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 [35] 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 [612]. 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 C2- and S2-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]:

nequ(x,y)=nmat(x,y)exp(xR)nmat(x,y)(1+xR)
with the bend radius R, the equivalent (tilted) refractive index profile nequ(x, y) and the refractive index profile of the unbent fiber denoted by nmat(x, y).

 figure: Fig. 1

Fig. 1 Section of the refractive index profile in the radial direction for (a) a straight fiber and (b) a bent fiber using conformal mapping. The horizontal red line corresponds to the fundamental mode’s effective mode index. The blue curve illustrates its intensity distribution.

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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 Δni in direction i, i ∈ {x,y,z}, needs to be considered [20]:

Δni=nin0=B2σiB1(σj+σk)
where ni and n0 are the refractive indices of the stressed and unstressed glass, σi,j,k are the components of the stress σ̂ in the direction denoted by the indices (i, j, k), which are a cyclical permutation of (x, y, z), and the photoelastic constants B1 = 4.22 × 10−6 MPa−1 and B2 = 0.65 × 10−6 MPa−1 for fused silica [21]. The stress σ̂ results from the strain ε̂ in the bent fiber. They are connected via the stress-strain relation that in the linear region can be described by Hooke’s law σ̂ = C̃ε̂ with the elasticity tensor C̃. In case of isotropic materials its only nonzero components read as C11=E(1ν)[(1+ν)(12ν)] and C12=Eν[(1+ν)(12ν)] 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 εz=Δll=xR[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 (εx = εy ≈ 0), the components of stress can be calculated according to σx=σy=C12xR and σz=C11xR. The refractive index of the material in transverse directions including the photoelastic effect therefore reads as:

nmat(x,y)=n0+Δnx,y=n0+xR[B2C12B1(C12+C11)]
with Δnx = Δny. 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 Reff defined as
Reff=R11n[B2C12+B1(C12+C11)]1.40R
yielding the final transformation
nequ=nmat(1+xR)=n0(1+xReff)
Note that the value of Reff = 1.40 R used here slightly differs from Reff = 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, B1, B2, 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:

2α[dBm]=10ln(10)π12κ2exp[2γ3(R+Rcore)eff3β22γRcore]em(R+Rcore)eff12γ32V2Km1(γRcore)Km+1(γRcore)
where R is the bending radius, Rcore is the radius of the fiber core, ncore and nclad are the refractive indices of the fiber core and cladding, k=2πλ is the free space propagation constant, β is the propagation constant of the guided mode in the unbent waveguide, κ=(ncore2k2β2)1/2 and γ=(β2nclad2k2)12 are the normalized propagation constants in core and cladding, respectively, V=(κ2+γ2)12=2πλRcorencore2nclad2 is the normalized frequency, Km±1 is the modified Bessel function of the second kind, and em is a scalar depending on the order of the mode LPmn (em = 2 for m = 0 and em = 1 otherwise). We calculated the propagation constant β using the analytical approach outlined in [26]. The effective bending radius Reff was calculated by Eq. (4). For reasonable loss estimations using Eq. (6) the fiber parameters Rcore 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 Rcore = 3.85 μm, whose determination is outlined in section 3, and nclad = nFusedSilica(λ = 1064nm) = 1.450 [28]. The modal power losses 2α are obtained from the imaginary parts of the effective mode indices Im(neff):

2α=20ln(10)2πλIm(neff)
with the free-space wavelength 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 LP01, the next higher-order modes LP11e,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 Rcore and numerical aperture NA, as accurately as possible. Our FEM calculations show that differences in Rcore 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 LP11 modes cut-off and the effective index ratio of the LP01 and the LP11 modes, and subsequently, used the FEM mode solver to find the combination of Rcore and NA, which predicts the measured cut-off wavelength and effective index ratio best.

The cut-off wavelength of the LP11 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 λc of the HOM. Using the V-Parameter of the LP11 mode cut-off Vc = 2.405 [30], a condition connecting Rcore, NA and λc can be deduced:

λc=2πRcoreNA2.405.
Therefore, the product Rcore NA is determined without knowing the parameters separately.

 figure: Fig. 2

Fig. 2 (a) Difference of transmission spectra of the investigated fiber with and without a bend around a mandrel yielding an LP11 mode cut-off at λc = 1209 nm. (b) The reflection spectrum from the fiber Bragg grating with peaks at λLP01 = 1079.86 nm and λLP11 = 1077.96 nm. Additionally the plots include (blue) the material refractive indices ncore, nclad including dispersion [28], as well as the effective mode indices nLP01, nLP11, and (black) the Bragg condition Eq. (9).

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To identify the exact values of both parameters, an additional measurement of the effective mode index ratio of the LP01 and the LP11 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 λLPmn. The Bragg condition [31]

λLPmn=2nLPmnΛ
connects the measured Bragg wavelength λLPmn with the corresponding effective index nLPmn by the grating period Λ. From the Bragg condition for the two involved modes LP01 and LP11 the relation
λLP01λLP11=nLP01nLP11
is deduced. Knowing the Bragg wavelengths from the reflection spectrum, Eq. (10) yields a second condition for the determination of the step-index fiber parameters. By variation of Rcore and NA, considering the constraints of Eq. (8) and Eq. (10), iterative FEM simulations finally found the pair Rcore = 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 ΔnDC can be estimated by ηΔnDC/n = Δλ/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), Δλ = 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 ΔnDC = 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 [3437]. 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]:

