## Field-flattened, ring-like propagation modes |

Optics Express, Vol. 21, Issue 10, pp. 12683-12698 (2013)

http://dx.doi.org/10.1364/OE.21.012683

Acrobat PDF (1266 KB)

### Abstract

We present a method for designing optical fibers that support field-flattened, ring-like higher order modes, and show that the effective and group indices of its modes can be tuned by adjusting the widths of the guide’s field-flattened layers or the average index of certain groups of layers. The approach provides a path to fibers that have simultaneously large mode areas and large separations between the propagation constants of their modes.

© 2013 OSA

## 1. Introduction

1. J. M. Fini and S. Ramachandran, “Natural bend-distortion immunity of higher-order-mode large-mode-area fibers,” Opt. Lett. **32**(7), 748–750 (2007). [CrossRef] [PubMed]

2. S. Ramachandran, J. M. Fini, M. Mermelstein, J. W. Nicholson, S. Ghalmi, and M. F. Yan, “Ultra-large effective-area, higher-order mode fibers: a new strategy for high-power lasers,” Laser Photonics Rev. **2**(6), 429–448 (2008). [CrossRef]

3. R. H. Stolen and C. Lin, “Self-phase-modulation in silica optical fibers,” Phys. Rev. A **17**(4), 1448–1453 (1978). [CrossRef]

4. A. K. Ghatak, I. C. Goyal, and R. Jindal, “Design of a waveguide refractive index profile to obtain a flat modal field,” Proc. SPIE **3666**, 40–44 (1999). [CrossRef]

8. C. Zhao, Z. Tang, Y. Ye, L. Shen, and D. Fan, “Design guidelines and characteristics for a kind of four-layer large flattened mode fibers,” Optik **119**(15), 749–754 (2008). [CrossRef]

9. N. N. Elkin, A. P. Napartovich, V. N. Troshchieva, and D. V. Vysotsky, “Modeling of large flattened mode area fiber lasers,” Laser Phys. **20**(2), 304–310 (2010). [CrossRef]

## 2. Definitions

### 2.1 Scaled quantities

*NA*, is defined as:where

_{flat}*n*is the refractive index of the cladding and

_{clad}*n*is the index of the layer or layers over which the field is to be flattened. The scaled radial coordinate, v, is defined as:where

_{flat}*λ*is the wavelength of the guided light and

*r*is the radial coordinate. The scaled refractive index profile,

*η*(v), is defined as:

*n*is usually chosen to be the minimum refractive index that can be well controlled. For silica fibers, the flattened layer might be lightly doped with an index-raising dopant such as germanium or doped with a rare-earth element along with index-raising and lowering dopants. Alternatively,

_{flat}*n*might be pure silica and the cladding might be lightly doped with an index depressing agent such as fluorine; in this case, the dopant only needs to extend to the penetration depth of the desired mode.

_{flat}*η*〉, is defined as:where

*η*and

_{i}*A*represent the scaled index and cross-sectional area of the

_{i}*i*

^{th}layer of the group. In the layer groups defined below, we sometimes constrain this value; 〈

*η*〉 sometimes alters the number of allowed modes or the guide’s intermodal spacings.

*A*, is given by Eq. (57):The scaled field is defined such that the physical field,

_{eff}*ψ*, is given from Eq. (50):where

*P*

_{0}is the power carried by the mode.

*η*is assumed to range between ± 10, which is achievable for germanium and fluorine-doped silica provided

*NA*is on the order of 0.06. In silica, other dopants might extend this range moderately, or in phosphate glasses or holey structures, various dopants or air holes can extend this range significantly. Moreover, in holey fibers

_{flat}*NA*might be controlled to a much smaller value, which would proportionally extend the range of

_{flat}*η*. A larger range of indices is generally advantageous, as it reduces the portion of the guide devoted to the stitching and matching groups described below.

### 2.2 Flattened layers

*n*=

_{flat}*n*) and the azimuthal order,

_{eff}*l*, must be equal to zero. Furthermore, it is necessary that a flattened layer be joined to appropriate stitching or termination groups, as defined below.

### 2.3 Stitching groups

*η*is 1 (from Eq. (3) since

_{flat}*n*(v) =

*n*), the minimum and maximum values of

_{flat}*η*are assumed to fall between ± 10, and the left edge of each group starts at v

_{0}= 0.5

*π*, an arbitrarily chosen value. The thicknesses of the layers that comprise the groups were determined numerically from Bessel solutions to the wave equation, as outlined in Appendix I.

