## General analysis of the mode interaction in multimode active fiber |

Optics Express, Vol. 20, Issue 6, pp. 6456-6471 (2012)

http://dx.doi.org/10.1364/OE.20.006456

Acrobat PDF (1267 KB)

### Abstract

General analysis of the mode interaction in multimode fiber is presented in this paper. By taking local gain into consideration, the general coupled mode equations in multimode active fiber are deduced in the model and the effect of various factors can be analyzed based on the general coupled mode equations. Analytical expression of the beam quality factor is deduced for the optical field emerging from the multimode active fiber. The evolution of the mode power and **M ^{2}** factor along the fiber are analyzed by numerical evaluations.

© 2012 OSA

## 1. Introduction

1. Y. Jeong, J. Nilsson, J. K. Sahu, D. B. S. Soh, C. Alegria, P. Dupriez, C. A. Codemard, D. N. Payne, R. Horley, L. M. B. Hickey, L. Wanzcyk, C. E. Chryssou, J. A. Alvarez-Chavez, and P. W. Turner, “Single-frequency, single-mode, plane-polarized ytterbium-doped fiber master oscillator power amplifier source with 264 W of output power,” Opt. Lett. **30**(5), 459–461 (2005). [CrossRef] [PubMed]

3. Y. Jeong, J. Nilsson, J. K. Sahu, D. N. Payne, R. Horley, L. M. B. Hickey, and P. W. Turner, “Power scaling of single-frequency ytterbium-doped fiber master-oscillator power-ampliﬁer sources up to 500 W,” IEEE J. Sel. Top. Quantum Electron. **13**(3), 546–551 (2007). [CrossRef]

2. A. Liem, J. Limpert, H. Zellmer, and A. Tünnermann, “100-W single-frequency master-oscillator fiber power amplifier,” Opt. Lett. **28**(17), 1537–1539 (2003). [CrossRef] [PubMed]

3. Y. Jeong, J. Nilsson, J. K. Sahu, D. N. Payne, R. Horley, L. M. B. Hickey, and P. W. Turner, “Power scaling of single-frequency ytterbium-doped fiber master-oscillator power-ampliﬁer sources up to 500 W,” IEEE J. Sel. Top. Quantum Electron. **13**(3), 546–551 (2007). [CrossRef]

4. A. V. Smith and J. J. Smith, “Mode instability in high power fiber amplifiers,” Opt. Express **19**(11), 10180–10192 (2011). [CrossRef] [PubMed]

9. C. Jauregui, T. Eidam, J. Limpert, and A. Tünnermann, “The impact of modal interference on the beam quality of high-power fiber amplifiers,” Opt. Express **19**(4), 3258–3271 (2011). [CrossRef] [PubMed]

**M**factor along the fiber are investigated by numerical simulations.

^{2}## 2. Models and theoretical analysis

*z*along the waveguide, i.e. the waveguide is no longer translationally invariant.

### 2.1 The general coupled mode equation

*n*≠

_{x}*n*≠

_{y}*n*Meanwhile, assume that all vector quantities in the model contain the implicit time dependence

_{z}.*E*in the perturbed waveguide is given by:

_{z}*k*th backward-propagating bound eigenmodes in the unperturbed waveguide, so:

*k*th forward-propagating bound eigenmode:where the coupling coefficients are given by:

*z*-dependent ordinary differential equations which can be solved using the forth-order Runge-Kutta method. Compared with the BPM, this model reduces the amount of calculation significantly, and describes the mode interaction intuitionisticly.

### 2.2 Local gain

*P*is the pump power;

_{p}*σ*(

_{a}*λ*) and

_{s}*σ*(

_{e}*λ*) is the stimulated absorption and emission cross-sections of the laser transition at the wavelength

_{s}*λ*;

_{s}*I*(

*P*) is a function of

_{p}*P*and different with that of other doped active fiber. It is worth noting that ∆(

_{p}*I*(

*P*) is too complicated to be listed here. As

_{p}*k*

_{0}is the vacuum wave number, and

### 2.3 Temperature

*2.1*to depict the thermally induced modal coupling.

