Abstract

This paper introduces a simple, analytical method for generalizing the behavior of bent, weakly-guided fibers and waveguides. It begins with a comprehensive study of the modes of the bent step-index fiber, which is later extended to encompass a wide range of more complicated waveguide geometries. The analysis is based on the introduction of a scaling parameter, analogous to the V-number for straight step-index fibers, for the bend radius. When this parameter remains constant, waveguides of different bend radii, numerical apertures and wavelengths will all propagate identical mode field distributions, except scaled in size. This allows the behavior of individual waveguides to be broadly extended, and is especially useful for generalizing the results of numerical simulations. The technique is applied to the bent step-index fiber in this paper to arrive at simple analytical formulae for the propagation constant and mode area, which are valid well beyond the transition to whispering-gallery modes. Animations illustrating mode deformation with respect to bending and curves describing polarization decoupling are also presented, which encompass the entire family of weakly-guided, step-index fibers.

© 2007 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |

  1. G. P. Agrawal, Fiber Optic Communication Systems, 2nd Edition. (Wiley, New York, 1997).
  2. C. H. Cox, Analog Optical Links, Theory and Practice, (Cambridge, 2004).
    [CrossRef]
  3. E. Udd, Fiber Optic Sensors: An Introduction for Engineers and Scientists, (Wiley, New York, 1991).
  4. F. T. S. Yu and S. Yin, Fiber Optic Sensors, (CRC, 2002).
    [CrossRef]
  5. J.-G. Werthen and M. Cohen, "The Power of Light: Photonic Power Innovations in Medical, Energy and Wireless Applications," Photonics Spectra 40, 68-72 (2006).
  6. M. J. F. Digonnet, Rare-Earth-Doped Fiber Lasers and Amplifiers, 2nd Edition, (Stanford, New York, 2001).
    [CrossRef]
  7. M. Marhic, Fiber Optical Parametric Amplifiers, Oscillators and Related Devices, (Cambridge, New York, 2007).
  8. S. J. Garth, "Birefringence in Bent Single-Mode Fibers," J. Lightwave Technol. 6, 445-449 (1988).
    [CrossRef]
  9. H.-G. Unger, Planar Optical Waveguides and Fibres (Oxford, 1977), Chap. 6.
  10. D. Marcuse, "Field Deformation and Loss Caused by Curvature of Optical Fibers," J. Opt. Soc. Am. 66, 311-320 (1976).
    [CrossRef]
  11. S. J. Garth, "Mode Behaviour on Bent Planar Dielectric Waveguides," IEE Proc.: Optoelectron. 142, 115-120 (1995).
    [CrossRef]
  12. T. Sørensen,  et al, "Spectral Macro-Bending Loss Considerations for Photonic Crystal Fibers," IEE Proc.: Optoelectron. 149, 206-210 (2002).
    [CrossRef]
  13. J. M. Fini, Bend-resistant design of conventional and microstructure fibers with very large mode area," Opt. Express 14, 69-81 (2006).
    [CrossRef] [PubMed]
  14. R. T. Schermer and J. H. Cole, "Improved bend loss formula verified for Optical Fiber by simulation and experiment," IEEE. J. Quantum Electron. 43, 899-909 (2007).
    [CrossRef]
  15. R. Scarmozzino et al, "Numerical techniques for modeling guided-wave photonic devices," IEEE J. Sel. Topics Quantum Electron. 6, 150-162 (2000).
    [CrossRef]
  16. M. Heiblum and J. H. Harris, "Analysis of Curved Optical Waveguides by Conformal Transformation," IEEE J. Quantum Electron. QE-11, 75-83 (1975).
    [CrossRef]
  17. D. Gloge, "Weakly Guiding Fibers," Appl. Opt. 10, 2252-2258 (1971).
    [CrossRef] [PubMed]
  18. A. Melloni,  et al, "Determination of Bend Mode Characteristics in Dielectric Waveguides," J. Lightw. Tech. 19, 571-577 (2001).
    [CrossRef]
  19. J. M. Fini, "Bend-Compensated Design of Large Mode Area Fibers," Opt. Lett. 31, 1963-1965 (2006).
    [CrossRef] [PubMed]
  20. 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]
  21. J. M. Fini, "Intuitive Modeling of Bend Distortion in Large Mode Area Fibers," Opt. Lett. 32, 1632-1634 (2007).
    [CrossRef] [PubMed]
  22. K. Nagano, S. Kawakami and S. Nishida, "Change of the Refractive Index in an Optical Fiber Due to External Forces," Appl. Opt. 17, 2080-2085 (1978).
    [CrossRef] [PubMed]
  23. D. Marcuse, Light Transmission Optics, 2nd Edition, (Van Nostrand, New York, 1982).
  24. R. T. Schermer is preparing a manuscript to be titled "Bend Loss in Weakly-Guided Fibers."
  25. A. W. Snyder and W. R. Young, "Modes of Optical Waveguides," J. Opt. Soc. Am. 68, 297-309 (1978).
    [CrossRef]
  26. J. R. Wait, "Electromagnetic Whispering Gallery Modes in a Dielectric Rod," Radio Science 2, 1005-1017 (1967).
  27. D. Marcuse, "Influence of Curvature on the Losses of Doubly-Clad Fibers," Appl. Opt. 21, 4208-4213 (1982).
    [CrossRef] [PubMed]
  28. 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]
  29. R. L. Farrow,  et al, "Peak-power limits on fiber amplifiers imposed by self-focusing," Opt. Lett. 23, 3423-3425 (2006).
    [CrossRef]
  30. M. E. Fermann, "Single-mode excitation of multimode fibers with ultrashort pulses," Opt. Lett. 23, 52-54 (1998).
    [CrossRef]
  31. S. Ramachandran,  et al, "Scaling to Ultra-Large-Aeff using Higher-Order Mode Fibers," in Proceedings of the 2006 Conference on Lasers and Electro-Optics, pp. CThAA2.
  32. M. Hotolenanu,  et al, "High order modes suppression in large mode area active fibers by controlling the radial distribution of the rare earth dopant," Proc. SPIE 6102, 61021T (2006).
    [CrossRef]
  33. H. L Offerhaus,  et al, "High-Energy Single-Transverse-Mode Q-Switched Fiber Laser based on Multimode Large Mode Area Erbium-Doped Fiber," Opt. Lett. 23, 1683 (1998).
    [CrossRef]
  34. U. Griebner,  et al, "Efficient Laser Operation with nearly diffraction-limited output from a diode-pumped heavily Nd-doped multimode fiber," Opt. Lett. 21, 266-268 (1996).
    [CrossRef] [PubMed]
  35. C. C. Renaud,  et al, "Compact High-Energy Q-Switched Cladding-Pumped Fiber Laser with a Tuning Range Over 40 nm," IEEE Photon. Technol. Lett. 11, 976-978 (1999).
    [CrossRef]
  36. R. Ulrich, S. C. Rashleigh and W. Eickhoff, "Bending-induced birefringence in single-mode fibers," Opt. Lett. 5, 273-275 (1980).
    [CrossRef] [PubMed]