#### 2.3.1 Half-wave stitching

_{0}, the ratio of the magnitude of the fields approaches unity and:where Δv is the scaled thickness of the layer,

*η*is the layer’s scaled index, the numeral one arises from the assumption that the layer is surrounded by field-flattened layers having

*η*= 1, and

*m*is an odd integer. This is also the condition for single layer, half-wave stitching in a one-dimensional slab waveguide (in that case, independent of v

_{0}).

*η*〉 = 1 for the group (see Eq. (4)) and to make

*ψ*= −1 and

*ψ*´ = 0 on the group’s right edge.

#### 2.3.2 Full-wave stitching

_{0}is increased an equation similar to Eq. (8) holds, but whose right-hand side is proportional to an even multiple of

*π*.

*ψ*= 1 and

*ψ*´ = 0 on the group’s right edge.

*ψ*

^{2}drops by a factor of two) within the third layer; we also require that, for the group, 〈

*η*〉 = 1 (see Eq. (4)). The thicknesses of the second portion of the third layer and of the remaining two layers are determined in the same fashion, but now with the constraint that

*ψ*= 1 and

*ψ*´ = 0 on the group’s right edge.

#### 2.3.3 Fractional-wave stitching

10. D. Jain, C. Baskiotis, and J. K. Sahu, “Mode area scaling with multi-trench rod-type fibers,” Opt. Express **21**(2), 1448–1455 (2013). [CrossRef] [PubMed]

### 2.4 Termination groups

_{01}mode and is at the cusp of supporting the LP

_{02}mode, that is, v = 1.23

*π.*It can be shown that its fundamental mode has a scaled effective area of 37.5; therefore, if the guide’s design operates at

*λ*= 1 μm and its core has a numerical aperture of 0.06, its effective area will be 260 μm

^{2}. It can also be shown that this mode has a scaled peak field of 0.219 = 1/√20.8. If the fiber carries 1 kW of power its peak field will be 2.61 W

^{1/2}/μm and its peak irradiance will be (2.61 W

^{1/2}/μm)

^{2}= 6.8 W/μm

^{2}. Note that the peak irradiance is 1.8 times higher than the simple ratio of the power to the effective area (37.5 ÷ 20.8). For flattened modes, this ratio is closer to unity, typically 1.15.

*η*is 1 (from Eq. (3) since

_{flat}*n*(v) =

*n*) and the minimum and maximum values of

_{flat}*η*are limited to ± 10. The thickness of the flattened layer is chosen so that each guide is on the cusp of allowing one axially-symmetric mode beyond the flattened mode. The thicknesses of the layers that comprise the groups were determined numerically from Bessel solutions to the wave equation, applying the constraints listed for each example.

*n*=

_{eff}*n*), the mode’s decay constant in the cladding is fixed and consequently the field in the cladding can only be reduced by reducing the field at the cladding interface – the purpose of the additional layers in Fig. 4(b) and Fig. 4(c).

_{flat}8. C. Zhao, Z. Tang, Y. Ye, L. Shen, and D. Fan, “Design guidelines and characteristics for a kind of four-layer large flattened mode fibers,” Optik **119**(15), 749–754 (2008). [CrossRef]

*η*〉 = 0.7), and to match the field’s slope at the cladding interface. Roughly 7% of the mode’s power is guided in the cladding.

*η*〉 = 0.7. The field at the cladding interface is −3% of the field in the flattened region, and 0.04% of the mode’s power is guided in the cladding, though now a significant power-fraction is guided by the termination group.

## 3. Example waveguides

*i*,

*v*and

*ix*), two three-layer half-wave stitching groups similar to those illustrated in Fig. 1(c) (

*ii-iv*and

*vi-viii*), and a two-layer termination group similar to the one in Fig. 4(b) (

*x*-

*xi*). Surrounding these layers is the cladding having

*η*= 0.

*η*〉 = 3.0, and the termination group has 〈

*η*〉 = 0.7. In Design B the flattened layers have equal widths, both stitching groups have 〈

*η*〉 = 2.4, and the termination group has 〈

*η*〉 = 0.7.

_{03}modes of Designs A and B to the LP

_{03}mode of a few-mode step index design, Design C. Design C is similar to the high-order mode fibers reported by others [2

2. S. Ramachandran, J. M. Fini, M. Mermelstein, J. W. Nicholson, S. Ghalmi, and M. F. Yan, “Ultra-large effective-area, higher-order mode fibers: a new strategy for high-power lasers,” Laser Photonics Rev. **2**(6), 429–448 (2008). [CrossRef]

_{03}and LP

_{13}modes of the three designs; when bent, the LP

_{03}’s will morph toward their respective LP

_{13}’s. Note that the power is more compactly packed in the flattened modes than in the step-index mode.