13. D. C. Brown and H. J. Hoffman, “Thermal, stress, and thermo-optic effects in high average power double-clad silica fiber lasers,” IEEE J. Sel. Top. Quantum Electron. **37**(2), 207–217 (2001). [CrossRef]

14. K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Thermo-optical effects in high-power ytterbium-doped fiber amplifiers,” Opt. Express **19**(24), 23965–23980 (2011). [CrossRef] [PubMed]

*∆n*(

_{β}*r*,

*z*) represents the index change caused by temperature only;

*∆n*(

_{ST,r,θ}*r*,

*z*) are the index changes caused by the thermal stress. Note that all the three refractive index changes are dependent on the critical heat transfer coefficient

*h*, and the concrete formulations can be found in references [13

_{c}13. D. C. Brown and H. J. Hoffman, “Thermal, stress, and thermo-optic effects in high average power double-clad silica fiber lasers,” IEEE J. Sel. Top. Quantum Electron. **37**(2), 207–217 (2001). [CrossRef]

14. K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Thermo-optical effects in high-power ytterbium-doped fiber amplifiers,” Opt. Express **19**(24), 23965–23980 (2011). [CrossRef] [PubMed]

### 2.4 Bend

15. M. B. Shemirani, W. Mao, R. A. Panicker, and J. M. Kahn, “Principal modes in graded-index multimode fiber in presence of spatial- and polarization-mode coupling,” J. Lightwave Technol. **27**(10), 1248–1261 (2009). [CrossRef]

*x-z*plane, and then the refractive index changes of the anisotropic media in each section can be given by [16]:where

*f*(

*z*) =

*R*(1-cos(

*z*

*/**R*)) for the macro bends;

*R*is the macro curvature radius of the fiber in this section. Provided that

*a*is radius of the core, it is available to neglect the stress-induced birefringence due to the curvature. Hence, the total changes caused by local gain, temperature and curvature can be collected and expressed as:

### 2.5 Beam quality factor

*x*-y polarizations, the transverse components of the electrical field can be decomposed into the linear combination of the corresponding polarized LP eigenmodes, respectively:Where {Ψ

*,*

_{j}*j*= 1,2,…,

*N*} represent the total LP eigenmodes in the fiber, and when

*j*= (

*m*,

*n*,

*h*)where

*n*

_{1}and

*n*

_{2}are the refractive indices of core and inner cladding, respectively;

*a*is radius of the core. Hence, the corresponding Fourier transform of

**M**factor can be calculated as follows:

^{2}_{0}is the impedance of the fiber.

## 3. Numerical simulation

_{01}and LP

_{11}modes in the multimode fiber while each mode has two orthogonal orientations. Considering the degeneracy of different azimuthal functions, hence there are LP

_{011x}, LP

_{011y}, LP

_{111x}, LP

_{111y}, LP

_{110x}and LP

_{110y}modes in the fiber. The beat length L

_{β}

^{01-11}between the LP

_{01}and LP

_{11}mode is about 0.0011m. Assume that the initial signal power injected into the amplifier is 0.01W, and different input modes possess the same power ratio with respect to the total signal power.

### 3.1 Local gain

**M**factor on the length of fiber are depicted in Fig. 1(b)-1(d).

^{2}17. M. Gong, Y. Yuan, C. Li, P. Yan, H. Zhang, and S. Liao, “Numerical modeling of transverse mode competition in strongly pumped multimode fiber lasers and amplifiers,” Opt. Express **15**(6), 3236–3246 (2007). [CrossRef] [PubMed]

*N*(

*r*) =

*N*(1-

*r*

^{2}),

*r*<1, where

*r*is the normalized radius and

*N*is assumed to be 2.43х10

^{26}which is the same with those of the first two.

**M**factor. When the doping distribution is flat doping with Г = 0.5, the LP

^{2}_{11}mode is suppressed significantly by the fundamental mode because the fundamental mode experiences larger gain than higher-order modes. As a result, the beam quality is improved with the increase of power of the fundamental mode. The same results are also presented when the doping distribution is parabolic doping. Inspired by this, various kinds of beam can be obtained provided that the corresponding doping distribution is designed reasonably.

_{01}and LP

_{11}modes achieves self-imaging [18

18. X. Zhu, A. Schülzgen, H. Li, L. Li, L. Han, J. V. Moloney, and N. Peyghambarian, “Detailed investigation of self-imaging in large-core multimode optical fibers for application in fiber lasers and amplifiers,” Opt. Express **16**(21), 16632–16645 (2008). [PubMed]

**M**factor should oscillate with a period of the beat length along the fiber. However, the oscillation of

^{2}**M**factor cannot be presented subtly in the Fig. 1(b)-1(d) due that the sampling step is much larger than the beat length. When the sampling step is far less than the beat length, the accurate evolution of mode power and

^{2}**M**factor for flat doping with Г = 1 are depicted in Fig. 2 , as well as other doping distributions.

