，高阶横模的临界曲率半径通常更小。通过调节曲率半径，可以对高阶模引入较高的损耗的同时不会显著影响基模。这在设计高功率光纤放大器和光纤激光器时采用多个横模的光纤可以实现更大的有效模场面积。 
可以采用等效折射率方法估测弯曲损耗的大小。其原理为在一定的有效折射率（考虑了横向不同位置处光程长度的改变量后）下计算模场分布。当外包层表面的反射光不会重新反射进入光纤纤芯的时候，该方法能够很方便的给出近似的弯曲损耗。一些更复杂的模型包含了以上效应，这样可以计算出随波长的变化情况，但是实际计算中比较复杂。 
弯曲损耗的大小有时与偏振有关。这个性质可以被用于从光纤激光器中获得一个稳定的单偏振态的输出。 
必须注意的是，弯曲不仅会引入额外的损耗，还会减小光纤的有效模场面积。这一性质对于大模场面积阶跃折射率光纤尤为显著。另外，弯曲还会导致双折射。

Definition: propagation losses in an optical fiber (or other waveguide) caused by bending

More general term: propagation losses

Bend losses are a frequently encountered problem in the context of waveguides, and in particular in fiber optics, since fibers can be easily bent. Bend losses mean that optical fibers exhibit additional propagation losses by coupling light from core modes (guided modes) to cladding modes when they are bent. Typically, these losses rise very quickly once a certain critical bend radius is reached. This critical radius can be very small (a few millimeters) for fibers with robust guiding characteristics (high numerical aperture), whereas it is much larger (often tens of centimeters) for single-mode fibers with large mode areas.

Generally, bend losses increase strongly for longer wavelengths, although the wavelength dependence is often strongly oscillatory due to interference with light reflected at the cladding/coating boundary, and/or at the outer coating surface. The increasing bend losses at longer wavelengths often limit the usable wavelength range of a single-mode fiber. For example, a fiber with a single-mode cut-off wavelength of 800 nm, as is suitable for operation in the 1-μm region, may not be usable at 1500 nm, because they would exhibit excessive bend losses. Note that even without macroscopic bending of a fiber, bend losses can occur as a result of microbends, i.e., microscopic disturbances in the fiber, which can be caused by imperfect fabrication conditions.

Figure 1: Amplitude distribution in a large mode area fiber, which is bent more and more strongly towards the right side. The fiber mode becomes substantially smaller and then very lossy; the light is coupled out into cladding modes. The numerical simulation has been done with the RP Fiber Power software.

In multimode fibers, bend losses are usually strongly mode-dependent. The critical bend radius is typically larger for higher-order transverse modes. By properly adjusting the bend radius, it is possible to introduce significant losses for higher-order modes without affecting the lowest-order mode. This can be useful e.g. for the design of high-power fiber amplifiers and fiber lasers where a higher effective mode area can be achieved when using a fiber with multiple transverse modes.

The magnitude of bend losses has some dependence on the polarization. This can be exploited, for example, for obtaining stable single-polarization emission from a fiber laser.

Photonic crystal fibers can have very low bend losses even far beyond the single-mode cutoff wavelength. Therefore, they can be “endlessly single-mode”, i.e., they exhibit usable single-mode characteristics over a very large wavelength range.

Note that bending not only introduces losses, but can also reduce the effective mode area. This is particularly true for large mode area step-index fibers. Also, bending induces birefringence [3, 5].

Estimating Bend Losses

For estimating the magnitude of bend loss, the equivalent index method [4] can be used. The basic idea behind this technique is to calculate the mode distributions for an effective index which contains a term accounting for the modified path lengths at different transverse positions. An elasto-optic correction term (taking into account local modifications of the refractive index by mechanical stress) leads to an effectively weaker “tilt” of the refractive index profile than when considering the geometrical effect alone [3, 9].

Such a method of calculating bend losses is convenient and usually a good approximation, provided that there is no light reflected e.g. from the outer cladding surface back to the fiber core. More sophisticated models (see e.g. Ref. [6]) can include such effects, and thus predict the full wavelength dependence, but are complicated to handle.

Bend-insensitive Fibers

In some application areas of fibers, and particularly in optical fiber communications, it is of interest to have a relatively bend-insensitive fibers. This is particularly important in the area of fiber to the home (FTTH), where it is problematic to demand that any tight bending is avoided in the fiber installation. The G.657 standard for telecom fibers defines the characteristics of such fibers, with the main categories A for use in access networks and B for short distances, but with even stronger bending tolerance – in category B3 allowing a bend radius down to only 5 mm. For comparison, the more general category G.652 allows a bend radius only down to 30 mm.

Bend-insensitive fibers can be made with different designs. It is common to have trench-assisted fiber designs, containing a ring (trench) with relatively low refractive index around the fiber core. Another possibility are certain designs of photonic crystal fibers, which may be called hole-assisted.

Note that there are trade-offs between bend insensitivity and other properties, such as low propagation losses in the straight form. However, lowest propagation losses are not essential e.g. for FTTH, where only a relatively short transmission distances need to be realized.

Bend Losses in Photonic Integrated Circuits

Bend losses are important not only in fiber optics, but also in the context of photonic integrated circuits. Compact circuits designs often require strong bending of waveguides on such chips, with desired values of the bend radius of a few microns only, but limited by the issue of bend losses. For strong bending and thus particularly compact designs, waveguides with a relatively high numerical aperture, i.e., with a large refractive index contrast (i.e., a high numerical aperture), are needed, e.g. based on silicon-on-insulator (SOI) technology. Those, however, are tentatively more sensitive to propagation losses by scattering at non-perfect boundaries between core and cladding.

Bibliography