## Abstract

We present a novel approach to directly measure the bend loss of individual modes in few-mode fibers based on the correlation filter technique. This technique benefits from a computer-generated hologram performing a modal decomposition, yielding the optical power of all propagating modes in the bent fiber. Results are compared with rigorous loss simulations and with common loss formulas for step-index fibers revealing high measurement fidelity. To the best of our knowledge, we demonstrate for the first time an experimental loss discrimination between index-degenerated modes.

© 2013 Optical Society of America

## 1. Introduction

Fiber bending is a widespread and common effect in all situations, be it in scientific or industrial environment, where fibers are used to guide light. In practice, nearly every fiber is bent to a certain degree, either with or without intention, which makes bending one of the most widespread effects impacting the performance of a fiber. In multimode optical fibers bending influences the individual modes differently due to their distinct properties like effective refractive index, intensity and phase distribution, etc. Hence, the fields of applications are manifold: fiber bending is employed in fiber lasers to suppress higher-order modes, to achieve effective single mode regime of multimode fibers [1, 2], and is of enormous significance for modern telecommunication, especially regarding mode multiplexing techniques [3–5] to enhance the capacity of today’s information transmission.

Various bending models exist in the literature as well as simulations dealing with the impact of bending onto individual fiber modes [6–12]. Whereas these models extensively cover the field of modal bend losses in multimode fibers, experiments are up to date limited to non-modally resolved measurements [13, 14] or to a quasi-single-mode regime [15].

In this work we present a novel approach to measure the bend loss of individual modes of multimode optical fibers. The method makes use of a computer-generated hologram to perform a modal decomposition of a given optical field, called the correlation filter technique. The modal decomposition is used to measure the power of each mode as a function of the bending diameter. Results are compared with rigorous bend simulations and with common loss formulas given in the literature [7, 9]. Besides other modal decomposition techniques, such as C^{2}- and S^{2}-imaging [16, 17], the correlation filter technique stands out due to its easy setup, real-time ability of the measurements, as well as the capability to distinguish degenerated modes (cf. section 5), and is therefore excellently suited to modally characterize fiber bending processes. The presented work is considered to be of significance for the field of the design and characterization of fibers and fiber lasers, specifically for verifying the value of various bend-resistant fiber designs, whose proposals are mainly based on simulations only [18, 19]. Moreover, the renaissance of mode multiplexed telecommunication will benefit from this work, since the modal transmission properties will strongly depend on bending.

The paper is structured as follows: section 2 shortly reviews the theoretical treatment of bent fibers in the fashion we used them in our analytical and numerical calculations. Section 3 introduces the investigated fiber and the determination of its parameters. Section 4 outlines the principles of the correlation filter method, cf. section 4.1, and illustrates the experimental setup, cf. section 4.2. Section 5 presents the discussion of the results and is followed by the summary, cf. section 6.

## 2. Bend modeling and simulation

#### 2.1. Theoretical treatment of bent fibers

Modeling of optical fibers typically relies on the translation invariance of the fiber’s refractive index distribution in the propagation direction, *z*, of light. When the fiber is curved this invariance no longer exists. However, by executing a proper mathematical coordinate transformation [11, 12], called conformal mapping, the curvature of the bent fiber can be taken into account by a tilted refractive index profile (Fig. 1) and the translation invariance in propagation direction can be retrieved. If the fiber’s cross section is set to the *x*-*y*-plane and the fiber is curved in the *x*-*z*-plane the well-known coordinate transformation formula reads as [12]:

*R*, the equivalent (tilted) refractive index profile

*n*

_{equ}(

*x*,

*y*) and the refractive index profile of the unbent fiber denoted by

*n*

_{mat}(

*x*,

*y*).