^{2}**M**factor is about 0.05 and not prominent.

^{2}### 3.2 Temperature

*increases or different cooling methods are employed leading to different heat transfer coefficients*

_{P}*h*. Impacts of different pump powers and heat transfer coefficients on mode interaction are discussed in this section. The fiber parameters are the same with those in section

_{c}*3.1*while the doping profile is the flat doping with Г = 1, and the simulated results of mode power and

**M**factor are shown in Fig. 3 .

^{2}**M**factor. The figures show that the period of mode interaction is about 1mm, which is equal to the beat length L

^{2}_{β}

^{01-11}. Because there are only two different propagation constants among the total LP modes, the phase distribution of the field achieves self-imaging after experiencing the beat length. Hence, the thermally induced refractive index change shows periodicity with a period of the beat length L

_{β}

^{01-11}, which can be interpreted as a series of micro wedges of higher refractive index [4

4. A. V. Smith and J. J. Smith, “Mode instability in high power fiber amplifiers,” Opt. Express **19**(11), 10180–10192 (2011). [CrossRef] [PubMed]

_{11}mode. If there are more LP modes with different propagation constants within the fiber, the periodicity of the thermally induced index change will change approximately as the least common multiple of all the different beat lengths within the fiber. In order to verify the viewpoint mentioned above, we assume that there are LP

_{01}, LP

_{11}, LP

_{21}and LP

_{02}modes within the fiber without consideration of the mode orientation. The beat lengths of any two modes are listed in Table 1 . These modes occupy the same power fraction and the other parameters remain unchanged. The corresponding results are depicted in Fig. 4 .

_{β}

^{21-02}. The curve of mode power (left) in Fig. 4 depicts that the period of mode interaction caused by the thermally induced index change is about 0.0057m, which is equal to the beat length L

_{β}

^{21-02}, the least common multiple of all the different beat lengths within the fiber. That is, the phase distribution of the field within the fiber achieves self-imaging after propagating through a length of L

_{β}

^{21-02}.

### 3.3 Micro bends

_{111}mode and satisfies

*f*(

*z*) =

*A*sin(

_{d}*k’z*) along the fiber where

*A*is the bend amplitude and

_{d}*3.1*while the doping profile is the flat doping with Г = 1. Figure 6 presents the simulated results for different bend amplitudes with the same frequency

*A*is twentieth of the radius of core, the oscillation period of mode power and

_{d}**M**factor is about 0.02m, which is two-tenfold compared with the beat length. Likewise, the similar result is presented in Fig. 6(b), where

^{2}*A*is a tenth of the radius of core. Moreover, by numerical simulations, the same result also applies with other bend amplitudes. It suggests that the spatial period of mode interaction within the fiber, in which energy transfer among different modes completes a cycle, cuts as the bend amplitude of micro bends increases. This phenomenon can be explained by an in-depth study on Eq. (20). Considering the micro bends within the fiber, so the

_{d}*z*-dependent item of the first section in

*C*, as well as other sections, presents as

_{kj}*iαf*(

*z*), where

*exp*(

*iαf*(

*z*)) within the mode coefficients

*b*(

_{j}*z*) and

*b*(

_{k}*z*). When

*f*(

*z*) =

*A*sin(

_{d}*k’z*), this item is expressed as:which oscillates approximately with a frequency proportional to

*A*. Hence the spatial period of mode interaction within the fiber cuts as the bend amplitude of micro bends increases. Because that the orientation of the micro bends is parallel to the LP

_{d}_{111}mode, the level of energy transfer between the LP

_{011}and LP

_{111}mode is much larger than that between the LP

_{011}and LP

_{110}mode, as well as the

**M**factor. Figure 7 shows the simulated results of mode power and

^{2}**M**factor for different spatial frequencies while maintaining the same bend amplitude.

^{2}_{01}and LP

_{11}modes, the mode interaction with each other becomes prominent and hence the

**M**factor oscillates in magnitude. Therefore, this conclusion can be generalized into the fiber supporting more bound modes. That is, mode interaction between a pair of modes is more prominent than that of others provided that the beat frequency between them is closest to the spatial frequency of micro bends.