2007

2006

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

J. M. Fini, "Bend-Compensated Design of Large Mode Area Fibers," Opt. Lett. 31, 1963-1965 (2006).
[CrossRef] [PubMed]

J.-G. Werthen and M. Cohen, "The Power of Light: Photonic Power Innovations in Medical, Energy and Wireless Applications," Photonics Spectra 40, 68-72 (2006).

R. L. Farrow,  et al, "Peak-power limits on fiber amplifiers imposed by self-focusing," Opt. Lett. 23, 3423-3425 (2006).
[CrossRef]

M. Hotolenanu,  et al, "High order modes suppression in large mode area active fibers by controlling the radial distribution of the rare earth dopant," Proc. SPIE 6102, 61021T (2006).
[CrossRef]

2002

T. Sørensen,  et al, "Spectral Macro-Bending Loss Considerations for Photonic Crystal Fibers," IEE Proc.: Optoelectron. 149, 206-210 (2002).
[CrossRef]

2001

A. Melloni,  et al, "Determination of Bend Mode Characteristics in Dielectric Waveguides," J. Lightw. Tech. 19, 571-577 (2001).
[CrossRef]

2000

R. Scarmozzino et al, "Numerical techniques for modeling guided-wave photonic devices," IEEE J. Sel. Topics Quantum Electron. 6, 150-162 (2000).
[CrossRef]

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]

1999

C. C. Renaud,  et al, "Compact High-Energy Q-Switched Cladding-Pumped Fiber Laser with a Tuning Range Over 40 nm," IEEE Photon. Technol. Lett. 11, 976-978 (1999).
[CrossRef]