Taking into account that there is an additional change in the refractive index due to the stress-optical effect caused by the local strain of the fiber in the curved region, the anisotropic photoelastic change of the refractive index Δ*n _{i}* in direction

*i*,

*i*∈ {

*x*,

*y*,

*z*}, needs to be considered [20]:

*n*and

_{i}*n*

_{0}are the refractive indices of the stressed and unstressed glass,

*σ*are the components of the stress

_{i,j,k}*σ*̂ in the direction denoted by the indices (

*i*,

*j*,

*k*), which are a cyclical permutation of (

*x*,

*y*,

*z*), and the photoelastic constants

*B*

_{1}= 4.22 × 10

^{−6}MPa

^{−1}and

*B*

_{2}= 0.65 × 10

^{−6}MPa

^{−1}for fused silica [21]. The stress

*σ*̂ results from the strain

*ε*̂ in the bent fiber. They are connected via the stress-strain relation that in the linear region can be described by Hooke’s law

*σ*̂ =

*C*̃

*ε*̂ with the elasticity tensor

*C*̃. In case of isotropic materials its only nonzero components read as ${C}_{11}=\raisebox{1ex}{$E\left(1-\nu \right)$}\!\left/ \!\raisebox{-1ex}{$\left[\left(1+\nu \right)\left(1-2\nu \right)\right]$}\right.$ and ${C}_{12}=\raisebox{1ex}{$E\nu $}\!\left/ \!\raisebox{-1ex}{$\left[\left(1+\nu \right)\left(1-2\nu \right)\right]$}\right.$ with

*ν*= 0.164 being the Poisson number and

*E*= 76GPa being the Young’s modulus for fused silica [21].

The dominant elongation component is given in the direction of the fiber axis (*z*-direction) by
${\epsilon}_{z}=\raisebox{1ex}{$\mathrm{\Delta}l$}\!\left/ \!\raisebox{-1ex}{$l$}\right.=\raisebox{1ex}{$x$}\!\left/ \!\raisebox{-1ex}{$R$}\right.$[22]. Here, *x* denotes the distance from the center of the fiber in the bending plane and *R* is the bend radius. Neglecting the transverse strain components (*ε _{x}* =

*ε*≈ 0), the components of stress can be calculated according to ${\sigma}_{x}={\sigma}_{y}=\raisebox{1ex}{${C}_{12}x$}\!\left/ \!\raisebox{-1ex}{$R$}\right.$ and ${\sigma}_{z}=\raisebox{1ex}{${C}_{11}x$}\!\left/ \!\raisebox{-1ex}{$R$}\right.$. The refractive index of the material in transverse directions including the photoelastic effect therefore reads as:

_{y}*n*= Δ

_{x}*n*. The above mentioned coordinate transformation is now applied to this modified refractive index distribution according to Eq. (1). Accordingly, the impact of the photoelastic effect can be considered by an effective radius of curvature

_{y}*R*

_{eff}defined as

*R*

_{eff}= 1.40

*R*used here slightly differs from

*R*

_{eff}= 1.28

*R*given in [7]. This is caused by the different theoretical approach, which we used in starting from the strain defined by Morey [20] as the origin of the photoelastic refractive index change instead of starting from the stress defined by Pockels [23]. In our case this leads to more consistent results between calculations and measurements.

In the calculations we omitted the presence of the coating. Hence, we calculated a core surrounded by a virtually infinite cladding. This decision relies on the fact, that we do not know the material parameters *n*, *B*_{1}, *B*_{2}, *E* and *ν* for the coating material, which are required for a reasonable consideration. In consequence, we neglect e.g. the Fresnel reflection at the interface between cladding and coating, which is known to lead to an oscillatory behavior of the bend loss as a function of the radius of curvature under specific circumstances [14,24]. However, we did not observe these oscillations in the conducted experiments (cf. section 5).

#### 2.2. Analytical loss calculation

Regarding an analytical treatment of the bend losses in step-index fibers, there exists a variety of approaches in the literature [7, 9, 14, 24, 25]. According to Marcuse *et al.*[9] and further developed by Schermer *et al.*[7], who took photoelastic effects into account, the bend loss can be computed by:

*R*is the bending radius,

*R*

_{core}is the radius of the fiber core,

*n*

_{core}and

*n*

_{clad}are the refractive indices of the fiber core and cladding, $k=\raisebox{1ex}{$2\pi $}\!\left/ \!\raisebox{-1ex}{$\lambda $}\right.$ is the free space propagation constant,

*β*is the propagation constant of the guided mode in the unbent waveguide, $\kappa ={\left({n}_{\text{core}}^{2}{k}^{2}-{\beta}^{2}\right)}^{1/2}$ and $\gamma ={\left({\beta}^{2}-{n}_{\text{clad}}^{2}{k}^{2}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$ are the normalized propagation constants in core and cladding, respectively, $V={\left({\kappa}^{2}+{\gamma}^{2}\right)}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}=\raisebox{1ex}{$2\pi $}\!\left/ \!\raisebox{-1ex}{$\lambda $}\right.{R}_{\text{core}}\sqrt{{n}_{\text{core}}^{2}-{n}_{\text{clad}}^{2}}$ is the normalized frequency,