^{2}## 4. Conclusion

**M**factor along the fiber have been analyzed by numerical simulations and the simulation results are consistent with the physical insights well. Hence, the model in the paper can deal with the problem of mode interaction under various factors and provide instructive suggestions when designing the fiber lasers and amplifiers.

^{2}## References and links

1. | Y. Jeong, J. Nilsson, J. K. Sahu, D. B. S. Soh, C. Alegria, P. Dupriez, C. A. Codemard, D. N. Payne, R. Horley, L. M. B. Hickey, L. Wanzcyk, C. E. Chryssou, J. A. Alvarez-Chavez, and P. W. Turner, “Single-frequency, single-mode, plane-polarized ytterbium-doped fiber master oscillator power amplifier source with 264 W of output power,” Opt. Lett. |

2. | A. Liem, J. Limpert, H. Zellmer, and A. Tünnermann, “100-W single-frequency master-oscillator fiber power amplifier,” Opt. Lett. |

3. | Y. Jeong, J. Nilsson, J. K. Sahu, D. N. Payne, R. Horley, L. M. B. Hickey, and P. W. Turner, “Power scaling of single-frequency ytterbium-doped fiber master-oscillator power-ampliﬁer sources up to 500 W,” IEEE J. Sel. Top. Quantum Electron. |

4. | A. V. Smith and J. J. Smith, “Mode instability in high power fiber amplifiers,” Opt. Express |

5. | A. V. Smith and J. J. Smith, “Mode competition in high power fiber amplifiers,” Opt. Express |

6. | N. Andermahr and C. Fallnich, “Interaction of transverse modes in a single-frequency few-mode fiber amplifier caused by local gain saturation,” Opt. Express |

7. | N. Andermahr and C. Fallnich, “Modeling of transverse mode interaction in large-mode-area fiber amplifiers,” Opt. Express |

8. | A. P. Napartovich and D. V. Vysotsky, “Theory of spatial mode competition in a fiber amplifier,” Phys. Rev. A |

9. | C. Jauregui, T. Eidam, J. Limpert, and A. Tünnermann, “The impact of modal interference on the beam quality of high-power fiber amplifiers,” Opt. Express |

10. | W. Snyder and J. D. Love, |

11. | N. N. Elkin, A. P. Napartovich, V. N. Troshchieva, and D. V. Vysotsky, in Proceedings of the Fourth Conference on Finite difference methods: Theory and Application (University Press, Rousse, Bulgaria, 2007) 167–172. |

12. | H. Lü, P. Zhou, H. Xiao, X. Wang, and Z. Jiang, “Space propagation model of Tm-doped fiber laser,” J. Opt. Soc. Am. A (Submitted to). |

13. | D. C. Brown and H. J. Hoffman, “Thermal, stress, and thermo-optic effects in high average power double-clad silica fiber lasers,” IEEE J. Sel. Top. Quantum Electron. |

14. | K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Thermo-optical effects in high-power ytterbium-doped fiber amplifiers,” Opt. Express |

15. | M. B. Shemirani, W. Mao, R. A. Panicker, and J. M. Kahn, “Principal modes in graded-index multimode fiber in presence of spatial- and polarization-mode coupling,” J. Lightwave Technol. |

16. | D. Marcuse, “Losses and impulse response of a parabolic index fiber with random bends,” Bell Syst. Tech. J. |

17. | M. Gong, Y. Yuan, C. Li, P. Yan, H. Zhang, and S. Liao, “Numerical modeling of transverse mode competition in strongly pumped multimode fiber lasers and amplifiers,” Opt. Express |

18. | X. Zhu, A. Schülzgen, H. Li, L. Li, L. Han, J. V. Moloney, and N. Peyghambarian, “Detailed investigation of self-imaging in large-core multimode optical fibers for application in fiber lasers and amplifiers,” Opt. Express |

**OCIS Codes**

(140.3430) Lasers and laser optics : Laser theory

(140.3510) Lasers and laser optics : Lasers, fiber

(140.3325) Lasers and laser optics : Laser coupling

**ToC Category:**

Lasers and Laser Optics

**History**

Original Manuscript: January 18, 2012

Revised Manuscript: February 17, 2012

Manuscript Accepted: February 20, 2012

Published: March 5, 2012

**Citation**

Haibin Lü, Pu Zhou, Xiaolin Wang, and Zongfu Jiang, "General analysis of the mode interaction in multimode active fiber," Opt. Express **20**, 6456-6471 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-6-6456

