Abstract

We report on a fast and experimentally easy technique for measuring the beam propagation ratio M2 of light guided by optical fibers. A holographic filter enables us to determine amplitudes and phases of the excited fiber eigenmodes. The coherent superposition of modes allows the reconstruction of the optical field. With this information at hand, we are able to simulate the free-space propagation of the beam and to perform a virtual caustic measurement. Associated beam propagation ratios M2 accurately agree with ISO-standard measurements.

© 2012 Optical Society of America

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References

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  1. A. E. Siegman, “New developments in laser resonators,” Proc. SPIE 1224, 2–14 (1990).
    [CrossRef]
  2. A. E. Siegman, “How to (maybe) measure laser beam quality,” in DPSS (Diode Pumped Solid State) Lasers: Applications and Issues (Optical Society of America, 1998), p. MQ1.
  3. S. Liao, M. Gong, and H. Zhang, “Theoretical calculation of beam quality factor of large-mode-area fiber amplifiers,” Laser Physics 19, 437–444 (2009).
    [CrossRef]
  4. ISO 11146-2, “Test methods for laser beam widths, divergence angles and beam propagation ratios. Part 2: General astigmatic beams” (ISO, 2005).
  5. T. Eidam, C. Wirth, C. Jauregui, F. Stutzki, F. Jansen, H.-J. Otto, O. Schmidt, T. Schreiber, J. Limpert, and A. Tünnermann, “Experimental observations of the threshold-like onset of mode instabilities in high power fiber amplifiers,” Opt. Express 19, 13218–13224 (2011).
    [CrossRef]
  6. 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, 3258–3271 (2011).
    [CrossRef]
  7. F. Stutzki, F. Jansen, C. Jauregui, J. Limpert, and A. Tünnermann, “Non-hexagonal large-pitch fibers for enhanced mode discrimination,” Opt. Express 19, 12081–12086 (2011).
    [CrossRef]
  8. H. Yoda, P. Polynkin, and M. Mansuripur, “Beam quality factor of higher order modes in a step-index fiber,” J. Lightwave Technol. 24, 1350–1355 (2006).
    [CrossRef]
  9. S. Wielandy, “Implications of higher-order mode content in large mode area fibers with good beam quality,” Opt. Express 15, 15402–15409 (2007).
    [CrossRef]
  10. V. A. Soifer and M. Golub, Laser Beam Mode Selection by Computer Generated Holograms (CRC Press, 1994).
  11. T. Kaiser, D. Flamm, S. Schröter, and M. Duparré, “Complete modal decomposition for optical fibers using CGH-based correlation filters,” Opt. Express 17, 9347–9356 (2009).
    [CrossRef]
  12. O. A. Schmidt, C. Schulze, D. Flamm, R. Brüning, T. Kaiser, S. Schröter, and M. Duparré, “Real-time determination of laser beam quality by modal decomposition,” Opt. Express 19, 6741–6748 (2011).
    [CrossRef]
  13. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).
  14. D. Gloge, “Weakly guiding fibers,” Appl. Opt. 10, 2252–2258 (1971).
    [CrossRef]
  15. D. Flamm, O. A. Schmidt, C. Schulze, J. Borchardt, T. Kaiser, S. Schröter, and M. Duparré, “Measuring the spatial polarization distribution of multimode beams emerging from passive step-index large-mode-area fibers,” Opt. Lett. 35, 3429–3431 (2010).
    [CrossRef]
  16. W. H. Lee, “Sampled Fourier transform hologram generated by computer,” Appl. Opt. 9, 639–643 (1970).
    [CrossRef]
  17. T. Grosjean, A. Sabac, and D. Courjon, “A versatile and stable device allowing the efficient generation of beams with radial, azimuthal or hybrid polarizations,” Opt. Commun. 252, 12–21 (2005).
    [CrossRef]
  18. F. Stutzki, F. Jansen, T. Eidam, A. Steinmetz, C. Jauregui, J. Limpert, and A. Tünnermann, “High average power large-pitch fiber amplifier with robust single-mode operation,” Opt. Lett. 36, 689–691 (2011).
    [CrossRef]
  19. C. Schulze, O. A. Schmidt, D. Flamm, S. Schröter, and M. Duparré, “Modal analysis of beams emerging from a multi-core fiber using computer-generated holograms,” Proc. SPIE 7914, 79142H (2011).
    [CrossRef]
  20. O. A. Schmidt, D. Flamm, and M. Duparré, “Modal decomposition for photonic crystal fibers using computer-generated holograms,” Proc. SPIE 7714, 77140W (2010).
    [CrossRef]

2011

2010

2009

S. Liao, M. Gong, and H. Zhang, “Theoretical calculation of beam quality factor of large-mode-area fiber amplifiers,” Laser Physics 19, 437–444 (2009).
[CrossRef]

T. Kaiser, D. Flamm, S. Schröter, and M. Duparré, “Complete modal decomposition for optical fibers using CGH-based correlation filters,” Opt. Express 17, 9347–9356 (2009).
[CrossRef]

2007

2006

2005

T. Grosjean, A. Sabac, and D. Courjon, “A versatile and stable device allowing the efficient generation of beams with radial, azimuthal or hybrid polarizations,” Opt. Commun. 252, 12–21 (2005).
[CrossRef]

1990

A. E. Siegman, “New developments in laser resonators,” Proc. SPIE 1224, 2–14 (1990).
[CrossRef]

1971

1970

Borchardt, J.