*K*

_{m}_{±1}is the modified Bessel function of the second kind, and

*e*

_{m}is a scalar depending on the order of the mode LP

_{mn}(

*e*

_{m}= 2 for m = 0 and

*e*

_{m}= 1 otherwise). We calculated the propagation constant

*β*using the analytical approach outlined in [26]. The effective bending radius

*R*

_{eff}was calculated by Eq. (4). For reasonable loss estimations using Eq. (6) the fiber parameters

*R*

_{core}and numerical aperture must be known with a precision exceeding standard fiber specifications. For this reason, they have been determined experimentally, as explained in the following section 3.

#### 2.3. Numerical loss computation

Numerical computation was done using the COMSOL Multiphysics^{®} software which provides a full-vectorial finite element method (FEM) mode solver. The geometry was separated into core and cladding, each characterized by a corresponding diameter and refractive index. The outer boundary region was realized by a perfectly matched layer (PML) [27], with a refractive index matched to the cladding region, for implementing losses into the simulation. The performance of the PML was improved by establishing a scattering boundary condition on the very edge of the simulation domain to avoid reflections.

The refractive index distribution in each simulation region was defined by the equivalent index profile of Eq. (1) including the photoelastic influence given by Eq. (3). The fiber under test was modeled using NA = 0.1202 and *R*_{core} = 3.85 μm, whose determination is outlined in section 3, and *n*_{clad} = *n*_{FusedSilica}(*λ* = 1064nm) = 1.450 [28]. The modal power losses 2*α* are obtained from the imaginary parts of the effective mode indices Im(*n*_{eff}):

*λ*.

## 3. Fiber properties

The fiber under test was a step-index fiber similar to the standard telecommunication fiber SMF28^{™} which is single-mode at 1550 nm wavelength. Beside the fundamental mode LP_{01}, the next higher-order modes LP_{11}* _{e,o}* are guided at 1064 nm, which is the measuring wavelength. For precise modeling of the fiber it is crucial to know its characteristic parameters, core radius

*R*

_{core}and numerical aperture NA, as accurately as possible. Our FEM calculations show that differences in

*R*

_{core}or NA of about 5% can result in bending losses that differ by more than one order of magnitude. Therefore, precise knowledge of these two parameters is essential for the loss simulations, necessitating to measure those quantities.

Since no fiber is an ideal step-index fiber, measuring the values of these two parameters directly with high accuracy is complicated. To overcome this problem we chose an indirect method where we measured two optically effective and easily accessible parameters, the LP_{11} modes cut-off and the effective index ratio of the LP_{01} and the LP_{11} modes, and subsequently, used the FEM mode solver to find the combination of *R*_{core} and NA, which predicts the measured cut-off wavelength and effective index ratio best.

The cut-off wavelength of the LP_{11} modes was determined according to the international standard [29]. This was done by comparing the transmission spectra of the unbent fiber and of the fiber with a bend around a mandrel. At specific wavelengths the higher-order modes (HOM) are very weakly guided and power carried by these modes is lost due to the bending. Consequently, the difference spectrum, see Fig. 2(a), shows loss maxima close to the defined cut-off wavelength *λ*_{c} of the HOM. Using the V-Parameter of the LP_{11} mode cut-off V_{c} = 2.405 [30], a condition connecting *R*_{core}, NA and *λ*_{c} can be deduced:

*R*

_{core}NA is determined without knowing the parameters separately.

To identify the exact values of both parameters, an additional measurement of the effective mode index ratio of the LP_{01} and the LP_{11} mode was performed. For this purpose a Bragg grating was inscribed into the fiber and the reflection spectrum shown in Fig. 2(b) was analyzed. In this reflection spectrum each of the two major peaks is associated with a specific mode and assigned to a specific Bragg wavelength *λ*_{LPmn}. The Bragg condition [31]

*λ*

_{LPmn}with the corresponding effective index

*n*

_{LPmn}by the grating period Λ. From the Bragg condition for the two involved modes LP

_{01}and LP

_{11}the relation

*R*

_{core}and NA, considering the constraints of Eq. (8) and Eq. (10), iterative FEM simulations finally found the pair

*R*

_{core}= 3.85 μm and NA = 0.1202 to match the experimental targets shown in Fig. 2 best.

