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

Thomas Kaiser; Daniel Flamm; Siegmund Schröter; Michael Duparré

doi:10.1364/OE.17.009347

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.

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Year
Author
Publication

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. 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]
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]
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]
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. 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]
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]

2009 (2)

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]

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]

2008 (1)

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]

2007 (3)

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

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]

2006 (1)

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]

2005 (2)

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]

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]

1998 (1)

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]

1970 (1)

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

Abouraddy, A. F.

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]

Cole, J.

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]

Duparré, M.

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]

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]

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]

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]

Duparré, M. R.

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]

Enk, S. J. van

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

Fink, Y.

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]

Golub, M.

V. A. Soifer and M. Golub, Laser Beam Mode Selection by Computer Generated Holograms (CRC Press, 1994).

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics ( McGraw-Hill Publishing Company, 1968).

Hodgson, N.

N. Hodgson and H. Weber, Laser Resonators and Beam Propagation (Springer, 2005).

Joannopoulos, J. D.

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]

Kaiser, T.

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]

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]

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]

Kauffmann, D.

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]

Kley, E.-B.

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]

Kowarschik, R. M.

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]

Lee, W. H.

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

Lüdge, B.

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]

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]

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]

Luedge, B.

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]

Mansuripur, M.

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]

Napartovich, A. P.

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

Nienhuis, G.

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

Pavelyev, V. S.

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]

Polynkin, P.

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]

Schermer, R.

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]

Schmidt, O. A.

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]

Schröter, S.

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]

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]

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]

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]

Shapira, O.

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]

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, 1986).

Soifer, V. A.

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]

V. A. Soifer and M. Golub, Laser Beam Mode Selection by Computer Generated Holograms (CRC Press, 1994).

Vysotsky, D. V.

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

Weber, H.

N. Hodgson and H. Weber, Laser Resonators and Beam Propagation (Springer, 2005).

Yariv, A.

A. Yariv, Optical Electronics in Modern Communications (Oxford University Press, 1997).

Yoda, H.

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]

Appl. Opt. (1)

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

IEEE J. Quantum Electron. (1)

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]

J. Lightwave Technol. (1)

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]

Phys. Rev. A (2)

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

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

Phys. Rev. Lett. (1)

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]

Proc. SPIE (5)

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]

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]

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]

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]

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]

Other (5)

A. Yariv, Optical Electronics in Modern Communications (Oxford University Press, 1997).

N. Hodgson and H. Weber, Laser Resonators and Beam Propagation (Springer, 2005).

A. E. Siegman, Lasers (University Science Books, 1986).

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).

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Fig. 1. Experimental setup for MODAN experiments. The setup is divided into an analytic branch containing the MODAN and a verification branch to check the results. MO1, MO2 - microscope objectives. L1, L2 - imaging lenses. FL - Fourier lens. The combination of MO2 and L1 causes a magnification of the near-field.

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Fig. 2. Working principle of optical correlation analysis. The beam emerging the fiber (d) can equivalently be described by the measured correlation pattern (a) which is processed by our software (b) to obtain a reconstructed field (c).

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Fig. 3. Modal decomposition of a beam with large amount of higher-order mode content. (a) – near-field of the investigated beam. (b) – reconstruction result. (c) modal excitation statistics for the field.

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Fig. 4. Phase resolving character of the MODAN method. (a) – investigated beam. (b) – result of the reconstruction shown with isophase lines. A first order singularity occurs. (c) – 3D-representation of the reconstructed vortex phase front.

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Fig. 5. Although the investigated field in (a) seems to be nearly perfect the fundamental mode, the excitation statistics (c) shows that ≈ 15% of the total power is propagating in higher-order modes and leads to a loss in beam quality [5].

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Tables (1)

Table 1. Most important properties of the test fiber used for verification of the MODAN method.

View Table

Equations (32)

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(Δ + k^{2}) = {\begin{array}{c} E (r) \\ H (r) \end{array}} = 0 .

[Δ_{⊥} + k_{0}^{2} n^{2} (u) - β^{2}] U (u) = 0,

U (u, z) = U (u) \exp (- iβz) .

U (u) = \sum_{n = 1}^{n_{max}} c_{n} ψ_{n} (u),

〈 ψ_{n}, ψ_{m} 〉 = ψ {∫∫}_{ℝ^{2}} d^{2} u ψ_{n}^{*} (u) ψ_{m} (u) = δ_{nm .}

c_{n} = ρ_{n} \exp (i ϕ_{n}) = 〈 ψ_{n}, U 〉 = {\int\int}_{ℝ^{2}} d^{2} u ψ_{n}^{*} (u) U (u),

\sum {∣ c_{n} ∣}^{2} = \sum ρ_{n}^{2} = 1 .

