## Mode resolved bend loss in few-mode optical fibers |

Optics Express, Vol. 21, Issue 3, pp. 3170-3181 (2013)

http://dx.doi.org/10.1364/OE.21.003170

Acrobat PDF (1237 KB)

### 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 OSA

## 1. Introduction

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). [CrossRef]

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). [CrossRef]

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). [CrossRef] [PubMed]

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]

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]

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]

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

^{2}- and S

^{2}-imaging [16

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]

## 2. Bend modeling and simulation

### 2.1. Theoretical treatment of bent fibers

*z*, of light. When the fiber is curved this invariance no longer exists. However, by executing a proper mathematical coordinate transformation [11

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]

*x*-

*y*-plane and the fiber is curved in the

*x*-

*z*-plane the well-known coordinate transformation formula reads as [12

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

*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*).

*n*in direction

_{i}*i*,

*i*∈ {

*x*,

*y*,

*z*}, needs to be considered [20

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

*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

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]

*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

*ν*= 0.164 being the Poisson number and

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

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]

*z*-direction) by

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

*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

_{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 yielding the final transformation Note that the value of

*R*

_{eff}= 1.40

*R*used here slightly differs from

*R*

_{eff}= 1.28

*R*given in [7

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]

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

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

*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

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

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

### 2.2. Analytical loss calculation

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]

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

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

*et al.*[9

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

*et al.*[7

**43**, 899–909 (2007). [CrossRef]

*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,

*β*is the propagation constant of the guided mode in the unbent waveguide,

*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

^{®}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

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

*R*

_{core}= 3.85 μm, whose determination is outlined in section 3, and

*n*

_{clad}=

*n*

_{FusedSilica}(

*λ*= 1064nm) = 1.450 [28

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

*α*are obtained from the imaginary parts of the effective mode indices Im(

*n*

_{eff}): with the free-space wavelength

*λ*.

## 3. Fiber properties

^{™}which is single-mode at 1550 nm wavelength. Beside the fundamental mode LP

_{01}, the next higher-order modes LP

_{11}

*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*

_{e,o}*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.

_{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.

_{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: Therefore, the product

*R*

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

_{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] connects the measured Bragg wavelength

*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 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

*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.

*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

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]

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]

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]

*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

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]

**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

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

*et al.*[32

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]

_{01}, LP

_{11}

*, and LP*

_{e}_{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.

*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

**17**, 9347–9356 (2009). [CrossRef] [PubMed]

*A*

_{0}is a constant, and

*T*̃ and

*Ũ*denote the Fourier transforms of

*T*and

*U*, respectively.

*f*-setup.

*T*(

**r**) of the hologram is given by [32

**17**, 9347–9356 (2009). [CrossRef] [PubMed]

^{*}” 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

*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:

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]

### 4.2. Experimental setup

*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.

_{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

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]

_{1}to record the beam intensity directly. The magnification was chosen to be

_{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.

*et al.*[38

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

## 5. Results and discussion

*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.

_{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.

_{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.

_{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}

*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*

_{e}_{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*

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

_{1}in Fig. 4), the measurement results shown in Fig. 5 can be easily converted into a common power loss representation: 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

*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*[

_{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).

_{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.

_{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

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]

_{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).

_{11}

*and the LP*

_{e}_{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}_{11}

*and LP*

_{e}_{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

_{11}

*and LP*

_{e}_{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

1. | J. P. Koplow, D. A. V. Kliner, and L. Goldberg, “Single-mode operation of a coiled multimode fiber amplifier,” Opt. Lett. |

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. |

3. | Z. Wang, N. Zhang, and X.-C. Yuan, “High-volume optical vortex multiplexing and de-multiplexing for free-space optical communication,” Opt. Express |

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 |

7. | R. Schermer and J. Cole, “Improved bend loss formula verified for optical fiber by simulation and experiment,” IEEE J. Quant. Electron. |

8. | D. Gloge, “Bending loss in multimode fibers with graded and ungraded core index,” Appl. Opt. |