This procedure is valid since the homogeneous contribution of the Bragg grating to the refractive index of the fiber core is negligible as verified by the following estimation. The grating induced average refractive index shift Δ*n*_{DC} can be estimated by *η*Δ*n*_{DC}/*n* = Δ*λ*/*λ*[31], where *η* is the overlap of the mode with the grating, which yields *η* ≈ 0.9 for the current fiber, *n* = 1.454 is the fiber core index (without grating), Δ*λ* = 0.106nm is the wavelength shift of both Bragg wavelengths, measured during the inscription process of the grating, and *λ* = 1081nm is the design wavelength. These parameters result in a refractive index change Δ*n*_{DC} = 1.6 × 10^{−4}. Taking this refractive index shift into account changes the best matching core radius and numerical aperture by only 1%. Particularly, since the product of core radius and numerical aperture is intentionally kept constant to conserve the measured cut-off wavelength (cf. Eq. (8)), the influence of the grating-induced refractive index change on the virgin fiber parameters and hence on the calculated modes and modal loss is negligible.

## 4. Measurement of modal bend loss

#### 4.1. Modal decomposition

Modal decomposition is a very versatile tool to characterize multimode laser beams yielding detailed information on the modes that compose the beam. A simple practical realization of such a modal decomposition is attained with the correlation filter technique [32]. Whereas the concept of this method is well-known [33], it has only recently been used in practical setups for various applications [34–37]. Modal decomposition of an optical field *U* implies its expansion into an orthonormal basis set. The basic functions in the case of a multimode fiber are the single transverse modes that are able to propagate in the given distribution of refractive index, which limits the number of modes to a finite set. Mathematically the modal decomposition is expressed as [32]:

**r**= (

*x*,

*y*),

*c*=

_{l}*ρ*

_{l}e^{iφl}is the complex expansion coefficient,

*ψ*(

_{l}**r**) is the

*l*mode with amplitude

^{th}*ρ*, phase

_{l}*φ*(with respect to a reference phase) and

_{l}*N*is the number of modes. To perform the modal decomposition, as described by Eq. (11), experimentally is the main task of the correlation filter method. The centerpiece of the technique is a computer-generated hologram (CGH, correlation filter) that is illuminated by the fiber beam. In the hologram the fields of the fiber modes are encoded using the coding technique of Lee

*et al.*[38]. For this purpose, the modes are calculated prior to the experiments based on the fiber’s parameters determined in section 3 by the aforementioned FEM mode solver (cf. section 2).

The hologram performs a correlation analysis of the beam with each encoded mode. This correlation is evaluated by recording the diffraction pattern (far field) with a CCD camera and measuring the intensities on the local optical axes of the correlation functions of each mode, which appear spatially separated in the far field. According to Kaiser *et al.*[32], the intensity on the local optical axes
${I}_{l}\propto {\rho}_{l}^{2}$ is called the correlation signal. As an example of the working principle, Fig. 3 shows two measured examples of the correlation analysis of a beam with the encoded modes, which are in this case the modes LP_{01}, LP_{11}* _{e}*, and LP

_{11}

*of the fiber specified in section 3. Illuminating such a hologram with the LP*

_{o}_{01}mode (Fig. 3(a)) results in a strong

*I*

_{LP01}-signal and zero

*I*

_{LP11}-signals (Fig. 3(b)). Similarly, the simulation of the diffraction of a LP

_{01}mode illuminating a transmission function encoded only with the LP

_{11}

*mode (Fig. 3(c)) shows a dark spot in the center of the diffraction pattern. In contrast, illuminating the same hologram with a (nearly) pure LP*

_{e}_{11}

*mode beam (Fig. 3(d)), a strong*

_{e}*I*

_{LP11e}-signal appears, where the

*I*

_{LP01}- and

*I*

_{LP11o}-signals are close to zero (Fig. 3(e)). Figure 3(f) depicts the simulation of a LP

_{11e}mode beam illuminating a hologram that only encodes the LP

_{11e}mode. Hence, in contrast to the illustration in Fig. 3(c), a bright spot appears at the center of the diffraction pattern.