_{13}modes of the flattened designs have essentially the same diameter as the inner rings of their corresponding LP

_{03}modes. The inner ring of the LP

_{13}mode for the step-index design, though, has a substantially larger diameter than its corresponding LP

_{03}mode. This suggests the latter’s mode will experience a larger shift in its centroid when that fiber is bent. The design of the high-order mode fiber in [2

2. S. Ramachandran, J. M. Fini, M. Mermelstein, J. W. Nicholson, S. Ghalmi, and M. F. Yan, “Ultra-large effective-area, higher-order mode fibers: a new strategy for high-power lasers,” Laser Photonics Rev. **2**(6), 429–448 (2008). [CrossRef]

*(the phase index-area spacing – essentially the radiance), defined by Eq. (59) in Appendix IV, for the modes of the three designs. The size-spacing products are an invariant of a design. Larger values are likely preferable, since they imply that larger-sized modes may be fabricated while keeping the intermodal spacing constant, and thus keeping the likelihood of intermodal coupling constant. Bear in mind that the effective area term in the Θ*

_{eff}*equation is the same for all of a design’s modes; for each design here, it is chosen to be the area of the design’s LP*

_{eff}_{03}mode.

*’s for the three highest-order symmetric modes, the LP*

_{eff}_{02}, LP

_{03}(flattened mode) and LP

_{04}(on the cusp of existence), have been made equal by choosing an appropriate thickness for the flattened layers and by choosing an appropriate value of 〈

*η*〉 (Eq. (3)) for each design’s stitching groups.

_{12}and LP

_{22}modes of B (red arrow in Fig. 7(b)).

*(group index-area spacing), defined by Eq. (62) in Appendix IV, for the modes of the three designs. The size-spacing products are an invariant of a design. Larger values are likely preferable, since they imply that larger-sized modes may be fabricated while the keeping the intermodal spacing constant, and thus keeping the likelihood of intermodal coupling constant. Keep in mind that the effective area term in Θ*

_{g}*equation is the same for all of a design’s modes; for each design, it is chosen to be the area of the design’s LP*

_{g}_{03}mode.

_{12}mode of B (red arrow in Fig. 8(b)), and that in A and C the flattened mode is the slowest axially-symmetric mode, while in B it is the fastest of all modes.

## 4. Discussion

## Appendix I: Bessel solutions

*ψ*represents the field of a guided mode,

*l*is the azimuthal order,

*n*(

*r*) is the index at radial coordinate

*r*,

*n*is the effective index (propagation constant) of the mode, and

_{eff}*λ*is the vacuum wavelength of the guided light. In the discussion that follows, we assume that the radial index profile varies in discreet steps, or layers.

*n*is the refractive index of the layer or layers in which the field will ultimately be flattened (in the method prescribed in this paper,

_{flat}*n*is chosen before the waveguide is designed). In these terms, the wave equation becomes:

_{flat}*ψ*and

*ζ*; we begin by determining analytic solutions for the field in layers whose index is greater than, less than, and equal to the propagation constant. Each analytic solution has two unknown constants, which can be determined by the boundary conditions.

*η*>

*η*(

_{eff}*n*>

*n*), the solution to the wave equation is:where

_{eff}*J*and

_{l}*Y*are oscillatory Bessel functions,

_{l}*A*and

*B*are unknown constants, and:

*ψ*and

*ζ*are known at some position

*v*

_{1}, such as at one of the layer’s boundaries, then

*A*and

*B*can be expressed:

*A*and

*B*were determined with the help of the identity [12]:Note that the derivatives of the Bessel functions can calculated exactly from the identities: In layers where

*η*<

*η*(

_{eff}*n*<

*n*) the solution to the wave equation is:where

_{eff}*I*and

_{l}*K*are exponentially growing and decaying modified Bessel functions and

_{l}*A*and

*B*are unknown constants. If

*ψ*and

*ζ*are known at some position

*v*

_{1}, such as at one of the layer’s boundaries, then

*A*and

*B*can be expressed: In determining

*A*and

*B*we used the identity:Note that the derivatives of the Bessel functions can be calculated exactly from the identities: In layers where

*η*=

*η*(

_{eff}*n*=

*n*) the wave equation reduces to:For

_{eff}*l*≠ 0 the solution is:and the constants

*A*and

*B*become: For

*l*= 0 the solution is:and the constants

*A*and

*B*become: Note that in Eq. (32), the field can be made independent of position by forcing the constant

*B*to zero (from Eq. (34), this is equivalent to making the field’s slope zero); thus a necessary condition is that

*n*=

*n*. Comparing Eq. (29) and Eq. (32) we see that the field can only be flattened if, in addition to

_{eff}*n*=

*n*, the azimuthal order,

_{eff}*l*, is also zero.