U(r)=l=1Nclψl(r),
where r = (x, y), cl = ρlel is the complex expansion coefficient, ψl(r) is the lth mode with amplitude ρl, phase φl (with respect to a reference phase) and 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 Ilρl2 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 LP01, LP11e, and LP11o of the fiber specified in section 3. Illuminating such a hologram with the LP01 mode (Fig. 3(a)) results in a strong ILP01-signal and zero ILP11-signals (Fig. 3(b)). Similarly, the simulation of the diffraction of a LP01 mode illuminating a transmission function encoded only with the LP11e 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 LP11e mode beam (Fig. 3(d)), a strong ILP11e-signal appears, where the ILP01- and ILP11o-signals are close to zero (Fig. 3(e)). Figure 3(f) depicts the simulation of a LP11e mode beam illuminating a hologram that only encodes the LP11e mode. Hence, in contrast to the illustration in Fig. 3(c), a bright spot appears at the center of the diffraction pattern.

 figure: Fig. 3

Fig. 3 Hologram illumination with pure modes and resulting correlation signals for the LP01, LP11e, and LP11o modes (dots and arrows mark the position of the intensity signals ILP01, ILP11e, and ILP11o being proportional to the mode powers ρLP012, ρLP11e2, and ρLP11o2). (a) Measured near field of a pure LP01 mode beam. (b) Corresponding measured correlation signals. (c) Simulation of the diffraction of a LP01 mode beam illuminating a hologram encoding the LP11e mode only, and propagation through a 2f -setup. (d) Measured near field of a pure LP11e mode beam. (e) Corresponding measured correlation signals. (f) Simulation of the diffraction of a LP11e mode beam illuminating a hologram encoding the LP11e mode only, and propagation through a 2f -setup. Intensities are normalized.

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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]:

C(r)=A0d2rT˜[2πλfr]U˜[2πλf(rr)]
where A0 is a constant, and T̃ 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 2f -setup.

To measure all modal powers ρl2 simultaneously from one diffraction pattern, the transmission function T(r) of the hologram is given by [32]:

T(r)=l=1Nψl*(r)eiKlr.
where “*” denotes the complex conjugation and Kl is a spatial frequency, that is different for each mode. Using this transmission function, the signal of the correlation of the unknown field 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 |C(r=Klλf2π)|2=ρl2. However, since the used hologram disregards the polarization of the modes, it is necessary to decompose the x- and y-component of the field, Ux and Uy, 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 x- and y-direction: ρl2=ρl,x2+ρl,y2[34]. In the following, we use ”modal power” and ”modal loss” refering to the total mode power ρl2.

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 (MO1) onto the fiber input facet. The focal length of MO1 (f(MO1) = 10mm) as well as the distance between the laser and the fiber (≈ 50cm) were adjusted to achieve reasonable power coupling.

 figure: Fig. 4

Fig. 4 Experimental setup. (a) LS - laser source, MO1,2 - microscope objectives, DB -bending diameter, P - polarizer, L1,2 - lenses, BS - beam splitter, CGH - computer-generated hologram, CCD1,2 - cameras. (b) Scheme of the metal plate with half-circle grooves. (c) Scheme of the fiber coiled around a mandrel with a stable loop at the end.

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The composition of the input mode mixture was controlled by transverse and lateral movement of the MO1 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 LP11 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 CCD1 to record the beam intensity directly. The magnification was chosen to be M=f(L1)f(MO2)=375mm4mm=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 (CCD2) after a 2f -setup (f(L2) = 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 LP11 modes initially.

 figure: Fig. 5

Fig. 5 Relative modal powers as a function of bending diameter DB (dashed lines to guide the eye). The mode intensity distributions are depicted on the right for the straight and bent fiber (DB = 1.5cm, bending in x-z-plane with bending center in direction of −x). The corresponding measured beam intensities (CCD1 in Fig. 4) as a function of bending diameter are shown in Media 1.

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As can be seen, the content of the fundamental mode LP01 increases substantially from 5% to 99% of the respective total power with stronger bending, whereas the two LP11 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 LP11 modes (symmetrical two-lobe beam), but is changing continuously into a fundamental Gaussian-like beam (high LP01 mode content) towards decreasing bending diameters.

Regarding the LP11 mode power curves in Fig. 5, it is interesting to note that the two LP11 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 LP11e are aligned in the direction of the bend, causing this mode to experience higher losses and to drop faster in power compared to mode LP11o, whose intensity lobes are orientated perpendicular (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 LP11e 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).

By integrating the measured near field intensities (CCD1 in Fig. 4), the measurement results shown in Fig. 5 can be easily converted into a common power loss representation:

2αl=10Llog10(ρl2(DB=30cm)ρl2(DB)).
Here, 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 (ρl2(DBmax=30cm)ρl2(DB=)), 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 2αl is defined as a positive quantity for power loss and not for amplitude loss. Since the loss of the fundamental mode LP01 is much lower than that of the LP11 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 LP01 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 2αl [ dBm] 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 LP11e and LP11o. 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 LP01 and in Fig. 6(b) for the LP11e and LP11o 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 LP01 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.

 figure: Fig. 6

Fig. 6 Modal power loss 2α as a function of bending diameter DB for (a) the fundamental mode LP01 and (b) the higher-order modes LP11e and LP11o. (CGH) modal decomposition measurements, (FEM) rigorous loss simulations by FEM, (ana) analytically calculated loss after Eq. (6).