Mathematically, the correlation function *C*(**r**) is achieved by the multiplication of the field under test *U*(**r**) with the transmission function *T*(**r**) of the hologram and Fourier transforming the resulting field using a lens of focal length *f* yielding [32]:

*A*

_{0}is a constant, and

*T*̃ and

*Ũ*denote the Fourier transforms of

*T*and

*U*, respectively.

Note that the integral relation of Eq. (12) is solved all-optically using the hologram and a subsequent 2*f* -setup.

To measure all modal powers
${\rho}_{l}^{2}$ simultaneously from one diffraction pattern, the transmission function *T*(**r**) of the hologram is given by [32]:

^{*}” denotes the complex conjugation and

**K**

*is a spatial frequency, that is different for each mode. Using this transmission function, the signal of the correlation of the unknown field*

_{l}*U*(

**r**) with each mode appears spatially separated in the diffraction pattern (far field) of the hologram, as depicted in Fig. 3(b) and (e), and the intensities on the local optical axes (dots in Fig. 3(b) and (e)) are ${\left|C\left(\mathbf{r}=\raisebox{1ex}{${\mathbf{K}}_{l}\lambda f$}\!\left/ \!\raisebox{-1ex}{$2\pi $}\right.\right)\right|}^{2}={\rho}_{l}^{2}$. However, since the used hologram disregards the polarization of the modes, it is necessary to decompose the

*x*- and

*y*-component of the field,

*U*and

_{x}*U*, separately according to Eq. (11), which is easily achieved by placing a polarizer in front of the hologram. Thus, the total modal power consists of the respective mode powers in

_{y}*x*- and

*y*-direction: ${\rho}_{l}^{2}={\rho}_{l,x}^{2}+{\rho}_{l,y}^{2}$[34]. In the following, we use ”modal power” and ”modal loss” refering to the total mode power ${\rho}_{l}^{2}$.

#### 4.2. Experimental setup

The experimental setup is depicted in Fig. 4(a). A very narrow linewidth Nd:YAG seed laser (*λ* = 1064nm, 50 mW power) was used to excite modes in the fiber under test by focusing the laser light with a microscope objective (MO_{1}) onto the fiber input facet. The focal length of MO_{1} (*f*(MO_{1}) = 10mm) as well as the distance between the laser and the fiber (≈ 50cm) were adjusted to achieve reasonable power coupling.

The composition of the input mode mixture was controlled by transverse and lateral movement of the MO_{1} relative to the fiber, as well as by using a binary phase plate, which shifts one half of the beam by *π* yielding an enhanced overlap with the LP_{11} modes [39]. The bending of a definite fiber section was adjusted by a metal plate with a set of milled half circles, in which the fiber was placed (Fig. 4(b)), or by coiling the fiber around mandrels of different diameter (Fig. 4(c)). In the latter case, the fiber was twofoldly coiled around the mandrel with a constant loop at the end, such that the fiber input and output facet were located on the same side. Both techniques provide an accurate adjustment of the bending diameter. It is important to note that the bending region was always followed by a straight (unperturbed) fiber section of > 0.5m. Hence the decomposition with the hologram was always performed into ideal (undeformed) fiber modes. Moreover, care was taken to minimize transition regions from bent to unbent fiber to avoid fiber sections of different bending diameter, which was done by approximating the geometry of Fig. 4(b) and (c). Bends necessary to guide the fiber to in- and outcoupling stage were > 30cm in diameter and thus neglectable in terms of modal loss as will be shown later.

The light exiting the fiber was imaged onto the CGH and to a CCD camera CCD_{1} to record the beam intensity directly. The magnification was chosen to be
$M=\raisebox{1ex}{$f\left({\text{L}}_{1}\right)$}\!\left/ \!\raisebox{-1ex}{$f\left({\text{MO}}_{2}\right)$}\right.=\raisebox{1ex}{$375\text{mm}$}\!\left/ \!\raisebox{-1ex}{$4\text{mm}$}\right.=94$. A previously placed polarizer was used to select one field component for modal decomposition with the CGH. The diffraction pattern of first order was recorded with a second CCD camera (CCD_{2}) after a 2*f* -setup (*f*(L_{2}) = 180mm), providing the powers of all modes from one camera frame only, according to section 4.1.