1998

1996

1995

S. J. Garth, "Mode Behaviour on Bent Planar Dielectric Waveguides," IEE Proc.: Optoelectron. 142, 115-120 (1995).
[CrossRef]

1988

S. J. Garth, "Birefringence in Bent Single-Mode Fibers," J. Lightwave Technol. 6, 445-449 (1988).
[CrossRef]

1982

1980

1978

1976

1975

M. Heiblum and J. H. Harris, "Analysis of Curved Optical Waveguides by Conformal Transformation," IEEE J. Quantum Electron. QE-11, 75-83 (1975).
[CrossRef]

1971

1967

J. R. Wait, "Electromagnetic Whispering Gallery Modes in a Dielectric Rod," Radio Science 2, 1005-1017 (1967).

Appl. Opt.

IEE Proc.: Optoelectron.

S. J. Garth, "Mode Behaviour on Bent Planar Dielectric Waveguides," IEE Proc.: Optoelectron. 142, 115-120 (1995).
[CrossRef]

T. Sørensen,  et al, "Spectral Macro-Bending Loss Considerations for Photonic Crystal Fibers," IEE Proc.: Optoelectron. 149, 206-210 (2002).
[CrossRef]

IEEE J. Quantum Electron.

M. Heiblum and J. H. Harris, "Analysis of Curved Optical Waveguides by Conformal Transformation," IEEE J. Quantum Electron. QE-11, 75-83 (1975).
[CrossRef]

IEEE J. Sel. Topics Quantum Electron.

R. Scarmozzino et al, "Numerical techniques for modeling guided-wave photonic devices," IEEE J. Sel. Topics Quantum Electron. 6, 150-162 (2000).
[CrossRef]

IEEE Photon. Technol. Lett.

C. C. Renaud,  et al, "Compact High-Energy Q-Switched Cladding-Pumped Fiber Laser with a Tuning Range Over 40 nm," IEEE Photon. Technol. Lett. 11, 976-978 (1999).
[CrossRef]

IEEE. J. Quantum Electron.

R. T. Schermer and J. H. Cole, "Improved bend loss formula verified for Optical Fiber by simulation and experiment," IEEE. J. Quantum Electron. 43, 899-909 (2007).
[CrossRef]

J. Lightw. Tech.

A. Melloni,  et al, "Determination of Bend Mode Characteristics in Dielectric Waveguides," J. Lightw. Tech. 19, 571-577 (2001).
[CrossRef]

J. Lightwave Technol.

S. J. Garth, "Birefringence in Bent Single-Mode Fibers," J. Lightwave Technol. 6, 445-449 (1988).
[CrossRef]

J. Opt. Soc. Am.

Opt. Express

Opt. Lett.

Photonics Spectra

J.-G. Werthen and M. Cohen, "The Power of Light: Photonic Power Innovations in Medical, Energy and Wireless Applications," Photonics Spectra 40, 68-72 (2006).

Proc. SPIE

M. Hotolenanu,  et al, "High order modes suppression in large mode area active fibers by controlling the radial distribution of the rare earth dopant," Proc. SPIE 6102, 61021T (2006).
[CrossRef]

Radio Science

J. R. Wait, "Electromagnetic Whispering Gallery Modes in a Dielectric Rod," Radio Science 2, 1005-1017 (1967).

Other

S. Ramachandran,  et al, "Scaling to Ultra-Large-Aeff using Higher-Order Mode Fibers," in Proceedings of the 2006 Conference on Lasers and Electro-Optics, pp. CThAA2.

D. Marcuse, Light Transmission Optics, 2nd Edition, (Van Nostrand, New York, 1982).

R. T. Schermer is preparing a manuscript to be titled "Bend Loss in Weakly-Guided Fibers."

M. J. F. Digonnet, Rare-Earth-Doped Fiber Lasers and Amplifiers, 2nd Edition, (Stanford, New York, 2001).
[CrossRef]

M. Marhic, Fiber Optical Parametric Amplifiers, Oscillators and Related Devices, (Cambridge, New York, 2007).

H.-G. Unger, Planar Optical Waveguides and Fibres (Oxford, 1977), Chap. 6.