9. | D. Marcuse, “Curvature loss formula for optical fibers,” J. Opt. Soc. Am. |

10. | D. Marcuse, “Field deformation and loss caused by curvature of optical fibers,” J. Opt. Soc. Am. |

11. | D. Marcuse, “Influence of curvature on the losses of doubly clad fibers,” Appl. Opt. |

12. | M. Heiblum and J. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quant. Electron. |

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. |

14. | L. Faustini and G. Martini, “Bend loss in single-mode fibers,” J. Lightwave Technol. |

15. | N. Shibata and M. Tsubokawa, “Bending loss measurement of LP11 mode in quasi-single-mode operation region,” Electron. Lett. |

16. | D. N. Schimpf, R. A. Barankov, and S. Ramachandran, “Cross-correlated (C2) imaging of fiber and waveguide modes,” Opt. Express |

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 |

18. | J. M. Fini, “Bend-resistant design of conventional and microstructure fibers with very large mode area,” Opt. Express |

19. | J. M. Fini and S. Ramachandran, “Natural bend-distortion immunity of higher-order-mode large-mode-area fibers,” Opt. Lett. |

20. | G. W. Scherer, “Stress-induced index profile distortion in optical waveguides,” Appl. Opt. |

21. | W. Primak and D. Post, “Photoelastic constants of vitreous silica and its elastic coefficient of refractive index,” J. Appl. Phys. |

22. | R. Ulrich, S. C. Rashleigh, and W. Eickhoff, “Bending-induced birefringence in single-mode fibers,” Opt. Lett. |

23. | F. Pockels, “Über die Änderung des optischen Verhaltens verschiedener Gläser durch elastische Deformation,” Ann. Phys. |

24. | H. Renner, “Bending losses of coated single-mode fibers: a simple approach,” J. Lightwave Technol. |

25. | J. Sakai, “Simplified bending loss formula for single-mode optical fibers,” Appl. Opt. |

26. | K. Okamoto, |

27. | J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys. |

28. | J. W. Fleming, “Dispersion in GeO2–SiO2 glasses,” Appl. Opt. |

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, |

31. | R. Kashyap, |

32. | T. Kaiser, D. Flamm, S. Schröter, and M. Duparré, “Complete modal decomposition for optical fibers using CGH-based correlation filters,” Opt. Express |

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. |

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. | D. Flamm, C. Schulze, R. Brüning, O. A. Schmidt, T. Kaiser, S. Schröter, and M. Duparré, “Fast M |

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. | C. Schulze, D. Naidoo, D. Flamm, O. A. Schmidt, A. Forbes, and M. Duparré, “Wavefront reconstruction by modal decomposition,” Opt. Express |

38. | W.-H. Lee, “Binary computer-generated holograms,” Appl. Opt. |

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. |

**OCIS Codes**

(060.2270) Fiber optics and optical communications : Fiber characterization

(060.2300) Fiber optics and optical communications : Fiber measurements

(060.2310) Fiber optics and optical communications : Fiber optics

(060.2400) Fiber optics and optical communications : Fiber properties

(120.3940) Instrumentation, measurement, and metrology : Metrology

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: November 12, 2012

Revised Manuscript: January 12, 2013

Manuscript Accepted: January 14, 2013

Published: February 1, 2013

**Citation**

Christian Schulze, Adrian Lorenz, Daniel Flamm, Alexander Hartung, Siegmund Schröter, Hartmut Bartelt, and Michael Duparré, "Mode resolved bend loss in few-mode optical fibers," Opt. Express **21**, 3170-3181 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-3-3170

