## Complete modal decomposition for optical fibers using CGH-based correlation filters

Optics Express, Vol. 17, Issue 11, pp. 9347-9356 (2009)

http://dx.doi.org/10.1364/OE.17.009347

Acrobat PDF (729 KB)

### Abstract

The description of optical fields in terms of their eigenmodes is an intuitive approach for beam characterization. However, there is a lack of unambiguous, pure experimental methods in contrast to numerical phase-retrieval routines, mainly because of the difficulty to characterize the phase structure properly, e.g. if it contains singularities. This paper presents novel results for the complete modal decomposition of optical fields by using computer-generated holographic filters. The suitability of this method is proven by reconstructing various fields emerging from a weakly multi-mode fiber (*V* ≈ 5) with arbitrary mode contents. Advantages of this approach are its mathematical uniqueness and its experimental simplicity. The method constitutes a promising technique for real-time beam characterization, even for singular beam profiles.

© 2009 Optical Society of America

## 1. Introduction

*physics*of the light field and an easy integration into the theoretical framework, e.g. laser mode competition and oscillations [2, 3

3. A. P. Napartovich and D. V. Vysotsky, “Theory of spatial mode competition in a fiber amplifier,” Phys. Rev. A **76**, 063801 (2007). [CrossRef]

4. R. Schermer and J. Cole, “Improved Bend Loss Formula Verified for Optical Fiber by Simulation and Experiment,” IEEE J. Quantum Electron. **43**, 899–909 (2007). [CrossRef]

5. H. Yoda, P. Polynkin, and M. Mansuripur, “Beam Quality Factor of Higher Order Modes in a Step-Index Fiber,” J. Lightwave Technol. **24**, 1350 (2006). [CrossRef]

*et al*. [6

6. O. Shapira, A. F. Abouraddy, J. D. Joannopoulos, and Y. Fink, “Complete Modal Decomposition for Optical Waveguides,” Phys. Rev. Lett. **94**, 143902 (2005). [CrossRef] [PubMed]

9. M. R. Duparré, V. S. Pavelyev, B. Luedge, E.-B. Kley, V. A. Soifer, and R. M. Kowarschik, “Generation, superposition, and separation of Gauss-Hermite modes by means of DOEs,” in *Diffractive and Holographic Device Technologies and Applications V*, I. Cindrich and S. H. Lee, eds., Proc. SPIE **3291**, p. 104–114 (1998). [CrossRef]

10. M. Duparré, B. Lüdge, and S. Schröter “On-line characterization of Nd:YAG laser beams by means of modal decomposition using diffractive optical correlation filters” in *Optical Design and Engineering II*,
Laurent Mazuray and Rolf Wartmann, eds., Proc. SPIE **5962**, p. 59622G (2005). [CrossRef]

*real-time*, e.g. just when a perturbation of the waveguiding device (fiber, resonator, etc.) takes place [11

11. T. Kaiser, B. Lüdge, S. Schröter, D. Kauffmann, and M. Duparré, “Detection of mode conversion effects in passive LMA fibres by means of optical correlation analysis,” in *Solid State Lasers and Amplifiers III*, J. A. Terry, T. Graf, and H. Jeĺinková, eds., Proc. SPIE **6998**, p. 69980J (2008). [CrossRef]

*V*≈ 5) LMA fiber, operated at

*λ*= 1064nm. To change the modal content, we intentionally applied different perturbations to the system. The possibility to reconstruct fields with various mode contents is proven as well as the ability to resolve phase singularities correctly, which is particularly useful in applications involving the orbital momentum of light [12

12. S. J. van Enk and G. Nienhuis, “Photons in polychromatic rotating modes,” Phys. Rev. A **76**, 053825 (2007). [CrossRef]

## 2. Modal decomposition for step-index fiber modes

10. M. Duparré, B. Lüdge, and S. Schröter “On-line characterization of Nd:YAG laser beams by means of modal decomposition using diffractive optical correlation filters” in *Optical Design and Engineering II*,
Laurent Mazuray and Rolf Wartmann, eds., Proc. SPIE **5962**, p. 59622G (2005). [CrossRef]

*incoherent*superposition of modes takes place. In the present paper, it will be illustrated for the

*coherent*mode superposition in weakly guiding optical fibers.