G. P. Agrawal, Fiber Optic Communication Systems, 2nd Edition. (Wiley, New York, 1997).

C. H. Cox, Analog Optical Links, Theory and Practice, (Cambridge, 2004).
[CrossRef]

E. Udd, Fiber Optic Sensors: An Introduction for Engineers and Scientists, (Wiley, New York, 1991).

F. T. S. Yu and S. Yin, Fiber Optic Sensors, (CRC, 2002).
[CrossRef]

Supplementary Material (2)

» Media 1: MOV (1940 KB)     
» Media 2: MOV (2360 KB)     

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (19)

Fig 1.
Fig 1.

Schematic diagram of the bent fiber (a), showing the bend radius R, core diameter 2a, and the cylindrical coordinates (ρ, ϕ, y). Also shown in (b) is the equivalent, straight fiber obtained by conformal mapping to the coordinate system (x, z, y) as indicated. Refractive index profiles are as indicated in Fig. 2. From [14].

Fig. 2.
Fig. 2.

Bent fiber refractive index profiles, corresponding to the two coordinate systems in Fig. 1. The fiber’s physical refractive index is shown in (a), neglecting stress. In this case, neff decreases with distance from the center of curvature, in order to maintain a mode with constant angular velocity. The index profile of the equivalent, straight fiber is also shown in (b), tilted with respect to (a) as a result of the coordinate transformation. From [14].

Fig. 3.
Fig. 3.

Normalized angular propagation constants of bent step-index fibers with the same Vnumber, 7.375, but different core sizes and numerical apertures. When plotted versus Reff in (a), each curve was distinct. However, when plotted versus in (b), the curves overlapped.

Fig 4.
Fig 4.

Mode field distributions for two different fibers, each with the same V=7.375 and =35.69. In each case the mode field distributions were identical, other than being scaled in size. The circular outline marks the core-cladding interface. The center of curvature was to the left of the figure. Subscripts “e” and “o” are added to the usual mode notation to differentiate between mode orientations which are even and odd, respectively, in the vertical direction normal to the plane of the bend. Simulated regions were much larger than shown.

Fig. 5.
Fig. 5.

Variation of the lowest-order fiber modes with bending, for V=7.375. Circular outlines mark the core-cladding interface, and the center of curvature was to the left of each plot. The subscripts “e” and “o” added to the names of the various modes denote whether each mode was even or odd, respectively, in the (vertical) direction normal to the plane of the bend. As the modes reach cutoff they disappear from the figure. (1940 kb). [Media 1]

Fig. 6.
Fig. 6.

Variation in the mode fields with bending, for V=29.5. The initial stages of mode deformation were similar to those shown in Fig. 5, although at larger values of ℜ. Each mode transitioned to a whispering gallery mode upon adequate bending, filling only a small fraction of the fiber core. (2360 kb). [Media 2]

Fig 7.
Fig 7.

Mode field distributions for fibers with different and V, but the same value (1-bs )/V=1. When this quantity was constant, the mode fields were similar, other than in their level of confinement to the core. Confinement improved with increasing V-number, as is typical of straight fibers. The value (1-bs )/V=1 corresponds to =trans /2, where trans is defined in Section 3.2.

Fig. 8.
Fig. 8.

Refractive index distribution and corresponding fundamental mode field profile |U| for a bent fiber. Shown versus x, through the center of the fiber (y=0). The mode fields are guided (oscillatory) where n>neff , and evanescent (decaying) where n<neff . With sufficient bending, the width of the guided region, and thus the mode, is reduced.

Fig. 9.
Fig. 9.

Effective mode area Abent of the fundamental (LP01) mode of various bent fibers. Mode area is normalized to that of the straight fiber Astraight for comparison. For all >trans the mode areas were essentially the same as those of the straight fiber. For <trans , they decreased steadily, following a path which was independent of fiber V-number. This variation in mode area followed the same trend as the variation in area of the guided region, indicated by the dashed line. Each curve was truncated where simulated radiation loss became excessive (over 10-4 dB/λ for typical fiber NA).

Fig. 10.
Fig. 10.

Effective mode areas of the lowest-order bent fiber modes, for V=29.5. Mode areas were normalized to those of the straight fiber for comparison. For all >2ℜtrans the effective areas were essentially the same as those of the straight fiber. For <trans , they decreased steadily, following a similar trend. Ripples in the curves for higher-order modes were related to significant reorientation of the mode fields in the whispering gallery region.

Fig. 11.
Fig. 11.