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The loss measurement of the LP11 modes (Fig. 6(b)) reveals a difference between the LP11e and LP11o modes. Typically, the losses of the LP11e mode are higher than the losses of the LP11o 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 LP11e and LP11o were discriminated. An earlier approach for measuring the LP11 mode losses by an interferometric technique [15] was not able to distinguish between the index degenerated LP11 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 LP11e and the LP11o modes but delivers only a single loss curve for both LP11 modes, which is fairly close to the measured values. In the FEM based simulation the LP11e and LP11o modes are separated in the same sequence as measured. Typically, the LP11e has higher losses than the LP11o mode.

Obviously, the simulated separation of the LP11e and LP11o modes is larger than the measured one. In the actual case FEM simulation and measurement match very well only for the LP11o mode. Regarding the LP11e 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.

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 LP11e and LP11o 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.

References and links

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5. S. V. Karpeev, V. S. Pavelyev, V. A. Soifer, S. N. Khonina, M. Duparré, B. Luedge, and J. Turunen, “Transverse mode multiplexing by diffractive optical elements,” Proc. SPIE 5854, Optical Technologies for Telecommunications , 1 (2005), doi: [CrossRef]  .

6. R. T. Schermer, “Mode scalability in bent optical fibers,” Opt. Express 15, 15674–15701 (2007). [CrossRef]   [PubMed]  

7. R. Schermer and J. Cole, “Improved bend loss formula verified for optical fiber by simulation and experiment,” IEEE J. Quant. Electron. 43, 899–909 (2007). [CrossRef]  

8. D. Gloge, “Bending loss in multimode fibers with graded and ungraded core index,” Appl. Opt. 11, 2506–2513 (1972). [CrossRef]   [PubMed]  

9. D. Marcuse, “Curvature loss formula for optical fibers,” J. Opt. Soc. Am. 66, 216–220 (1976). [CrossRef]  

10. D. Marcuse, “Field deformation and loss caused by curvature of optical fibers,” J. Opt. Soc. Am. 66, 311–320 (1976). [CrossRef]  

11. D. Marcuse, “Influence of curvature on the losses of doubly clad fibers,” Appl. Opt. 21, 4208–4213 (1982). [CrossRef]   [PubMed]  

12. M. Heiblum and J. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quant. Electron. 11, 75–83 (1975). [CrossRef]  

13. A. Harris and P. Castle, “Bend loss measurements on high numerical aperture single-mode fibers as a function of wavelength and bend radius,” J. Lightwave Technol. 4, 34–40 (1986). [CrossRef]  

14. L. Faustini and G. Martini, “Bend loss in single-mode fibers,” J. Lightwave Technol. 15, 671–679 (1997). [CrossRef]  

15. N. Shibata and M. Tsubokawa, “Bending loss measurement of LP11 mode in quasi-single-mode operation region,” Electron. Lett. 21, 1042–1043 (1985). [CrossRef]  

16. D. N. Schimpf, R. A. Barankov, and S. Ramachandran, “Cross-correlated (C2) imaging of fiber and waveguide modes,” Opt. Express 19, 13008–13019 (2011) [CrossRef]   [PubMed]  

17. J. W. Nicholson, A. D. Yablon, S. Ramachandran, and S. Ghalmi, “Spatially and spectrally resolved imaging of modal content in large-mode-area fibers,” Opt. Express 16, 7233–7243 (2008) [CrossRef]   [PubMed]  

18. J. M. Fini, “Bend-resistant design of conventional and microstructure fibers with very large mode area,” Opt. Express 14, 69–81 (2006). [CrossRef]   [PubMed]  

19. J. M. Fini and S. Ramachandran, “Natural bend-distortion immunity of higher-order-mode large-mode-area fibers,” Opt. Lett. 32, 748–750 (2007). [CrossRef]   [PubMed]  

20. G. W. Scherer, “Stress-induced index profile distortion in optical waveguides,” Appl. Opt. 19, 2000–2006 (1980). [CrossRef]   [PubMed]  

21. W. Primak and D. Post, “Photoelastic constants of vitreous silica and its elastic coefficient of refractive index,” J. Appl. Phys. 30, 779 –788 (1959). [CrossRef]  

22. R. Ulrich, S. C. Rashleigh, and W. Eickhoff, “Bending-induced birefringence in single-mode fibers,” Opt. Lett. 5, 273–275 (1980). [CrossRef]   [PubMed]  

23. F. Pockels, “Über die Änderung des optischen Verhaltens verschiedener Gläser durch elastische Deformation,” Ann. Phys. 312, 745–771 (1902). [CrossRef]  

24. H. Renner, “Bending losses of coated single-mode fibers: a simple approach,” J. Lightwave Technol. 10, 544–551 (1992). [CrossRef]  

25. J. Sakai, “Simplified bending loss formula for single-mode optical fibers,” Appl. Opt. 18, 951–952 (1979). [CrossRef]   [PubMed]  

26. K. Okamoto, Fundamentals of Optical Waveguides, Second Edition (Optics and Photonics Series) (Academic Press, 2005).

27. J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994). [CrossRef]  

28. J. W. Fleming, “Dispersion in GeO2–SiO2 glasses,” Appl. Opt. 23, 4486–4493 (1984). [CrossRef]   [PubMed]  

29. IEC, “Optical fibres - Part 1-44: Measurement methods and test procedures - Cut-off wavelength (IEC 60793-1-44:2011),” (2012).

30. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, 1996).