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### References

- Y. Jeong, J. Nilsson, J. K. Sahu, D. B. S. Soh, C. Alegria, P. Dupriez, C. A. Codemard, D. N. Payne, R. Horley, L. M. B. Hickey, L. Wanzcyk, C. E. Chryssou, J. A. Alvarez-Chavez, and P. W. Turner, “Single-frequency, single-mode, plane-polarized ytterbium-doped fiber master oscillator power amplifier source with 264 W of output power,” Opt. Lett.30(5), 459–461 (2005). [CrossRef] [PubMed]
- A. Liem, J. Limpert, H. Zellmer, and A. Tünnermann, “100-W single-frequency master-oscillator fiber power amplifier,” Opt. Lett.28(17), 1537–1539 (2003). [CrossRef] [PubMed]
- Y. Jeong, J. Nilsson, J. K. Sahu, D. N. Payne, R. Horley, L. M. B. Hickey, and P. W. Turner, “Power scaling of single-frequency ytterbium-doped fiber master-oscillator power-ampliﬁer sources up to 500 W,” IEEE J. Sel. Top. Quantum Electron.13(3), 546–551 (2007). [CrossRef]
- A. V. Smith and J. J. Smith, “Mode instability in high power fiber amplifiers,” Opt. Express19(11), 10180–10192 (2011). [CrossRef] [PubMed]
- A. V. Smith and J. J. Smith, “Mode competition in high power fiber amplifiers,” Opt. Express19(12), 11318–11329 (2011). [CrossRef] [PubMed]
- N. Andermahr and C. Fallnich, “Interaction of transverse modes in a single-frequency few-mode fiber amplifier caused by local gain saturation,” Opt. Express16(12), 8678–8684 (2008). [CrossRef] [PubMed]
- N. Andermahr and C. Fallnich, “Modeling of transverse mode interaction in large-mode-area fiber amplifiers,” Opt. Express16(24), 20038–20046 (2008). [CrossRef] [PubMed]
- A. P. Napartovich and D. V. Vysotsky, “Theory of spatial mode competition in a fiber amplifier,” Phys. Rev. A76(6), 063801 (2007). [CrossRef]
- C. Jauregui, T. Eidam, J. Limpert, and A. Tünnermann, “The impact of modal interference on the beam quality of high-power fiber amplifiers,” Opt. Express19(4), 3258–3271 (2011). [CrossRef] [PubMed]
- W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1983).
- N. N. Elkin, A. P. Napartovich, V. N. Troshchieva, and D. V. Vysotsky, in Proceedings of the Fourth Conference on Finite difference methods: Theory and Application (University Press, Rousse, Bulgaria, 2007) 167–172.
- H. Lü, P. Zhou, H. Xiao, X. Wang, and Z. Jiang, “Space propagation model of Tm-doped fiber laser,” J. Opt. Soc. Am. A (Submitted to).
- D. C. Brown and H. J. Hoffman, “Thermal, stress, and thermo-optic effects in high average power double-clad silica fiber lasers,” IEEE J. Sel. Top. Quantum Electron.37(2), 207–217 (2001). [CrossRef]
- K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Thermo-optical effects in high-power ytterbium-doped fiber amplifiers,” Opt. Express19(24), 23965–23980 (2011). [CrossRef] [PubMed]
- M. B. Shemirani, W. Mao, R. A. Panicker, and J. M. Kahn, “Principal modes in graded-index multimode fiber in presence of spatial- and polarization-mode coupling,” J. Lightwave Technol.27(10), 1248–1261 (2009). [CrossRef]
- D. Marcuse, “Losses and impulse response of a parabolic index fiber with random bends,” Bell Syst. Tech. J.52, 1423–1437 (1973).
- M. Gong, Y. Yuan, C. Li, P. Yan, H. Zhang, and S. Liao, “Numerical modeling of transverse mode competition in strongly pumped multimode fiber lasers and amplifiers,” Opt. Express15(6), 3236–3246 (2007). [CrossRef] [PubMed]
- X. Zhu, A. Schülzgen, H. Li, L. Li, L. Han, J. V. Moloney, and N. Peyghambarian, “Detailed investigation of self-imaging in large-core multimode optical fibers for application in fiber lasers and amplifiers,” Opt. Express16(21), 16632–16645 (2008). [PubMed]

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