Relationship between core area and the whispering gallery transition bend radius, trans , in step-index fiber. For each particular mode, the bend radius where the whispering gallery transition occurs must increase as Acore 3/2 . However, with increasing mode order, trans is reduced significantly. The fundamental minimum value of trans for a given core area is marked by the dashed line, which corresponds bs =½, and thus btrans =0. Along this boundary, trans increases as Acore 1/2 . The shaded region marks where bend loss becomes prohibitive, for typical fiber NA (the dotted line indicates where loss was of the order 10-7 dB/λ0).

Fig. 12.
Fig. 12.

Variation in propagation constant β with bending, for a variety of modes and Vnumbers. Where ℜ>ℜtrans, the propagation constant increased roughly to second order in R-1. For the LP11e mode, the change was significantly less than other simulated modes. By normalizing the vertical axis by (k0NA)2(1-bs)/bs, the curves for similar modes were made to overlap rather well, regardless of V-number. The curves all converge in the whispering gallery region.

Fig. 13.
Fig. 13.

(a). Simulated data from the whispering gallery region of Fig. 12, plotted along different axes. The dashed line marks where the horizontal and vertical axes are equal, and matched the data well for all mode and fibers simulated. (b) Variation in the normalized angular propagation constant with bending, for a variety of modes and V-numbers. All data corresponds to the perturbation region, >trans . The dashed line indicates where the horizontal and vertical axes are equal, and matched the data extremely well for all fibers and modes simulated.

Fig. 14.
Fig. 14.

Refractive index profiles of similar bent fibers, neglecting stress, assuming cylindrical symmetry. Also shown is the variation in the modal effective index in the ρ-direction when bent to the same bend radius. All modes indicated are confined to the region from -a to a, and reach the whispering gallery transition at the same bend radius as shown. Variation in neff from fiber to fiber due to differences in the refractive index profiles have been omitted for the purpose of illustration.

Fig. 15.
Fig. 15.

Non-degeneracy of different orientations, LPmne and LPmno, of various LPmn modes in the bent step-index fiber. These mode pairs are degenerate in the straight fiber (bLPo =bLPe ). In almost all cases bLPo was greater than bLPe , with the exception the LP21 mode at large /trans . The vertical axis was normalized by (1-bs ) to provide curves which were relatively independent of V-number.

Fig. 16.
Fig. 16.

Comparison between the change in b caused by induced birefringence, Δ(by -bx ), and that resulting from mode orientation, |bLPo -bLPe |. The vertical axis was normalized by (1-bs )(afiber /a)2 NA 2 to provide relatively universal curves.

Fig. 17.
Fig. 17.

Relative power in the minor polarization for various modes of the bent step-index fiber, for the case (|bLPo -bLPe |≫|Δ(by -bx )|), and NA/ncore =0.0656. The curves are also valid for other numerical apertures after scaling the power in the minor polarization by (NA/ncore )4.. With decreasing bend radius, power in the minor polarization drops off rapidly. The modes are essentially linearly polarized by the whispering gallery transition. In the whispering gallery region, the modes remain linearly polarized, with the exception of a particular bend radius where the LP21e and LP21o modes become degenerate. Normalizing the vertical axis by V -2 also provided curves which were relatively independent of V-number.

Fig. 18.
Fig. 18.

Relative power in the minor polarization when induced birefringence dominates, for various modes of the bent step-index fiber. Behavior is similar to that in Fig. 17, except that polarization decoupling begins at larger values of /trans with increased mode order. The modes are essentially linearly polarized both approaching and in the whispering gallery region.

Fig. 19.
Fig. 19.

Variation in the normalized angular propagation constant with V-number, for various modes of the straight step-index fiber. When small, (1-bs ) varies approximately as V -2.

Tables (1)

Tables Icon

Table 1 Appendix G: Table of symbols

Equations (53)

Equations on this page are rendered with MathJax. Learn more.

