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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 3 — Feb. 11, 2013
  • pp: 3170–3181
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Mode resolved bend loss in few-mode optical fibers

Christian Schulze, Adrian Lorenz, Daniel Flamm, Alexander Hartung, Siegmund Schröter, Hartmut Bartelt, and Michael Duparré  »View Author Affiliations


Optics Express, Vol. 21, Issue 3, pp. 3170-3181 (2013)
http://dx.doi.org/10.1364/OE.21.003170


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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

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

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

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]

], and is of enormous significance for modern telecommunication, especially regarding mode multiplexing techniques [3

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

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] .

] to enhance the capacity of today’s information transmission.

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

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

, 12

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

], 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

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

]:
nequ(x,y)=nmat(x,y)exp(xR)nmat(x,y)(1+xR)
(1)
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).

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.

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

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

]:
Δni=nin0=B2σiB1(σj+σk)
(2)
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

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]

]. 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

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]

].

The dominant elongation component is given in the direction of the fiber axis (z-direction) by εz=Δll=xR[22

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

]. 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)]
(3)
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
(4)
yielding the final transformation
nequ=nmat(1+xR)=n0(1+xReff)
(5)
Note that the value of Reff = 1.40 R used here slightly differs from Reff = 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]

]. This is caused by the different theoretical approach, which we used in starting from the strain defined by Morey [20

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

] as the origin of the photoelastic refractive index change instead of starting from the stress defined by Pockels [23

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

]. 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

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

,24

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

]. 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

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

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

, 14

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

, 24

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

, 25

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

]. According to Marcuse et al.[9

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

] and further developed by Schermer et al.[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]

], 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)
(6)
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

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

]. 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

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

], 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.

3. Fiber properties

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

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

]. 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

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

], a condition connecting Rcore, NA and λc can be deduced:
λc=2πRcoreNA2.405.
(8)
Therefore, the product Rcore NA is determined without knowing the parameters separately.

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).

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

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

]
λLPmn=2nLPmnΛ
(9)
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
(10)
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.

4. Measurement of modal bend loss

4.1. Modal decomposition

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

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]

], 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.

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.

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

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]

]:
C(r)=A0d2rT˜[2πλfr]U˜[2πλf(rr)]
(12)
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.

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.

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.

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

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

], 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

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.

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.

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

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

]) and accordingly, the highest loss when the surrounding cladding index exceeds the mode’s effective index (cf. Fig. 1).

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.

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).

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

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. 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]

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]

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

  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. Express19, 482–492 (2011). [CrossRef] [PubMed]
  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. Express15, 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. Express19, 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. Express16, 7233–7243 (2008) [CrossRef] [PubMed]
  18. J. M. Fini, “Bend-resistant design of conventional and microstructure fibers with very large mode area,” Opt. Express14, 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. Express17, 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. Express20, 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]

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