Brüning, R.

Courjon, D.

T. Grosjean, A. Sabac, and D. Courjon, “A versatile and stable device allowing the efficient generation of beams with radial, azimuthal or hybrid polarizations,” Opt. Commun. 252, 12–21 (2005).
[CrossRef]

Duparré, M.

Eidam, T.

Flamm, D.

Gloge, D.

Golub, M.

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

Gong, M.

S. Liao, M. Gong, and H. Zhang, “Theoretical calculation of beam quality factor of large-mode-area fiber amplifiers,” Laser Physics 19, 437–444 (2009).
[CrossRef]

Goodman, J. W.

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

Grosjean, T.

T. Grosjean, A. Sabac, and D. Courjon, “A versatile and stable device allowing the efficient generation of beams with radial, azimuthal or hybrid polarizations,” Opt. Commun. 252, 12–21 (2005).
[CrossRef]

Jansen, F.

Jauregui, C.

Kaiser, T.

Lee, W. H.

Liao, S.

S. Liao, M. Gong, and H. Zhang, “Theoretical calculation of beam quality factor of large-mode-area fiber amplifiers,” Laser Physics 19, 437–444 (2009).
[CrossRef]

Limpert, J.

Mansuripur, M.

Otto, H.-J.

Polynkin, P.

Sabac, A.

T. Grosjean, A. Sabac, and D. Courjon, “A versatile and stable device allowing the efficient generation of beams with radial, azimuthal or hybrid polarizations,” Opt. Commun. 252, 12–21 (2005).
[CrossRef]

Schmidt, O.

Schmidt, O. A.

O. A. Schmidt, C. Schulze, D. Flamm, R. Brüning, T. Kaiser, S. Schröter, and M. Duparré, “Real-time determination of laser beam quality by modal decomposition,” Opt. Express 19, 6741–6748 (2011).
[CrossRef]

C. Schulze, O. A. Schmidt, D. Flamm, S. Schröter, and M. Duparré, “Modal analysis of beams emerging from a multi-core fiber using computer-generated holograms,” Proc. SPIE 7914, 79142H (2011).
[CrossRef]

O. A. Schmidt, D. Flamm, and M. Duparré, “Modal decomposition for photonic crystal fibers using computer-generated holograms,” Proc. SPIE 7714, 77140W (2010).
[CrossRef]

D. Flamm, O. A. Schmidt, C. Schulze, J. Borchardt, T. Kaiser, S. Schröter, and M. Duparré, “Measuring the spatial polarization distribution of multimode beams emerging from passive step-index large-mode-area fibers,” Opt. Lett. 35, 3429–3431 (2010).
[CrossRef]

Schreiber, T.

Schröter, S.

Schulze, C.

Siegman, A. E.

A. E. Siegman, “New developments in laser resonators,” Proc. SPIE 1224, 2–14 (1990).
[CrossRef]

A. E. Siegman, “How to (maybe) measure laser beam quality,” in DPSS (Diode Pumped Solid State) Lasers: Applications and Issues (Optical Society of America, 1998), p. MQ1.

Soifer, V. A.

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

Steinmetz, A.

Stutzki, F.

Tünnermann, A.

Wielandy, S.

Wirth, C.

Yoda, H.

Zhang, H.

S. Liao, M. Gong, and H. Zhang, “Theoretical calculation of beam quality factor of large-mode-area fiber amplifiers,” Laser Physics 19, 437–444 (2009).
[CrossRef]

Appl. Opt.

J. Lightwave Technol.

Laser Physics

S. Liao, M. Gong, and H. Zhang, “Theoretical calculation of beam quality factor of large-mode-area fiber amplifiers,” Laser Physics 19, 437–444 (2009).
[CrossRef]

Opt. Commun.

T. Grosjean, A. Sabac, and D. Courjon, “A versatile and stable device allowing the efficient generation of beams with radial, azimuthal or hybrid polarizations,” Opt. Commun. 252, 12–21 (2005).
[CrossRef]

Opt. Express

Opt. Lett.