**u**denotes the transverse coordinates. The spatial dependence of the field then reads as

*U*(

**u**) that is normalized to unit power can be regarded as a superposition of eigensolutions

*ψ*(

_{n}**u**) of Eq. (2)

*c*are given by

_{n}*a*denotes the fiber core radius and

*C*is chosen so that

*ψ*(

**u**) fulfills the normalization condition. The values for

*U*and

*W*can be calculated from the characteristic equations

*V*=

*ak*

_{0}(

*n*

^{2}

_{core}-

*n*

^{2}

_{clad})

^{1/2}being the well-known

*V*-parameter of the fiber.

*J*means the BESSEL-function of first kind and

_{l}*l*

^{th}order, whereas

*K*means the modified BESSEL-function of second kind and

_{l}*l*

^{th}order.

## 3. Optical correlation analysis

11. T. Kaiser, B. Lüdge, S. Schröter, D. Kauffmann, and M. Duparré, “Detection of mode conversion effects in passive LMA fibres by means of optical correlation analysis,” in *Solid State Lasers and Amplifiers III*, J. A. Terry, T. Graf, and H. Jeĺinková, eds., Proc. SPIE **6998**, p. 69980J (2008). [CrossRef]

14. W. H. Lee, “Sampled Fourier Transform Hologram Generated by Computer,” Appl. Opt. **9**, 639–643 (1970). [CrossRef] [PubMed]

*f*. This yields [7]

*A*

_{0}being a constant factor. By applying the convolution theorem

*c*. A detector, placed in the center of the output focal plane, measures a signal which is proportional to ∣

_{n}*c*∣

_{n}^{2}=

*ρ*

_{n}^{2}.

**Multibranch operation and intermodal phase measurement**The transmission function of the holographic element can be modified by using angular multiplexing so that more than one channel or interferometric superpositions of them can be realized simultaneously. If the modified transmission function reads as

**V**

_{n}. In the MODAN plane, we obtain now

*U*(

**u**) and the mode field distributions

*ψ*. Each of them is shifted transverse by

_{n}**V**

*/*

_{n}, f*k*

_{0}. If the combination of

**V**

*/ and*

_{n}f*k*

_{0}is chosen properly, the contributions from different channels do not overlap due to their strong decay for ∣

**u**∣ ≫ ∣

**V**

_{n}∣

*f*/

*k*

_{0}.

**u**

*=*

_{n}**V**

*/*

_{n}f*k*

_{0}, for which we obtain

*ψ*in the investigated beam. We refer to the signals encoded on different spatial carrier frequencies as the “branches” of the correlation pattern. Note that the information about

_{n}*all*modes is obtained

*simultaneously*by an experimentally simple optical 2

*f*-setup.

*ρ*, a field reconstruction is possible by applying numerical phase retrieval algorithms, to obtain the correct intermodal phases [15

_{n}15. T. Kaiser, S. Schröter, and M. Duparré, “Modal decomposition in step-index fibers by optical correlation analysis,” in *Laser Resonators and Beam Control XI*, A. V. Kudryashov, A. H. Paxton, V. S. Ilchenko, and L. Aschke, eds., Proc. SPIE **7194**, p. 719407 (2009). [CrossRef]

*measured directly*by introducing an interferometric superposition of channels within a branch of the MODAN. This is of particular importance, since many laser beam parameters like the beam propagation ratio M

^{2}and the beam pointing stability depend strongly on the intermodal phase [5

5. H. Yoda, P. Polynkin, and M. Mansuripur, “Beam Quality Factor of Higher Order Modes in a Step-Index Fiber,” J. Lightwave Technol. **24**, 1350 (2006). [CrossRef]

*ψ*are added with transmission functions reading as

_{n}*ψ*

_{0}(

**u**) denotes a specific mode, which phase is used as reference (most commonly the fundamental mode of the system). Therefore,

*ϕ*

_{0}is a meaningless total phase and can be chosen as

*ϕ*

_{0}= 0.

*n*- 2) branches must be comprised in the hologram to obtain the complete modal decomposition for a system with

*n*modes on a purely experimental basis. The number of branches would be limited basically by the SNR of angular multiplexing. In our case, the practical limit was given by the dimensions and resolution of the CCD chip. Here, one needs to find an optimum where the extends of the correlation signals are not to small but still separated from each other. So far, we have investigated systems with up to 26 angle-multiplexed branches in one hologram, which allows the investigation of a weakly multi-modal fiber with

*V*≈ 5, as described in this paper.

## 4. Experimental verification

*f*-setup, which is a major advantage of this method. However, accurate adjustment of the optical components relative to each other is required.

*spatial*eigenstates. The most important parameters of the used fiber are given in Table 1.