V = a k o NA ,
NA = n core 2 n clad 2 ,
b s = n eff ( s ) 2 n clad 2 n core 2 n clad 2 ,
R eff k clad ( NA n clad ) 3 ,
R eff ( silica ) 1.27 R .
β = ( 2 π n eff λ 0 ) ρ = R ,
b = n eff ( real ) 2 n clad 2 n core 2 n clad 2 ρ = R + a .
b = ( Re ( β ) k o ) 2 ( R R + a ) 2 n clad 2 n core 2 n clad 2 ,
n bent 2 n straight 2 ( 1 + 2 x R eff ) ,
W effx = R eff N A 2 ( 1 b ) 2 n core 2 ,
trans ( 4 V 1 b ) ( n core n clad ) 2 4 V 1 b .
b trans 2 b s 1 ,
trans ( 2 V 1 b s ) .
1 1 b s J m + 1 ( V 1 b s ) J m + 1 ( V 1 b s ) + 1 b s K m + 1 ( V b s ) K m + 1 ( V b s ) = m + 1 V b s ( 1 b s )
A core = π a 2 = V 2 λ 0 2 4 π N A 2 ,
R eff ( trans ) = ( 2 a 1 b s ) ( n core NA ) 2 .
Re ( β ) 2 β s 2 + ( k 0 NA ) 2 ( 1 b s b s ) [ ( trans 1 ) ( trans 1 ) 2 3 ] ( < trans )
b b s 2 V ( > trans )
b b s ( 1 b s b s ) [ 1 + ( trans 1 ) 2 3 ( trans ) ( 1 b s ) ] . ( < trans )
U T ( r T ' ) = U T ( r T ) M ,
r T r T M .
n straight 2 ( r T , M ) = n ref 2 ( M ) + g 1 ( r T ) k 0 2 M 2 .
R eff ( M ) k o 2 M 3 n ref 2 ( M ) ,
n eff 2 ( M ) = n ref 2 ( M ) + c 1 ( k 0 M ) 2 ,
R eff M [ n straight 2 ( r T ) n eff 2 ( M ) ] = g 2 ( r T ) ,
( 2 + μ ε ω 2 ) E = [ 1 ε ( ε ) E ] 1 μ ( μ ) × ( × E ) .
E ( r ) = E o U ( r T ) exp ( j β z ) ,
( T 2 + μ ε ω 2 β 2 ) U T = T [ 1 ε ( T ε ) U T ] 1 μ ( T μ ) × ( T × U T )
( T 2 + μ ε ω 2 β 2 ) U z = j β [ 1 ε ( T ε ) U T ] 1 μ ( T μ ) [ ( T U z ) + β U T ]
( T 2 + μ ε ω 2 β 2 ) U i 0 ,
U ( r T ' ) U ( r T )
r T ' r T M ,
[ μ ( r T ' ) ε ( r T ' ) ω 2 β 2 ] = 1 M 2 [ μ ( r T ) ε ( r T ) ω 2 β 2 ] .
U T ( r T ' ) = U T ( r T ) M
U z ( r T ' ) = β ' U z ( r T ) β M .
( k o M ) 2 n 2 ( r T ' , M ) n eff 2 ( M ) = g 3 ( r T ' )
n 2 ( r T ' , M ) = n ref 2 ( M ) + g 1 ( r T ' ) ( k 0 M ) 2
n eff 2 ( M ) = n ref 2 ( M ) + c 1 ( k 0 M ) 2
n mapped 2 ( r T ) = n 2 ( r T ) exp ( 2 x R ) .
n mapped 2 ( r T ' , M ) [ n ref 2 ( M ) + g 1 ( r T ' ) ( k 0 M ) 2 ] [ 1 + ( 2 Mx R ) ] ,
n ref 2 ( M ) g 1 ( r T ' ) k 0 2 M 2 ,
R ( M ) k o 2 M 3 n ref 2 ( M ) ,
a k clad ( n clad NA ) .
R eff k clad ( n clad NA ) 3 ,
Δ ( β y β x ) = 1 4 k 0 n core 3 ( p 12 p 11 ) ( 1 + ν ) ( a fiber R ) 2 ,
Δ ( b y b x ) 0.035 ( trans ) 2 ( a fiber a ) 2 ( 1 b s ) 2 NA 2 ,
U H c e U LPmne + c o U LPmno ,
U 2 dx dy = 1 .
P min or P total 1 1 + ( Λ + Λ 2 + 1 ) 2 ,
Λ = ( n core NA ) 2 V 2 ( b LPix b LPjy ) 2 a 2 Core ( T U LPi ) ( U LPj n ̂ ) ds ( i = e , o ; j = o , e ) .
A guided A core = [ 1 2 1 π ( 1 W effx a ) 2 W effx a ( W effx a ) 2 ]
× [ 1 2 1 π sin 1 ( 2 W effx a ( W effx a ) 2 ) ] ,
W effx a 1 + trans 1 b s [ ( 1 trans ) ( trans ) 1 3 ( 1 trans ) 2 3 ]

Metrics