Macrobending Loss

Detailed macrobending loss calculation using the curvature radiation model for single-mode fiber.

Bend Radius (mm)Wavelength (nm)Core Radius (µm)Core NACore Index n₁Number of TurnsElasto-optic correction (R_eff = R/(1 − p_e) = 1.255 R)

Loss per turn

4.72e-3 dB

0.0501 dB/m of bent fibre

Total loss (1 turn)

4.72e-3 dB

V-number

2.071

LP11 cutoff λc = 1335 nm

Mode-field diameter

10.08 µm

Marcuse 1977 Gaussian fit

Cladding index n₂

1.4440

n₂ = √(n₁² − NA²)

Loss per Turn vs Bend Radius

Loss per Turn vs Wavelength (R = 15 mm)

Model

α = √π κ² exp(−2γ³R / 3β²) / (2 γ^(3/2) V² √R K₁²(W)) [1/m, power]

κ = U/a, γ = W/a, β² = (n₁k)² − κ², U J₁(U)/J₀(U) = W K₁(W)/K₀(W)

Marcuse's curvature-loss formula for the LP01 mode of a step-index fibre (J. Opt. Soc. Am. 66, 216, 1976), with the exact LP01 eigenvalues. Loss per turn = 4.343 α · 2πR dB.

Textbook approximation. The elasto-optic correction is on by default: bending strains the glass, so silica acts as if R were larger (R/(1 − p_e) ≈ 1.25 R with the Bertholds–Dändliker photo-elastic constants; Schermer & Cole 2007 fit 1.28 R to measured SMF-28 loss). The loss falls by about 18× at the defaults; untick it for Marcuse's formula as published. The model leaves out reflections from the cladding and coating (ripples in R and λ), and the transition loss where each bend starts and ends. ITU-T G.657 (08/2024) specifies bend loss at R = 5, 7.5, 10 and 15 mm.

Model and references

Textbook approximation. A standard approximation from the literature (for example paraxial, plane-wave or Gaussian-beam), valid within the range the page states. About the tiers

Marcuse's asymptotic curvature loss (R much larger than a) for the LP01 mode of a step-index, weakly guiding fibre, with the elasto-optic correction R_eff = R/(1 − p_e) ≈ 1.25R on by default. Not modelled: coating reflections, transition loss.

Marcuse D. (1976). Curvature loss formula for optical fibers. J. Opt. Soc. Am. 66, 216. Curvature loss formula for an LP_ν mode, used here for LP01. doi:10.1364/josa.66.000216
Snyder A. W., Love J. D. (1983). Optical Waveguide Theory, bend-loss chapter. Chapman and Hall. Normalized power attenuation coefficient of the fundamental mode of a step profile.
Gloge D. (1971). Weakly Guiding Fibers. Appl. Opt. 10, 2252. Eigenvalue equation of the LP01 mode (U and W). doi:10.1364/ao.10.002252
Marcuse D. (1977). Loss Analysis of Single-Mode Fiber Splices. Bell Syst. Tech. J. 56, 703–718. Gaussian fit of the mode-field radius. doi:10.1002/j.1538-7305.1977.tb00534.x
Nagano K., Kawakami S., Nishida S. (1978). Change of the refractive index in an optical fiber due to external forces. Appl. Opt. 17, 2080. Elasto-optic equivalent bend radius. doi:10.1364/ao.17.002080
Schermer R. T., Cole J. H. (2007). Improved bend loss formula verified for optical fiber by simulation and experiment. IEEE J. Quantum Electron. 43, 899. R_eff ≈ 1.28R for silica. doi:10.1109/jqe.2007.903364

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Macrobending Loss

Detailed macrobending loss calculation using the curvature radiation model for single-mode fiber.

Bend Radius (mm)Wavelength (nm)Core Radius (µm)Core NACore Index n₁Number of TurnsElasto-optic correction (R_eff = R/(1 − p_e) = 1.255 R)

Loss per turn

4.72e-3 dB

0.0501 dB/m of bent fibre

Total loss (1 turn)

4.72e-3 dB

V-number

2.071

LP11 cutoff λc = 1335 nm

Mode-field diameter

10.08 µm

Marcuse 1977 Gaussian fit

Cladding index n₂

1.4440

n₂ = √(n₁² − NA²)

Loss per Turn vs Bend Radius

Loss per Turn vs Wavelength (R = 15 mm)

Model

α = √π κ² exp(−2γ³R / 3β²) / (2 γ^(3/2) V² √R K₁²(W)) [1/m, power]

κ = U/a, γ = W/a, β² = (n₁k)² − κ², U J₁(U)/J₀(U) = W K₁(W)/K₀(W)

Marcuse's curvature-loss formula for the LP01 mode of a step-index fibre (J. Opt. Soc. Am. 66, 216, 1976), with the exact LP01 eigenvalues. Loss per turn = 4.343 α · 2πR dB.

Model and references

Textbook approximation. A standard approximation from the literature (for example paraxial, plane-wave or Gaussian-beam), valid within the range the page states. About the tiers

Marcuse D. (1976). Curvature loss formula for optical fibers. J. Opt. Soc. Am. 66, 216. Curvature loss formula for an LP_ν mode, used here for LP01. doi:10.1364/josa.66.000216
Snyder A. W., Love J. D. (1983). Optical Waveguide Theory, bend-loss chapter. Chapman and Hall. Normalized power attenuation coefficient of the fundamental mode of a step profile.
Gloge D. (1971). Weakly Guiding Fibers. Appl. Opt. 10, 2252. Eigenvalue equation of the LP01 mode (U and W). doi:10.1364/ao.10.002252
Marcuse D. (1977). Loss Analysis of Single-Mode Fiber Splices. Bell Syst. Tech. J. 56, 703–718. Gaussian fit of the mode-field radius. doi:10.1002/j.1538-7305.1977.tb00534.x
Nagano K., Kawakami S., Nishida S. (1978). Change of the refractive index in an optical fiber due to external forces. Appl. Opt. 17, 2080. Elasto-optic equivalent bend radius. doi:10.1364/ao.17.002080
Schermer R. T., Cole J. H. (2007). Improved bend loss formula verified for optical fiber by simulation and experiment. IEEE J. Quantum Electron. 43, 899. R_eff ≈ 1.28R for silica. doi:10.1109/jqe.2007.903364

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