Proc. SPIE

C. Schulze, O. A. Schmidt, D. Flamm, S. Schröter, and M. Duparré, “Modal analysis of beams emerging from a multi-core fiber using computer-generated holograms,” Proc. SPIE 7914, 79142H (2011).
[CrossRef]

O. A. Schmidt, D. Flamm, and M. Duparré, “Modal decomposition for photonic crystal fibers using computer-generated holograms,” Proc. SPIE 7714, 77140W (2010).
[CrossRef]

A. E. Siegman, “New developments in laser resonators,” Proc. SPIE 1224, 2–14 (1990).
[CrossRef]

Other

A. E. Siegman, “How to (maybe) measure laser beam quality,” in DPSS (Diode Pumped Solid State) Lasers: Applications and Issues (Optical Society of America, 1998), p. MQ1.

ISO 11146-2, “Test methods for laser beam widths, divergence angles and beam propagation ratios. Part 2: General astigmatic beams” (ISO, 2005).

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

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

Supplementary Material (1)

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Figures (7)

Fig. 1.
Fig. 1.

Numerical M 2 determination of a beam superposition of step-index fiber modes LP 01 and LP 02 : intensity distributions with stated weightings and different intermodal phase differences Δ ϕ [9].

Fig. 2.
Fig. 2.

Intensity measurements (left) and corresponding reconstructions (right) of an imaged beam (magnification: 37.5) emerging from a step-index LMA fiber ( V 4 ) for different propagation lengths. Additionally, the barycenter and the normalized variance ellipse for every intensity distribution is plotted. The relative power amount of all higher-order modes is 35%. For all propagation lengths, the two-dimensional cross correlation coefficient as a quality measure for the reconstruction is higher than 0.93 and expresses strong conformity.

Fig. 3.
Fig. 3.

Experimental setup. MO, microscope objective; NP, nanopositioning unit; PP, phase plate; P, polarizer; L, imaging lens; CL, cylindrical lens; BS, beam splitter; FL, Fourier lens.

Fig. 4.
Fig. 4.

(a) ISO conform caustic measurement. Second-order moments x 2 , y 2 depending on z -position. The inset depicts the near-field recording. Resulting beam propagation ratio: M eff , ISO 2 = 1.33 . (b) Modal power spectrum. The dominating modes are the fundamental mode guiding 41% and the mode LP 11 e guiding 58% of the total power, respectively. (c) Modally resolved phase differences. The phase difference between dominating modes LP 01 and LP 11 e is 1.4 rad . Resulting beam propagation ratio of the VCM: M eff , VCM 2 = 1.35 .

Fig. 5.
Fig. 5.

ISO conform caustic measurement. Second-order moments x 2 , y 2 depending on z -position. The inset depicts the near-field recording. Resulting beam propagation ratio: M eff , ISO 2 = 1.02 . The relative modal power guided in the fundamental mode is 98%. Resulting beam propagation ratio determined by the VCM: M eff , VCM 2 = 1.03 .

Fig. 6.
Fig. 6.

Comparison of the M eff 2 parameters of seven beams with differing modal content measured according to ISO standard and VCM, respectively. The insets show the measured near-field intensity (left) and the reconstructed intensity (right) of the respective beam.

Fig. 7.
Fig. 7.

M eff 2 parameter on a time scale of 14.2 s taken with a rate of 30 Hz. The insets depict measured near-field intensities at selected points in time. The fiber movement is schematically shown on top of the graph. After the horizontal coupling process was completed at 7 s , we reversed the scanning movement and returned to the starting position. A similar coupling process with an online monitoring (8 Hz) of the measured and the reconstructed near-field intensity, as well as the corresponding M eff 2 value, can be seen in (Media 1).

Tables (1)

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Table 1. Fiber Properties of Investigated LMA Fibers A and B

Equations (10)

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[ 2 + ε r ( r ) k 2 ] ψ l ( r ) = β l 2 ψ l ( r ) ,
U ( r ) = l l max c l ψ l ( r ) .
c l = [ c l x c l y ] = [ ϱ l x exp ( i ϕ l x ) ϱ l y exp ( i ϕ l y ) ] ,
l l max ( ϱ l x 2 + ϱ l y 2 ) = 1 .
U j ( r ) = U ( r ) · e j = l l max c l j ψ l ( r ) .
T ( r ) = ψ l * ( r ) ,
T l cos = ( ψ 0 * + ψ l * ) / 2 ,
T l sin = ( ψ 0 * + i ψ l * ) / 2 .
Δ ϕ l j = tan 1 [ 2 I l j sin I 0 ( ϱ l j 2 ϱ 0 j 2 ) 2 I l j cos I 0 ( ϱ l j 2 ϱ 0 j 2 ) ] [ π , π ] .
U ( x , y ; z ) = F { H ˜ ( k x , k y ; z ) U ˜ ( k x , k y ; z = 0 ) } ,

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