*λ*

_{0}= 1064 nm), using free-space coupling between the incident beam and the fiber. Behind the end facet, a polarizer was used to select the intended polarization state. An imaging system consisting of a suitable microscope objective MO2 and a lens L1 magnified the imaged near-field on the HOE by a factor of 37.6. The diffracted light in the first diffraction order passed a further lens (FL) to carry out the FOURIER transform. After the image was recorded by a CCD camera, it was evaluated by a specific software created at our institute. The modal content was changed by slightly misaligning the adjustment between laser and fiber to obtain a variety of different multimodal excitations. In a second branch behind a beam splitter, the near- and far-field was recorded by a second CCD camera.

*μ*m while a minimum feature size of 600 nm was used for microlithography. The hologram contained (3 × 8 - 2) = 22 branches for the measurement of the intermodal weights

*ρ*

_{n}^{2}and phases

*ϕ*of the 8 spatial LP eigenmodes of the fiber.

_{n}**V**

_{n},

**V**

^{cos}

_{n}and

**V**

^{sin}

_{n}define the points of interest in the FOURIER domain. The remaining problem is to find the absolute coordinate system, where these coordinates hold. This can be achieved by measuring the first order moments of the adjustment channels. The four points define a rectangular region. Within this region, the location of the points of interest relative to the corners is also known from the spatial frequencies and so, the coordinate system is fixed. There, the intensity has to be read out to obtain the information about the modal coefficients

*c*=

_{n}*ρ*exp(

_{n}*iϕ*). With this information, it is possible to reconstruct the field emerging the fiber, including amplitude and phase distribution. The result of the reconstruction shown in Fig. 2(c) can be compared to the directly measured near-field intensity in Fig. 2(d) to verify the correct functionality of the system.

_{n}*ρ*

_{n}^{2}and intermodal phases

*ϕ*in real-time becomes possible.

_{n}12. S. J. van Enk and G. Nienhuis, “Photons in polychromatic rotating modes,” Phys. Rev. A **76**, 053825 (2007). [CrossRef]

^{st}-order phase singularity. The vortex phase front is shown in 3D-representation in Fig. 4(c).

_{02}mode and an equal excitation of LP

^{e}

_{11}and LP

^{o}

_{11}account for the beam profile, which explains the radial symmetry of the beam itself. A comparison between optical correlation analysis and standard

*M*

^{2}method is given in Ref. [16

16. O. A. Schmidt, T. Kaiser, B. Lüdge, S. Schröter, and M. Duparré, “Laser-beam characterization by means of modal decomposition versus M2 method,” in *Laser Resonators and Beam Control XI*, A. V. Kudryashov, A. H. Paxton, V. S. Ilchenko, and L. Aschke, eds., Proc. SPIE **7194**, p. 71940C (2009). [CrossRef]

## 5. Conclusions

*ρ*

_{n}^{2}and the intermodal phases

*ϕ*can be measured simultaneously in different angle-multiplexed branches of the correlation pattern. Naturally, one would choose the fundamental LP

_{n}_{01}mode as reference for the phase measurement. However, a problem might occur if this reference mode is not excited sufficiently, although this situation is unlikely. By reviewing Eq. (23), it is obvious that the intermodal phase becomes immeasurable in this case. However, this can be overcome by introducing a second reference mode on additional branches of the hologram.

## References and links

1. | N. Hodgson and H. Weber, |

2. | A. E. Siegman, |

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

4. | R. Schermer and J. Cole, “Improved Bend Loss Formula Verified for Optical Fiber by Simulation and Experiment,” IEEE J. Quantum Electron. |

5. | H. Yoda, P. Polynkin, and M. Mansuripur, “Beam Quality Factor of Higher Order Modes in a Step-Index Fiber,” J. Lightwave Technol. |

6. | O. Shapira, A. F. Abouraddy, J. D. Joannopoulos, and Y. Fink, “Complete Modal Decomposition for Optical Waveguides,” Phys. Rev. Lett. |

7. | J. W. Goodman, |

8. | V. A. Soifer and M. Golub, |

9. | M. R. Duparré, V. S. Pavelyev, B. Luedge, E.-B. Kley, V. A. Soifer, and R. M. Kowarschik, “Generation, superposition, and separation of Gauss-Hermite modes by means of DOEs,” in |

10. | M. Duparré, B. Lüdge, and S. Schröter “On-line characterization of Nd:YAG laser beams by means of modal decomposition using diffractive optical correlation filters” in |

11. | T. Kaiser, B. Lüdge, S. Schröter, D. Kauffmann, and M. Duparré, “Detection of mode conversion effects in passive LMA fibres by means of optical correlation analysis,” in |