The hologram itself was a binary amplitude-only filter in which the transmission function, described by Eq. (13), was encoded according to the coding technique of Lee *et al.*[38], and which was specifically designed for the fiber of section 3. The fabrication of the filter involved the patterning of a photoresist mask by direct writing laser lithography and the subsequent dry etching of a subjacent 60 nm thick chromium layer on a glass substrate.

One might expect the need for a very precise adaption of the fiber parameters and the correlation filter. However, the distribution of the power density of the modes is usually very slowly varying with respect to changing NA, core diameter or wavelength, resulting in a very slowly varying correlation coefficient too. For example, based on the determined fiber parameters, a change of the NA by 10% would change the value of the overlap integral of the two involved fundamental modes from 1 to 0.9988. This is well below the measurement error (relative mode power) of about 2% and hence negligible. Even large deviances in fiber diameter can be handled, provided the different fibers have roughly the same V-parameter, by adapting the magnification of the 4f-setup.

## 5. Results and discussion

Performing a modal decomposition with the setup described in section 4.2 allows measuring the modal powers for a variety of different bending diameters. In a first bending experiment the fiber was placed in the previously mentioned grooved metal plate, in which half circles of definite diameter were milled (Fig. 4(b)). Bending the fiber as described in section 3 starting from an initial diameter of 20 cm down to 0.5 cm results in a change of relative modal content (relative to the total power at each bending point) as depicted in Fig. 5. The bending plane was, as in all following experiments, the *x*-*z*-plane (see coordinate system in Fig. 4). The input coupling of the light was adjusted in such a way that nearly all power was propagating in the LP_{11} modes initially.

As can be seen, the content of the fundamental mode LP_{01} increases substantially from 5% to 99% of the respective total power with stronger bending, whereas the two LP_{11} modes decrease in relative content. Note, that due to the normalization of the mode powers to 100% for each bending diameter, the total power loss is not contained in Fig. 5.

The change of the beam intensity as a function of the bending diameter can be viewed in Fig. 5 (
Media 1). As can be seen, the modal content is initially dominated by the LP_{11} modes (symmetrical two-lobe beam), but is changing continuously into a fundamental Gaussian-like beam (high LP_{01} mode content) towards decreasing bending diameters.

Regarding the LP_{11} mode power curves in Fig. 5, it is interesting to note that the two LP_{11} modes, which are degenerate in an ideal straight fiber, behave differently with respect to bending. The difference in trend is caused by the asymmetry induced by the bend (*x*-*z*-plane). Hence, the intensity lobes of mode LP_{11}* _{e}* are aligned in the direction of the bend, causing this mode to experience higher losses and to drop faster in power compared to mode LP

_{11}

*, whose intensity lobes are orientated perpendicular (*

_{o}*y*-direction). This behavior can be easily understood in terms of the ”tilted” index profile (Fig. 1), which is discussed in section 2. The index profile is lifted at the side opposing the bending center, yielding a shift of intensity of any guided mode to that side, due to a better guidance. The LP

_{11}

*mode exhibits the most power in that region (cf. Fig. 5), yielding the highest field deformation (see also [10]) and accordingly, the highest loss when the surrounding cladding index exceeds the mode’s effective index (cf. Fig. 1).*

_{e}By integrating the measured near field intensities (CCD_{1} in Fig. 4), the measurement results shown in Fig. 5 can be easily converted into a common power loss representation:

*L*is the circumference of the half-circle grooves and the measured modal powers at the largest bending diameter of 30cm were used to approximate the situation of an unbent fiber $\left({\rho}_{l}^{2}\left({D}_{\text{B}}^{\text{max}}=30\text{cm}\right)\approx {\rho}_{l}^{2}\left({D}_{\text{B}}=\infty \right)\right)$, which is reasonable since the modal powers already reached a constant value above 20 cm in bending diameter as seen in Fig. 5. Note that 2

*α*is defined as a positive quantity for power loss and not for amplitude loss. Since the loss of the fundamental mode LP

_{l}_{01}is much lower than that of the LP

_{11}modes, the circumference of the half-circle grooves (Fig. 4(b)) was too short to provide a measurable change in the absolute power of the fundamental mode. For the measurement of the LP