31. R. Kashyap, Fiber Bragg Gratings (Optics and Photonics) (Academic Press, 1999).

32. T. Kaiser, D. Flamm, S. Schröter, and M. Duparré, “Complete modal decomposition for optical fibers using CGH-based correlation filters,” Opt. Express 17, 9347–9356 (2009). [CrossRef]   [PubMed]  

33. M. A. Golub, A. M. Prokhorov, I. N. Sisakian, and V. A. Soifer, “Synthesis of spatial filters for investigation of the transverse mode composition of coherent radiation,” Sov. J. Quantum Electron. 9, 1866–1868 (1982).

34. D. Flamm, O. A. Schmidt, C. Schulze, J. Borchardt, T. Kaiser, S. Schröter, and M. Duparré, “Measuring the spatial polarization distribution of multimode beams emerging from passive step-index large-mode-area fibers,” Opt. Lett. 35, 3429–3431 (2010). [CrossRef]   [PubMed]  

35. D. Flamm, C. Schulze, R. Brüning, O. A. Schmidt, T. Kaiser, S. Schröter, and M. Duparré, “Fast M2 measurement for fiber beams based on modal analysis,” Appl. Opt. 51, 987–993 (2012). [CrossRef]   [PubMed]  

36. D. Flamm, D. Naidoo, C. Schulze, A. Forbes, and M. Duparré, “Mode analysis with a spatial light modulator as a correlation filter,” Opt. Lett. 37, 2478–2480 (2012). [CrossRef]   [PubMed]  

37. C. Schulze, D. Naidoo, D. Flamm, O. A. Schmidt, A. Forbes, and M. Duparré, “Wavefront reconstruction by modal decomposition,” Opt. Express 20, 19714–19725 (2012). [CrossRef]   [PubMed]  

38. W.-H. Lee, “Binary computer-generated holograms,” Appl. Opt. 18, 3661–3669 (1979). [CrossRef]   [PubMed]  

39. T. Grosjean, A. Sabac, and D. Courjon, “A versatile and stable device allowing the efficient generation of beams with radial, azimuthal or hybrid polarizations,” Opt. Commun. 252, 12–21 (2005). [CrossRef]  

References

  • View by:

  1. J. P. Koplow, D. A. V. Kliner, and L. Goldberg, “Single-mode operation of a coiled multimode fiber amplifier,” Opt. Lett. 25, 442–444 (2000).
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  2. F. D. Teodoro, J. P. Koplow, S. W. Moore, and D. A. V. Kliner, “Diffraction-limited, 300-kw peak-power pulses from a coiled multimode fiber amplifier,” Opt. Lett. 27, 518–520 (2002).
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  3. Z. Wang, N. Zhang, and X.-C. Yuan, “High-volume optical vortex multiplexing and de-multiplexing for free-space optical communication,” Opt. Express 19, 482–492 (2011).
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  4. S. V. Karpeev, V. S. Pavelyev, V. A. Soifer, L. L. Doskolovich, M. R. Duparré, and B. Luedge, “Mode multiplexing by diffractive optical elements in optical telecommunication,” Proc. SPIE 5480, Laser Optics 2003: Diode Lasers and Telecommunication Systems, 153 (2004), doi:.
    [Crossref]
  5. S. V. Karpeev, V. S. Pavelyev, V. A. Soifer, S. N. Khonina, M. Duparré, B. Luedge, and J. Turunen, “Transverse mode multiplexing by diffractive optical elements,” Proc. SPIE 5854, Optical Technologies for Telecommunications, 1 (2005), doi:.
    [Crossref]
  6. R. T. Schermer, “Mode scalability in bent optical fibers,” Opt. Express 15, 15674–15701 (2007).
    [Crossref] [PubMed]
  7. R. Schermer and J. Cole, “Improved bend loss formula verified for optical fiber by simulation and experiment,” IEEE J. Quant. Electron. 43, 899–909 (2007).
    [Crossref]
  8. D. Gloge, “Bending loss in multimode fibers with graded and ungraded core index,” Appl. Opt. 11, 2506–2513 (1972).
    [Crossref] [PubMed]
  9. D. Marcuse, “Curvature loss formula for optical fibers,” J. Opt. Soc. Am. 66, 216–220 (1976).
    [Crossref]
  10. D. Marcuse, “Field deformation and loss caused by curvature of optical fibers,” J. Opt. Soc. Am. 66, 311–320 (1976).
    [Crossref]
  11. D. Marcuse, “Influence of curvature on the losses of doubly clad fibers,” Appl. Opt. 21, 4208–4213 (1982).
    [Crossref] [PubMed]
  12. M. Heiblum and J. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quant. Electron. 11, 75–83 (1975).
    [Crossref]
  13. A. Harris and P. Castle, “Bend loss measurements on high numerical aperture single-mode fibers as a function of wavelength and bend radius,” J. Lightwave Technol. 4, 34–40 (1986).
    [Crossref]
  14. L. Faustini and G. Martini, “Bend loss in single-mode fibers,” J. Lightwave Technol. 15, 671–679 (1997).
    [Crossref]
  15. N. Shibata and M. Tsubokawa, “Bending loss measurement of LP11 mode in quasi-single-mode operation region,” Electron. Lett. 21, 1042–1043 (1985).
    [Crossref]
  16. D. N. Schimpf, R. A. Barankov, and S. Ramachandran, “Cross-correlated (C2) imaging of fiber and waveguide modes,” Opt. Express 19, 13008–13019 (2011)
    [Crossref] [PubMed]
  17. J. W. Nicholson, A. D. Yablon, S. Ramachandran, and S. Ghalmi, “Spatially and spectrally resolved imaging of modal content in large-mode-area fibers,” Opt. Express 16, 7233–7243 (2008)
    [Crossref] [PubMed]
  18. J. M. Fini, “Bend-resistant design of conventional and microstructure fibers with very large mode area,” Opt. Express 14, 69–81 (2006).
    [Crossref] [PubMed]
  19. J. M. Fini and S. Ramachandran, “Natural bend-distortion immunity of higher-order-mode large-mode-area fibers,” Opt. Lett. 32, 748–750 (2007).
    [Crossref] [PubMed]
  20. G. W. Scherer, “Stress-induced index profile distortion in optical waveguides,” Appl. Opt. 19, 2000–2006 (1980).
    [Crossref] [PubMed]
  21. W. Primak and D. Post, “Photoelastic constants of vitreous silica and its elastic coefficient of refractive index,” J. Appl. Phys. 30, 779 –788 (1959).
    [Crossref]
  22. R. Ulrich, S. C. Rashleigh, and W. Eickhoff, “Bending-induced birefringence in single-mode fibers,” Opt. Lett. 5, 273–275 (1980).
    [Crossref] [PubMed]
  23. F. Pockels, “Über die Änderung des optischen Verhaltens verschiedener Gläser durch elastische Deformation,” Ann. Phys. 312, 745–771 (1902).
    [Crossref]
  24. H. Renner, “Bending losses of coated single-mode fibers: a simple approach,” J. Lightwave Technol. 10, 544–551 (1992).
    [Crossref]
  25. J. Sakai, “Simplified bending loss formula for single-mode optical fibers,” Appl. Opt. 18, 951–952 (1979).
    [Crossref] [PubMed]
  26. K. Okamoto, Fundamentals of Optical Waveguides, Second Edition (Optics and Photonics Series) (Academic Press, 2005).
  27. J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
    [Crossref]
  28. J. W. Fleming, “Dispersion in GeO2–SiO2 glasses,” Appl. Opt. 23, 4486–4493 (1984).
    [Crossref] [PubMed]
  29. IEC, “Optical fibres - Part 1-44: Measurement methods and test procedures - Cut-off wavelength (IEC 60793-1-44:2011),” (2012).
  30. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, 1996).
  31. R. Kashyap, Fiber Bragg Gratings (Optics and Photonics) (Academic Press, 1999).
  32. T. Kaiser, D. Flamm, S. Schröter, and M. Duparré, “Complete modal decomposition for optical fibers using CGH-based correlation filters,” Opt. Express 17, 9347–9356 (2009).
    [Crossref] [PubMed]
  33. M. A. Golub, A. M. Prokhorov, I. N. Sisakian, and V. A. Soifer, “Synthesis of spatial filters for investigation of the transverse mode composition of coherent radiation,” Sov. J. Quantum Electron. 9, 1866–1868 (1982).
  34. D. Flamm, O. A. Schmidt, C. Schulze, J. Borchardt, T. Kaiser, S. Schröter, and M. Duparré, “Measuring the spatial polarization distribution of multimode beams emerging from passive step-index large-mode-area fibers,” Opt. Lett. 35, 3429–3431 (2010).
    [Crossref] [PubMed]
  35. D. Flamm, C. Schulze, R. Brüning, O. A. Schmidt, T. Kaiser, S. Schröter, and M. Duparré, “Fast M2 measurement for fiber beams based on modal analysis,” Appl. Opt. 51, 987–993 (2012).
    [Crossref] [PubMed]
  36. D. Flamm, D. Naidoo, C. Schulze, A. Forbes, and M. Duparré, “Mode analysis with a spatial light modulator as a correlation filter,” Opt. Lett. 37, 2478–2480 (2012).
    [Crossref] [PubMed]
  37. C. Schulze, D. Naidoo, D. Flamm, O. A. Schmidt, A. Forbes, and M. Duparré, “Wavefront reconstruction by modal decomposition,” Opt. Express 20, 19714–19725 (2012).
    [Crossref] [PubMed]
  38. W.-H. Lee, “Binary computer-generated holograms,” Appl. Opt. 18, 3661–3669 (1979).
    [Crossref] [PubMed]
  39. T. Grosjean, A. Sabac, and D. Courjon, “A versatile and stable device allowing the efficient generation of beams with radial, azimuthal or hybrid polarizations,” Opt. Commun. 252, 12–21 (2005).
    [Crossref]

2012 (3)

2011 (2)

2010 (1)

2009 (1)

2008 (1)

2007 (3)

2006 (1)

2005 (2)

S. V. Karpeev, V. S. Pavelyev, V. A. Soifer, S. N. Khonina, M. Duparré, B. Luedge, and J. Turunen, “Transverse mode multiplexing by diffractive optical elements,” Proc. SPIE 5854, Optical Technologies for Telecommunications, 1 (2005), doi:.
[Crossref]

T. Grosjean, A. Sabac, and D. Courjon, “A versatile and stable device allowing the efficient generation of beams with radial, azimuthal or hybrid polarizations,” Opt. Commun. 252, 12–21 (2005).
[Crossref]

2002 (1)

2000 (1)

1997 (1)

L. Faustini and G. Martini, “Bend loss in single-mode fibers,” J. Lightwave Technol. 15, 671–679 (1997).
[Crossref]

1994 (1)

J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[Crossref]

1992 (1)

H. Renner, “Bending losses of coated single-mode fibers: a simple approach,” J. Lightwave Technol. 10, 544–551 (1992).
[Crossref]

1986 (1)

A. Harris and P. Castle, “Bend loss measurements on high numerical aperture single-mode fibers as a function of wavelength and bend radius,” J. Lightwave Technol. 4, 34–40 (1986).
[Crossref]

1985 (1)

N. Shibata and M. Tsubokawa, “Bending loss measurement of LP11 mode in quasi-single-mode operation region,” Electron. Lett. 21, 1042–1043 (1985).
[Crossref]

1984 (1)

1982 (2)

M. A. Golub, A. M. Prokhorov, I. N. Sisakian, and V. A. Soifer, “Synthesis of spatial filters for investigation of the transverse mode composition of coherent radiation,” Sov. J. Quantum Electron. 9, 1866–1868 (1982).