## Appendix II: Transfer matrices

*A*and

*B*can be substituted into the original expressions for

*ψ*and the corresponding expressions for

*ζ*to obtain transfer matrices,

**M**, that relate

*ψ*and

*ζ*at position

*v*

_{2}to their known values at position

*v*

_{1}:In all cases, the matrices can be written in the form:where

*x*

_{1}is the quantity

*x*, defined by Eq. (16), evaluated at position v

_{1}and index

*η*

_{12}(the index between v

_{1}and v

_{2}), and

*x*

_{2}is

*x*evaluated at v

_{2}and index

*η*

_{12}.

*η*>

*η*(

_{eff}*n*>

*n*):In layers where

_{eff}*η*<

*η*:In layers where

_{eff}*η*=

*η*and

_{eff}*l*≠ 0:In layers where

*η*=

*η*and

_{eff}*l*= 0:The transfer matrix solution to the wave equation for a step-like fiber then becomes:where the quantity Ω is defined as:and Ω

*is (from Eq. (39)):where*

_{clad}*x*is the term

_{clad}*x*, as defined by Eq. (16), evaluated at position v

*and index*

_{clad}*η*= 0. Note that the Bessel derivates can be calculated from Eq. (27). Ω

_{clad}*is similarly calculated from Eq. (35), Eq. (36), Eq. (37), or Eq. (38) at the core’s boundary.*

_{core}**M**is the product of the matrices that represent the layers between the core and cladding; it takes advantage of the fact that

*ψ*and

*ζ*are continuous across layer boundaries. For a given waveguide, the propagation constant

*η*is determined iteratively – that is, by varying its value until the transfer matrix solution is satisfied.

_{eff}## Appendix III: Mode normalization

*const*) term of Eq. (46) to make the power carried by a mode equal to some preselected value,

*P*

_{0}:Define

*ψ*such that:Then normalization reduces to setting:

_{scaled}*η*≠

*η*(

_{eff}*n*≠

*n*):For

_{eff}*η*=

*η*and

_{eff}*l*= 0:For

*η*=

*η*and

_{eff}*l*= 1:And finally, for

*η*=

*η*and

_{eff}*l*≥ 2:

*ψ*are continuous across interfaces, this reduces to

## Appendix IV: Size-spacing products

*, is also fixed:The right-most term is found through substitution; note that though it was derived from scaling arguments, it consists only of quantities that can be directly measured, and that since Θ*

_{eff}*is fixed, if a mode’s size is increased, its effective index necessarily approaches the cladding index. Since this holds for all modes, it follows that as a desired mode’s size is increased, the effective indices of all modes necessarily approach each other.*

_{eff}*n*, is also important. Using an integral form of the group index [13] it can be shown that:and following arguments similar to those that led to Θ

_{g}*, it can be shown that the following quantity is also fixed for each mode of a waveguide:where*

_{eff}*n*is the product of a mode’s phase and group indices. Like Θ

_{eff}n_{g}*, this is a strict invariant of a design (within the strictures of the weak-guiding approximation), but unfortunately the separations between the Θ*

_{eff,}*’s are not obvious indicators of the separations between the group indices. The following term is more transparent:where the right hand side has been found by substitution. Since Θ*

_{eff,g}*depends on*

_{g}*A*it is not a true invariant of the guide. However, if the mode’s area is sufficiently large (usually the case for high power laser applications) then the third term can be neglected and since Θ

_{eff}*and Θ*

_{eff}_{eff,}

*are true invariants, then Θ*

_{g}*is approximately invariant.*

_{g}## Acknowledgments

## References and links

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

2. | S. Ramachandran, J. M. Fini, M. Mermelstein, J. W. Nicholson, S. Ghalmi, and M. F. Yan, “Ultra-large effective-area, higher-order mode fibers: a new strategy for high-power lasers,” Laser Photonics Rev. |

3. | R. H. Stolen and C. Lin, “Self-phase-modulation in silica optical fibers,” Phys. Rev. A |

4. | A. K. Ghatak, I. C. Goyal, and R. Jindal, “Design of a waveguide refractive index profile to obtain a flat modal field,” Proc. SPIE |