_{01}losses, the fiber was coiled several times around mandrels of definite diameter (Fig. 4(c)) instead of placing it into the grooved metal plate. For each bending diameter different bending lengths were realized by different numbers of applied coils and a length-dependent change in transmitted power was recorded. Afterwards, the modal loss 2

*α*[ $\raisebox{1ex}{$\text{dB}$}\!\left/ \!\raisebox{-1ex}{$\text{m}$}\right.$] was determined from a logarithmic fit of the power data. This procedure is equally valid as the previously mentioned one described by Eq. (14) and yields consistent results regarding the bending losses of the higher-order modes LP

_{l}_{11e}and LP

_{11o}. Note that this technique is inherently independent of bends that occur away from the mandrels, since only relative power changes are recorded and no absolute power values (the remaining fiber sections are kept constant).

The obtained loss values are depicted in Fig. 6(a) for the LP_{01} and in Fig. 6(b) for the LP_{11e} and LP_{11o} modes. In all cases they are compared to the FEM simulations and to the losses predicted by the analytical approach (cf. section 2). Concerning the LP_{01} mode (Fig. 6(a)) the analytically obtained losses are more than one order of magnitude below the experimental ones. This difference can be attributed to the fact, that the analytical approach does not consider mode field deformation during bending. In contrast, the FEM based simulations take these mode field deformations into account and show excellent agreement with the measured losses. The slightly different slope between the FEM based losses and the measured values probably arises from the assumption of an ideal step-index profile for the simulations. A real refractive index profile typically is a smoothed version of the ideal step-index profile due to diffusion phenomena, or contains other non-uniformities, such as a central dip, which results from dopant depletion, and refractive index changes in the region immediately surrounding the core due to variations in dopant concentration.

The loss measurement of the LP_{11} modes (Fig. 6(b)) reveals a difference between the LP_{11e} and LP_{11o} modes. Typically, the losses of the LP_{11e} mode are higher than the losses of the LP_{11o} mode. The reason of this behavior is the different field orientation as explained above while discussing the evolution of relative modal power with bending shown in Fig. 5. To the best of our knowledge this is the first time, that the losses of the higher-order modes LP_{11e} and LP_{11o} were discriminated. An earlier approach for measuring the LP_{11} mode losses by an interferometric technique [15] was not able to distinguish between the index degenerated LP_{11} modes and was limited to a quasi single-mode regime. The average uncertainties of the measurements in Fig. 6 are about 15% of the respective loss values, both, for the logarithmic fitting and using Eq. (14).

Since the analytical approach given by Eq. (6) does not consider the field distribution but only the effective index it is not able to distinguish between the LP_{11}* _{e}* and the LP

_{11}

*modes but delivers only a single loss curve for both LP*

_{o}_{11}modes, which is fairly close to the measured values. In the FEM based simulation the LP

_{11}

*and LP*

_{e}_{11}

*modes are separated in the same sequence as measured. Typically, the LP*

_{o}_{11}

*has higher losses than the LP*

_{e}_{11}

*mode.*

_{o}Obviously, the simulated separation of the LP_{11}* _{e}* and LP

_{11}

*modes is larger than the measured one. In the actual case FEM simulation and measurement match very well only for the LP*

_{o}_{11}

*mode. Regarding the LP*

_{o}_{11}

*mode especially for small bending radii the measured values are constantly lower than the simulated ones. This again might be due to the fact that an idealized step-index profile is used to describe the refractive index distribution of the optical fiber.*

_{e}## 6. Summary

We presented a novel technique to measure the bending loss of individual modes in few-mode optical fibers. To the best of our knowledge, it is the first time of a technique being able to discriminate between different modes even in the presence of degenerated propagation constants, e.g., between the LP_{11}* _{e}* and LP

_{11}

*modes. The technique is based on a computer-generated hologram, which performs a correlation of the beam emerging from the fiber with the modes encoded in the hologram. In consequence, the power of every single mode within the fiber is measured as a function of the bending diameter. The resulting modal bending losses have been compared to FEM simulations by means of the conformal mapping approach, including the photoelastic effect, and to common analytical loss formulas for step-index fibers, revealing improved agreement with the FEM calculations. The achieved results are considered to be of great significance in all fields of applications where fibers need to be bent, such as the design and characterization of fibers and fiber lasers and in modern telecommunication strategies including mode multiplexed information transmission.*

_{o}## References and links

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