D. Marcuse, “Influence of curvature on the losses of doubly clad fibers,” Appl. Opt. 21, 4208–4213 (1982).
[Crossref] [PubMed]

1980 (2)

1979 (2)

1976 (2)

1975 (1)

M. Heiblum and J. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quant. Electron. 11, 75–83 (1975).
[Crossref]

1972 (1)

1959 (1)

W. Primak and D. Post, “Photoelastic constants of vitreous silica and its elastic coefficient of refractive index,” J. Appl. Phys. 30, 779 –788 (1959).
[Crossref]

1902 (1)

F. Pockels, “Über die Änderung des optischen Verhaltens verschiedener Gläser durch elastische Deformation,” Ann. Phys. 312, 745–771 (1902).
[Crossref]

Barankov, R. A.

Berenger, J.-P.

J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. 114, 185–200 (1994).
[Crossref]

Borchardt, J.

Brüning, R.

Castle, P.

A. Harris and P. Castle, “Bend loss measurements on high numerical aperture single-mode fibers as a function of wavelength and bend radius,” J. Lightwave Technol. 4, 34–40 (1986).
[Crossref]

Cole, J.

R. Schermer and J. Cole, “Improved bend loss formula verified for optical fiber by simulation and experiment,” IEEE J. Quant. Electron. 43, 899–909 (2007).
[Crossref]

Courjon, D.

T. Grosjean, A. Sabac, and D. Courjon, “A versatile and stable device allowing the efficient generation of beams with radial, azimuthal or hybrid polarizations,” Opt. Commun. 252, 12–21 (2005).
[Crossref]

Doskolovich, L. L.

S. V. Karpeev, V. S. Pavelyev, V. A. Soifer, L. L. Doskolovich, M. R. Duparré, and B. Luedge, “Mode multiplexing by diffractive optical elements in optical telecommunication,” Proc. SPIE 5480, Laser Optics 2003: Diode Lasers and Telecommunication Systems, 153 (2004), doi:.
[Crossref]

Duparré, M.

Duparré, M. R.

S. V. Karpeev, V. S. Pavelyev, V. A. Soifer, L. L. Doskolovich, M. R. Duparré, and B. Luedge, “Mode multiplexing by diffractive optical elements in optical telecommunication,” Proc. SPIE 5480, Laser Optics 2003: Diode Lasers and Telecommunication Systems, 153 (2004), doi:.
[Crossref]

Eickhoff, W.

Faustini, L.

L. Faustini and G. Martini, “Bend loss in single-mode fibers,” J. Lightwave Technol. 15, 671–679 (1997).
[Crossref]

Fini, J. M.

Flamm, D.

Fleming, J. W.

Forbes, A.

Ghalmi, S.

Gloge, D.

Goldberg, L.

Golub, M. A.

M. A. Golub, A. M. Prokhorov, I. N. Sisakian, and V. A. Soifer, “Synthesis of spatial filters for investigation of the transverse mode composition of coherent radiation,” Sov. J. Quantum Electron. 9, 1866–1868 (1982).

Grosjean, T.

T. Grosjean, A. Sabac, and D. Courjon, “A versatile and stable device allowing the efficient generation of beams with radial, azimuthal or hybrid polarizations,” Opt. Commun. 252, 12–21 (2005).
[Crossref]

Harris, A.

A. Harris and P. Castle, “Bend loss measurements on high numerical aperture single-mode fibers as a function of wavelength and bend radius,” J. Lightwave Technol. 4, 34–40 (1986).
[Crossref]

Harris, J.

M. Heiblum and J. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quant. Electron. 11, 75–83 (1975).
[Crossref]

Heiblum, M.

M. Heiblum and J. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quant. Electron. 11, 75–83 (1975).
[Crossref]

Kaiser, T.

Karpeev, S. V.

S. V. Karpeev, V. S. Pavelyev, V. A. Soifer, S. N. Khonina, M. Duparré, B. Luedge, and J. Turunen, “Transverse mode multiplexing by diffractive optical elements,” Proc. SPIE 5854, Optical Technologies for Telecommunications, 1 (2005), doi:.
[Crossref]

S. V. Karpeev, V. S. Pavelyev, V. A. Soifer, L. L. Doskolovich, M. R. Duparré, and B. Luedge, “Mode multiplexing by diffractive optical elements in optical telecommunication,” Proc. SPIE 5480, Laser Optics 2003: Diode Lasers and Telecommunication Systems, 153 (2004), doi:.
[Crossref]

Kashyap, R.

R. Kashyap, Fiber Bragg Gratings (Optics and Photonics) (Academic Press, 1999).

Khonina, S. N.