ψ (u) = R (r) Φ (φ),

R (r) = C \cdot {\begin{array}{c} \frac{J_{l} (Ur / a)}{J_{l} (U)} r \leq a \\ \frac{K_{l} (Wr / a)}{K_{l} (W)} r > a, \end{array} and Φ (φ) = {\begin{array}{c} \cos (lφ) for “ even ” modes \\ \sin (lφ) for “ odd ” modes, \end{array}

U \frac{J_{n + 1} (U)}{J_{n} (U)} = W \frac{K_{n + 1} (W)}{K_{n} (W)} and U^{2} + W^{2} = V^{2},

U = a \sqrt{k_{0}^{2} n_{core}^{2} - β^{2}}

W = a \sqrt{β^{2} - k_{0}^{2} n_{clad}^{2}},

T (u) = ψ_{n}^{*} (u),

U (u) = \sum_{n}^{n_{max}} c_{n} ψ_{n} (u),

W_{0} (u) = ψ_{n}^{*} (u) U (u) .

W_{f} (u) = \frac{k_{0}}{2 π i f} \exp (2 i k_{0} f) {∫∫}_{ℝ^{2}} d^{2} u' W_{0} (u') \exp [- i \frac{k_{0}}{f} u' u]

= \underset{{≗A}_{0}}{\underset{︸}{- 2 πi \frac{k_{0}}{f} \exp (2 i k_{0} f)}} {\tilde{W}}_{0} (\frac{k_{0}}{f} u),

𝓕 = {f (x) g (x)} = [\tilde{f} * \tilde{g}] (k)

W_{f} (u) = A_{0} {∫∫}_{ℝ^{2}} d^{2} u' {\tilde{ψ}}_{n}^{*} (\frac{k_{0}}{f} u') \tilde{U} (\frac{k_{0}}{f} [u - u']) .

W_{f} (0) = A_{0} 〈 ψ_{n}, U 〉 \propto c_{n} .

T (u) = \sum_{n}^{n_{max}} ψ_{n}^{*} (u) \exp (i V_{n} u),

W_{0} (u) = {\sum_{n} ψ}_{n}^{*} (u) U (u) \exp (i V_{n} u),

𝓕 {f (x) \exp (i v_{0} x)} = \tilde{f} (k - v_{0}),

W_{f} (u) = A_{0} \sum_{n} [{∫∫}_{ℝ^{2}} d^{2} u' {\tilde{ψ}}_{n}^{*} (\frac{k_{0}}{f} u') \tilde{U} (\frac{k_{0}}{f} [u - u']) - V_{n}],

{∣ W_{f} (u_{n} = V_{n} \frac{f}{k_{0}}) ∣}^{2} = {∣ A_{0} ∣}^{2} {∣ 〈 ψ_{n}, W (u) 〉 ∣}^{2} \propto {∣ c_{n} ∣}^{2} = ρ_{n}^{2},

T_{n}^{\cos} (u) = \frac{1}{\sqrt{2}} [ψ_{0}^{*} (u) + ψ_{n}^{*} (u)] \exp (i V_{n}^{\cos} u)

T_{n}^{\sin} (u) = \frac{1}{\sqrt{2}} [ψ_{0}^{*} (u) + {iψ}_{n}^{*} (u)] \exp (i V_{n}^{\sin} u),

{∣ W_{f} (u_{n} = V_{n}^{\cos} \frac{f}{k_{0}}) ∣}^{2} = \frac{1}{2} {∣ A_{0} (c_{0} + c_{n}) ∣}^{2}

= \frac{1}{2} {∣ A_{0} ∣}^{2} [ρ_{0}^{2} + ρ_{n}^{2} + 2 ρ_{0} ρ_{n} \cos (ϕ_{n} - ϕ_{0})],

{∣ W_{f} (u_{n} = V_{n}^{\sin} \frac{f}{k_{0}}) ∣}^{2} = \frac{1}{2} {∣ A_{0} (c_{0} + i c_{n}) ∣}^{2}

= \frac{1}{2} {∣ A_{0} ∣}^{2} [ρ_{0}^{2} + ρ_{n}^{2} + 2 ρ_{0} ρ_{n} \sin (ϕ_{n} - ϕ_{0})] .

T_{n}^{adj} (u) = \exp (i V_{n}^{adj} u) .

core diameter 2a	25μm
cladding diameter	240μm
NA	0.07
fiber length	≈ 10m
λ _excit	1064nm
resulting V parameter	5.17
number of guided LP modes	16

Abstract

References

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