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### References

- 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). [CrossRef]
- 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). [CrossRef]
- Z. Wang, N. Zhang, and X.-C. Yuan, “High-volume optical vortex multiplexing and de-multiplexing for free-space optical communication,” Opt. Express19, 482–492 (2011). [CrossRef] [PubMed]
- 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]
- 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]
- R. T. Schermer, “Mode scalability in bent optical fibers,” Opt. Express15, 15674–15701 (2007). [CrossRef] [PubMed]
- 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]
- D. Gloge, “Bending loss in multimode fibers with graded and ungraded core index,” Appl. Opt.11, 2506–2513 (1972). [CrossRef] [PubMed]
- D. Marcuse, “Curvature loss formula for optical fibers,” J. Opt. Soc. Am.66, 216–220 (1976). [CrossRef]
- D. Marcuse, “Field deformation and loss caused by curvature of optical fibers,” J. Opt. Soc. Am.66, 311–320 (1976). [CrossRef]
- D. Marcuse, “Influence of curvature on the losses of doubly clad fibers,” Appl. Opt.21, 4208–4213 (1982). [CrossRef] [PubMed]
- M. Heiblum and J. Harris, “Analysis of curved optical waveguides by conformal transformation,” IEEE J. Quant. Electron.11, 75–83 (1975). [CrossRef]
- 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]
- L. Faustini and G. Martini, “Bend loss in single-mode fibers,” J. Lightwave Technol.15, 671–679 (1997). [CrossRef]
- N. Shibata and M. Tsubokawa, “Bending loss measurement of LP11 mode in quasi-single-mode operation region,” Electron. Lett.21, 1042–1043 (1985). [CrossRef]
- D. N. Schimpf, R. A. Barankov, and S. Ramachandran, “Cross-correlated (C2) imaging of fiber and waveguide modes,” Opt. Express19, 13008–13019 (2011) [CrossRef] [PubMed]
- 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. Express16, 7233–7243 (2008) [CrossRef] [PubMed]
- J. M. Fini, “Bend-resistant design of conventional and microstructure fibers with very large mode area,” Opt. Express14, 69–81 (2006). [CrossRef] [PubMed]
- 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]
- G. W. Scherer, “Stress-induced index profile distortion in optical waveguides,” Appl. Opt.19, 2000–2006 (1980). [CrossRef] [PubMed]
- 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]
- R. Ulrich, S. C. Rashleigh, and W. Eickhoff, “Bending-induced birefringence in single-mode fibers,” Opt. Lett.5, 273–275 (1980). [CrossRef] [PubMed]
- F. Pockels, “Über die Änderung des optischen Verhaltens verschiedener Gläser durch elastische Deformation,” Ann. Phys.312, 745–771 (1902). [CrossRef]
- H. Renner, “Bending losses of coated single-mode fibers: a simple approach,” J. Lightwave Technol.10, 544–551 (1992). [CrossRef]
- J. Sakai, “Simplified bending loss formula for single-mode optical fibers,” Appl. Opt.18, 951–952 (1979). [CrossRef] [PubMed]
- K. Okamoto, Fundamentals of Optical Waveguides, Second Edition (Optics and Photonics Series) (Academic Press, 2005).
- J.-P. Berenger, “A perfectly matched layer for the absorption of electromagnetic waves,” J. Comput. Phys.114, 185–200 (1994). [CrossRef]
- J. W. Fleming, “Dispersion in GeO2–SiO2 glasses,” Appl. Opt.23, 4486–4493 (1984). [CrossRef] [PubMed]
- IEC, “Optical fibres - Part 1-44: Measurement methods and test procedures - Cut-off wavelength (IEC 60793-1-44:2011),” (2012).
- A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, 1996).
- R. Kashyap, Fiber Bragg Gratings (Optics and Photonics) (Academic Press, 1999).
- T. Kaiser, D. Flamm, S. Schröter, and M. Duparré, “Complete modal decomposition for optical fibers using CGH-based correlation filters,” Opt. Express17, 9347–9356 (2009). [CrossRef] [PubMed]
- 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. 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]
- 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]
- 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]
- C. Schulze, D. Naidoo, D. Flamm, O. A. Schmidt, A. Forbes, and M. Duparré, “Wavefront reconstruction by modal decomposition,” Opt. Express20, 19714–19725 (2012). [CrossRef] [PubMed]
- W.-H. Lee, “Binary computer-generated holograms,” Appl. Opt.18, 3661–3669 (1979). [CrossRef] [PubMed]
- 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]

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