S. V. Karpeev, V. S. Pavelyev, V. A. Soifer, S. N. Khonina, M. Duparré, B. Luedge, and J. Turunen, “Transverse mode multiplexing by diffractive optical elements,” Proc. SPIE 5854, Optical Technologies for Telecommunications, 1 (2005), doi:.
[Crossref]

Kliner, D. A. V.

Koplow, J. P.

Lee, W.-H.

Love, J. D.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, 1996).

Luedge, B.

S. V. Karpeev, V. S. Pavelyev, V. A. Soifer, S. N. Khonina, M. Duparré, B. Luedge, and J. Turunen, “Transverse mode multiplexing by diffractive optical elements,” Proc. SPIE 5854, Optical Technologies for Telecommunications, 1 (2005), doi:.
[Crossref]

S. V. Karpeev, V. S. Pavelyev, V. A. Soifer, L. L. Doskolovich, M. R. Duparré, and B. Luedge, “Mode multiplexing by diffractive optical elements in optical telecommunication,” Proc. SPIE 5480, Laser Optics 2003: Diode Lasers and Telecommunication Systems, 153 (2004), doi:.
[Crossref]

Marcuse, D.

Martini, G.

L. Faustini and G. Martini, “Bend loss in single-mode fibers,” J. Lightwave Technol. 15, 671–679 (1997).
[Crossref]

Moore, S. W.

Naidoo, D.

Nicholson, J. W.

Okamoto, K.

K. Okamoto, Fundamentals of Optical Waveguides, Second Edition (Optics and Photonics Series) (Academic Press, 2005).

Pavelyev, V. S.

S. V. Karpeev, V. S. Pavelyev, V. A. Soifer, S. N. Khonina, M. Duparré, B. Luedge, and J. Turunen, “Transverse mode multiplexing by diffractive optical elements,” Proc. SPIE 5854, Optical Technologies for Telecommunications, 1 (2005), doi:.
[Crossref]

S. V. Karpeev, V. S. Pavelyev, V. A. Soifer, L. L. Doskolovich, M. R. Duparré, and B. Luedge, “Mode multiplexing by diffractive optical elements in optical telecommunication,” Proc. SPIE 5480, Laser Optics 2003: Diode Lasers and Telecommunication Systems, 153 (2004), doi:.
[Crossref]

Pockels, F.

F. Pockels, “Über die Änderung des optischen Verhaltens verschiedener Gläser durch elastische Deformation,” Ann. Phys. 312, 745–771 (1902).
[Crossref]

Post, D.

W. Primak and D. Post, “Photoelastic constants of vitreous silica and its elastic coefficient of refractive index,” J. Appl. Phys. 30, 779 –788 (1959).
[Crossref]

Primak, W.

W. Primak and D. Post, “Photoelastic constants of vitreous silica and its elastic coefficient of refractive index,” J. Appl. Phys. 30, 779 –788 (1959).
[Crossref]

Prokhorov, A. M.

M. A. Golub, A. M. Prokhorov, I. N. Sisakian, and V. A. Soifer, “Synthesis of spatial filters for investigation of the transverse mode composition of coherent radiation,” Sov. J. Quantum Electron. 9, 1866–1868 (1982).

Ramachandran, S.

Rashleigh, S. C.

Renner, H.

H. Renner, “Bending losses of coated single-mode fibers: a simple approach,” J. Lightwave Technol. 10, 544–551 (1992).
[Crossref]

Sabac, A.

T. Grosjean, A. Sabac, and D. Courjon, “A versatile and stable device allowing the efficient generation of beams with radial, azimuthal or hybrid polarizations,” Opt. Commun. 252, 12–21 (2005).
[Crossref]

Sakai, J.

Scherer, G. W.

Schermer, R.

R. Schermer and J. Cole, “Improved bend loss formula verified for optical fiber by simulation and experiment,” IEEE J. Quant. Electron. 43, 899–909 (2007).
[Crossref]

Schermer, R. T.

Schimpf, D. N.

Schmidt, O. A.

Schröter, S.

Schulze, C.

Shibata, N.

N. Shibata and M. Tsubokawa, “Bending loss measurement of LP11 mode in quasi-single-mode operation region,” Electron. Lett. 21, 1042–1043 (1985).
[Crossref]

Sisakian, I. N.

M. A. Golub, A. M. Prokhorov, I. N. Sisakian, and V. A. Soifer, “Synthesis of spatial filters for investigation of the transverse mode composition of coherent radiation,” Sov. J. Quantum Electron. 9, 1866–1868 (1982).

Snyder, A. W.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, 1996).

Soifer, V. A.

S. V. Karpeev, V. S. Pavelyev, V. A. Soifer, S. N. Khonina, M. Duparré, B. Luedge, and J. Turunen, “Transverse mode multiplexing by diffractive optical elements,” Proc. SPIE 5854, Optical Technologies for Telecommunications, 1 (2005), doi:.
[Crossref]

M. A. Golub, A. M. Prokhorov, I. N. Sisakian, and V. A. Soifer, “Synthesis of spatial filters for investigation of the transverse mode composition of coherent radiation,” Sov. J. Quantum Electron. 9, 1866–1868 (1982).

S. V. Karpeev, V. S. Pavelyev, V. A. Soifer, L. L. Doskolovich, M. R. Duparré, and B. Luedge, “Mode multiplexing by diffractive optical elements in optical telecommunication,” Proc. SPIE 5480, Laser Optics 2003: Diode Lasers and Telecommunication Systems, 153 (2004), doi:.
[Crossref]

Teodoro, F. D.