12. | S. J. van Enk and G. Nienhuis, “Photons in polychromatic rotating modes,” Phys. Rev. A |

13. | A. Yariv, |

14. | W. H. Lee, “Sampled Fourier Transform Hologram Generated by Computer,” Appl. Opt. |

15. | T. Kaiser, S. Schröter, and M. Duparré, “Modal decomposition in step-index fibers by optical correlation analysis,” in |

16. | O. A. Schmidt, T. Kaiser, B. Lüdge, S. Schröter, and M. Duparré, “Laser-beam characterization by means of modal decomposition versus M2 method,” in |

**OCIS Codes**

(060.2270) Fiber optics and optical communications : Fiber characterization

(090.1970) Holography : Diffractive optics

(090.2890) Holography : Holographic optical elements

(120.3940) Instrumentation, measurement, and metrology : Metrology

(140.3295) Lasers and laser optics : Laser beam characterization

**ToC Category:**

Holography

**History**

Original Manuscript: March 30, 2009

Revised Manuscript: May 13, 2009

Manuscript Accepted: May 13, 2009

Published: May 19, 2009

**Citation**

Thomas Kaiser, Daniel Flamm, Siegmund Schröter, and Michael Duparré, "Complete modal decomposition for
optical fibers using CGH-based
correlation filters," Opt. Express **17**, 9347-9356 (2009)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-11-9347

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### References

- N. Hodgson and H. Weber, Laser Resonators and Beam Propagation (Springer, 2005).
- A. E. Siegman, Lasers (University Science Books, 1986).
- A. P. Napartovich and D. V. Vysotsky, "Theory of spatial mode competition in a fiber amplifier," Phys. Rev. A 76, 063801 (2007). [CrossRef]
- R. Schermer and J. Cole, "Improved Bend Loss Formula Verified for Optical Fiber by Simulation and Experiment," IEEE J. Quantum Electron. 43, 899-909 (2007). [CrossRef]
- H. Yoda, P. Polynkin, and M. Mansuripur, "Beam Quality Factor of Higher Order Modes in a Step-Index Fiber," J. Lightwave Technol. 24, 1350 (2006). [CrossRef]
- O. Shapira, A. F. Abouraddy, J. D. Joannopoulos, and Y. Fink, "Complete Modal Decomposition for Optical Waveguides," Phys. Rev. Lett. 94, 143902 (2005). [CrossRef] [PubMed]
- J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill Publishing Company, 1968).
- V. A. Soifer and M. Golub, Laser Beam Mode Selection by Computer Generated Holograms (CRC Press, 1994).
- M. R. Duparre, V. S. Pavelyev, B. Luedge, E.-B. Kley, V. A. Soifer, and R. M. Kowarschik, "Generation, superposition, and separation of Gauss-Hermite modes by means of DOEs," in Diffractive and Holographic Device Technologies and Applications V, I. Cindrich and S. H. Lee, eds., Proc. SPIE 3291, 104-114 (1998). [CrossRef]
- M. Duparre, B. Ludge and S. Schroter, "On-line characterization of Nd:YAG laser beams by means of modal decomposition using diffractive optical correlation filters," Proc. SPIE 5962, 59622G (2005). [CrossRef]
- T. Kaiser, B. Ludge, S. Schroter, D. Kauffmann, and M. Duparre, "Detection of mode conversion effects in passive LMA fibres by means of optical correlation analysis," Proc. SPIE 6998, 69980J (2008). [CrossRef]
- S. J. van Enk and G. Nienhuis, "Photons in polychromatic rotating modes," Phys. Rev. A 76, 053825 (2007). [CrossRef]
- A. Yariv, Optical Electronics in Modern Communications (Oxford University Press, 1997).
- W. H. Lee, "Sampled Fourier Transform Hologram Generated by Computer," Appl. Opt. 9, 639-643 (1970). [CrossRef] [PubMed]
- T. Kaiser, S. Schroter, and M. Duparre, "Modal decomposition in step-index fibers by optical correlation analysis," in Laser Resonators and Beam Control XI, A. V. Kudryashov, A. H. Paxton, V. S. Ilchenko, and L. Aschke, eds., Proc. SPIE 7194, 719407 (2009). [CrossRef]
- O. A. Schmidt, T. Kaiser, B. Ludge, S. Schroter, and M. Duparre, "Laser-beam characterization by means of modal decomposition versus M2 method," Proc. SPIE 7194, 71940C (2009). [CrossRef]

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