5. | J. W. Dawson, R. Beach, I. Jovanovic, B. Wattellier, Z. Liao, S. A. Payne, and C. P. J. Barty, “Large flattened mode optical fiber for reduction of non-linear effects in optical fiber lasers,” Proc. SPIE |

6. | W. Torruellas, Y. Chen, B. McIntosh, J. Farroni, K. Tankala, S. Webster, D. Hagan, M. J. Soileau, M. Messerly, and J. Dawson. “High peak power ytterbium-doped fiber amplifiers,” Proc SPIE |

7. | B. Ward, C. Robin, and M. Culpepper, “Photonic crystal fiber designs for power scaling of single-polarization amplifiers,” Proc. SPIE |

8. | C. Zhao, Z. Tang, Y. Ye, L. Shen, and D. Fan, “Design guidelines and characteristics for a kind of four-layer large flattened mode fibers,” Optik |

9. | N. N. Elkin, A. P. Napartovich, V. N. Troshchieva, and D. V. Vysotsky, “Modeling of large flattened mode area fiber lasers,” Laser Phys. |

10. | D. Jain, C. Baskiotis, and J. K. Sahu, “Mode area scaling with multi-trench rod-type fibers,” Opt. Express |

11. | A. Yariv, |

12. | M. Abramowitz and I. Stegun, |

13. | A. W. Snyder and J. D. Love, |

**OCIS Codes**

(060.4005) Fiber optics and optical communications : Microstructured fibers

(060.3510) Fiber optics and optical communications : Lasers, fiber

**ToC Category:**

Fiber Optics and Optical Communications

**History**

Original Manuscript: March 29, 2013

Manuscript Accepted: April 28, 2013

Published: May 16, 2013

**Citation**

Michael J. Messerly, Paul H. Pax, Jay W. Dawson, Raymond J. Beach, and John E. Heebner, "Field-flattened, ring-like propagation modes," Opt. Express **21**, 12683-12698 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-10-12683

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### References

- J. M. Fini and S. Ramachandran, “Natural bend-distortion immunity of higher-order-mode large-mode-area fibers,” Opt. Lett.32(7), 748–750 (2007). [CrossRef] [PubMed]
- S. Ramachandran, J. M. Fini, M. Mermelstein, J. W. Nicholson, S. Ghalmi, and M. F. Yan, “Ultra-large effective-area, higher-order mode fibers: a new strategy for high-power lasers,” Laser Photonics Rev.2(6), 429–448 (2008). [CrossRef]
- R. H. Stolen and C. Lin, “Self-phase-modulation in silica optical fibers,” Phys. Rev. A17(4), 1448–1453 (1978). [CrossRef]
- A. K. Ghatak, I. C. Goyal, and R. Jindal, “Design of a waveguide refractive index profile to obtain a flat modal field,” Proc. SPIE3666, 40–44 (1999). [CrossRef]
- J. W. Dawson, R. Beach, I. Jovanovic, B. Wattellier, Z. Liao, S. A. Payne, and C. P. J. Barty, “Large flattened mode optical fiber for reduction of non-linear effects in optical fiber lasers,” Proc. SPIE5335, 132–139 (2004). [CrossRef]
- W. Torruellas, Y. Chen, B. McIntosh, J. Farroni, K. Tankala, S. Webster, D. Hagan, M. J. Soileau, M. Messerly, and J. Dawson. “High peak power ytterbium-doped fiber amplifiers,” Proc SPIE 6102, 61020N (2006).
- B. Ward, C. Robin, and M. Culpepper, “Photonic crystal fiber designs for power scaling of single-polarization amplifiers,” Proc. SPIE6453, 645307, 645307-9 (2007). [CrossRef]
- C. Zhao, Z. Tang, Y. Ye, L. Shen, and D. Fan, “Design guidelines and characteristics for a kind of four-layer large flattened mode fibers,” Optik119(15), 749–754 (2008). [CrossRef]
- N. N. Elkin, A. P. Napartovich, V. N. Troshchieva, and D. V. Vysotsky, “Modeling of large flattened mode area fiber lasers,” Laser Phys.20(2), 304–310 (2010). [CrossRef]
- D. Jain, C. Baskiotis, and J. K. Sahu, “Mode area scaling with multi-trench rod-type fibers,” Opt. Express21(2), 1448–1455 (2013). [CrossRef] [PubMed]
- A. Yariv, Optical Electronics, 3rd Edition, (Holt, Rinehart and Winston, 1985).
- M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, 1972).
- A. W. Snyder and J. D. Love, Optical Waveguide Theory p.644 (Chapman and Hall Ltd, 1983).

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