Tsubokawa, M.

N. Shibata and M. Tsubokawa, “Bending loss measurement of LP11 mode in quasi-single-mode operation region,” Electron. Lett. 21, 1042–1043 (1985).
[Crossref]

Turunen, J.

S. V. Karpeev, V. S. Pavelyev, V. A. Soifer, S. N. Khonina, M. Duparré, B. Luedge, and J. Turunen, “Transverse mode multiplexing by diffractive optical elements,” Proc. SPIE 5854, Optical Technologies for Telecommunications, 1 (2005), doi:.
[Crossref]

Ulrich, R.

Wang, Z.

Yablon, A. D.

Yuan, X.-C.

Zhang, N.

Ann. Phys. (1)

F. Pockels, “Über die Änderung des optischen Verhaltens verschiedener Gläser durch elastische Deformation,” Ann. Phys. 312, 745–771 (1902).
[Crossref]

Appl. Opt. (7)

Electron. Lett. (1)

N. Shibata and M. Tsubokawa, “Bending loss measurement of LP11 mode in quasi-single-mode operation region,” Electron. Lett. 21, 1042–1043 (1985).
[Crossref]

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Supplementary Material (1)

Media 1: MPEG (174 KB)     

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Figures (6)

Fig. 1
Fig. 1 Section of the refractive index profile in the radial direction for (a) a straight fiber and (b) a bent fiber using conformal mapping. The horizontal red line corresponds to the fundamental mode’s effective mode index. The blue curve illustrates its intensity distribution.
Fig. 2
Fig. 2 (a) Difference of transmission spectra of the investigated fiber with and without a bend around a mandrel yielding an LP11 mode cut-off at λc = 1209 nm. (b) The reflection spectrum from the fiber Bragg grating with peaks at λLP01 = 1079.86 nm and λLP11 = 1077.96 nm. Additionally the plots include (blue) the material refractive indices ncore, nclad including dispersion [28], as well as the effective mode indices nLP01, nLP11, and (black) the Bragg condition Eq. (9).
Fig. 3
Fig. 3 Hologram illumination with pure modes and resulting correlation signals for the LP01, LP11e, and LP11o modes (dots and arrows mark the position of the intensity signals ILP01, ILP11e, and ILP11o being proportional to the mode powers ρ LP 01 2, ρ LP 11 e 2, and ρ LP 11 o 2). (a) Measured near field of a pure LP01 mode beam. (b) Corresponding measured correlation signals. (c) Simulation of the diffraction of a LP01 mode beam illuminating a hologram encoding the LP11e mode only, and propagation through a 2f -setup. (d) Measured near field of a pure LP11e mode beam. (e) Corresponding measured correlation signals. (f) Simulation of the diffraction of a LP11e mode beam illuminating a hologram encoding the LP11e mode only, and propagation through a 2f -setup. Intensities are normalized.
Fig. 4
Fig. 4 Experimental setup. (a) LS - laser source, MO1,2 - microscope objectives, DB -bending diameter, P - polarizer, L1,2 - lenses, BS - beam splitter, CGH - computer-generated hologram, CCD1,2 - cameras. (b) Scheme of the metal plate with half-circle grooves. (c) Scheme of the fiber coiled around a mandrel with a stable loop at the end.
Fig. 5
Fig. 5 Relative modal powers as a function of bending diameter DB (dashed lines to guide the eye). The mode intensity distributions are depicted on the right for the straight and bent fiber (DB = 1.5cm, bending in x-z-plane with bending center in direction of −x). The corresponding measured beam intensities (CCD1 in Fig. 4) as a function of bending diameter are shown in Media 1.
Fig. 6
Fig. 6 Modal power loss 2α as a function of bending diameter DB for (a) the fundamental mode LP01 and (b) the higher-order modes LP11e and LP11o. (CGH) modal decomposition measurements, (FEM) rigorous loss simulations by FEM, (ana) analytically calculated loss after Eq. (6).

Equations (14)

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n equ ( x , y ) = n mat ( x , y ) exp ( x R ) n mat ( x , y ) ( 1 + x R )
Δ n i = n i n 0 = B 2 σ i B 1 ( σ j + σ k )
n mat ( x , y ) = n 0 + Δ n x , y = n 0 + x R [ B 2 C 12 B 1 ( C 12 + C 11 ) ]
R eff = R 1 1 n [ B 2 C 12 + B 1 ( C 12 + C 11 ) ] 1.40 R
n equ = n mat ( 1 + x R ) = n 0 ( 1 + x R eff )
2 α [ dB m ] = 10 ln ( 10 ) π 1 2 κ 2 exp [ 2 γ 3 ( R + R core ) eff 3 β 2 2 γ R core ] e m ( R + R core ) eff 1 2 γ 3 2 V 2 K m 1 ( γ R core ) K m + 1 ( γ R core )
2 α = 20 ln ( 10 ) 2 π λ Im ( n eff )
λ c = 2 π R core NA 2.405 .
λ LPmn = 2 n LPmn Λ
λ LP 01 λ LP1 1 = n LP 01 n LP 11
U ( r ) = l = 1 N c l ψ l ( r ) ,
C ( r ) = A 0 d 2 r T ˜ [ 2 π λ f r ] U ˜ [ 2 π λ f ( r r ) ]
T ( r ) = l = 1 N ψ l * ( r ) e i K l r .
2 α l = 10 L log 10 ( ρ l 2 ( D B = 30 cm ) ρ l 2 ( D B ) ) .

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