Abstract

The modal decomposition of an arbitrary optical field may be done without regard to the spatial scale of the chosen basis functions, but this generally leads to a large number of modes in the expansion. While this may be considered as mathematically correct, it is not efficient and not physically representative of the underlying field. Here we demonstrate a modal decomposition approach that requires no a priori knowledge of the spatial scale of the modes, but nevertheless leads to an optimised modal expansion. We illustrate the power of the method by successfully decomposing beams from a diode-pumped solid state laser resonator into an optimised Laguerre-Gaussian mode set. Our experimental results, which are in agreement with theory, illustrate the versatility of the approach.

© 2012 Optical Society of America

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References

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  2. M. A. Golub, A. M. Prokhorov, I. N. Sisakian, and V. A. Soifer “Synthesis of spatial filters for investigation of the transverse mode composition of coherent radiation,” Sov. J. Quantum Electron. 9, 1866–1868 (1982).
  3. E. Tervonen, J. Turunen, and A. Friberg, “Transverse laser mode structure determination from spatial coherence measurements: Experimental results,” Appl. Phys. B. 49, 409–414 (1989).
    [Crossref]
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  11. I. A. Litvin, A. Dudley, and A. Forbes, “Poynting vector and orbital angular momentum density of superpositions of bessel beams,” Opt. Express 19, 16760–16771 (2011).
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  12. A. Dudley, I. A. Litvin, and A. Forbes, “Quantitative measurement of the orbital angular momentum density of light,” Appl. Opt. 51, 823–833 (2012).
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  13. C. Schulze, D. Naidoo, D. Flamm, O. A. Schmidt, A. Forbes, and M. Duparré, “Wavefront reconstruction by modal decomposition,” Opt. Express 20, 19714–19725 (2012).
    [Crossref] [PubMed]
  14. I. A. Litvin, A. Dudley, F. S. Roux, and A. Forbes, “Azimuthal decomposition with digital holograms,” Opt. Express 20, 10996–11004 (2012).
    [Crossref] [PubMed]
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  17. D. Flamm, C. Schulze, R. Brüning, O. A. Schmidt, T. Kaiser, S. Schröter, and M. Duparré, “Fast M2 measurement for fiber beams based on modal analysis,” Appl. Opt. 51, 987–993 (2012).
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2012 (6)

2011 (2)

2010 (1)

2009 (1)

2003 (1)

2000 (1)

1999 (1)

1995 (1)

1989 (1)

E. Tervonen, J. Turunen, and A. Friberg, “Transverse laser mode structure determination from spatial coherence measurements: Experimental results,” Appl. Phys. B. 49, 409–414 (1989).
[Crossref]

1982 (1)

M. A. Golub, A. M. Prokhorov, I. N. Sisakian, and V. A. Soifer “Synthesis of spatial filters for investigation of the transverse mode composition of coherent radiation,” Sov. J. Quantum Electron. 9, 1866–1868 (1982).

1966 (1)

Arrizon, A.

Borchardt, J.

Borghi, R.

Brüning, R.

Cutolo, A.

Dudley, A.

Duparré, M.

Flamm, D.

Forbes, A.

Friberg, A.

E. Tervonen, J. Turunen, and A. Friberg, “Transverse laser mode structure determination from spatial coherence measurements: Experimental results,” Appl. Phys. B. 49, 409–414 (1989).
[Crossref]

Golub, M. A.

M. A. Golub, A. M. Prokhorov, I. N. Sisakian, and V. A. Soifer “Synthesis of spatial filters for investigation of the transverse mode composition of coherent radiation,” Sov. J. Quantum Electron. 9, 1866–1868 (1982).

Goodman, J. W.

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

Gori, F.

Guattari, G.

Isernia, T.

Izzo, I.

Kaiser, T.

Kirk, A. G.

Kogelnik, H.

Li, T.

Litvin, I. A.

Naidoo, D.

Pierri, R.

Prokhorov, A. M.

M. A. Golub, A. M. Prokhorov, I. N. Sisakian, and V. A. Soifer “Synthesis of spatial filters for investigation of the transverse mode composition of coherent radiation,” Sov. J. Quantum Electron. 9, 1866–1868 (1982).

Roux, F. S.

Santarsiero, M.

Schmidt, O. A.

Schröter, S.

Schulze, C.

Siegman, A.

A. 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.

Sisakian, I. N.

M. A. Golub, A. M. Prokhorov, I. N. Sisakian, and V. A. Soifer “Synthesis of spatial filters for investigation of the transverse mode composition of coherent radiation,” Sov. J. Quantum Electron. 9, 1866–1868 (1982).

Soifer, V. A.

M. A. Golub, A. M. Prokhorov, I. N. Sisakian, and V. A. Soifer “Synthesis of spatial filters for investigation of the transverse mode composition of coherent radiation,” Sov. J. Quantum Electron. 9, 1866–1868 (1982).

Tervonen, E.

E. Tervonen, J. Turunen, and A. Friberg, “Transverse laser mode structure determination from spatial coherence measurements: Experimental results,” Appl. Phys. B. 49, 409–414 (1989).
[Crossref]

Turunen, J.

E. Tervonen, J. Turunen, and A. Friberg, “Transverse laser mode structure determination from spatial coherence measurements: Experimental results,” Appl. Phys. B. 49, 409–414 (1989).
[Crossref]

Wei, H.

Xue, X.

Zeni, L.

Appl. Opt. (5)

Appl. Phys. B. (1)

E. Tervonen, J. Turunen, and A. Friberg, “Transverse laser mode structure determination from spatial coherence measurements: Experimental results,” Appl. Phys. B. 49, 409–414 (1989).
[Crossref]

J. Opt. Soc. Am. A (1)

Opt. Express (5)

Opt. Lett. (4)

Sov. J. Quantum Electron. (1)

M. A. Golub, A. M. Prokhorov, I. N. Sisakian, and V. A. Soifer “Synthesis of spatial filters for investigation of the transverse mode composition of coherent radiation,” Sov. J. Quantum Electron. 9, 1866–1868 (1982).

Other (3)

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

A. 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, “ISO 11146-1:2005 Test methods for laser beam widths, divergence angles and beam propagation ratios Part 1: Stigmatic and simple astigmatic beams,” (2005).

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Figures (6)
Fig. 1
Fig. 1 Schematic experimental setup of the end-pumped Nd:YAG resonator, where the output beam is 1:1 imaged onto a camera (CCD1) and a Spatial Light Modulator (SLM), whose diffraction pattern is observed in the far field with CCD2. M1,2: curved (R = 500mm) and flat mirror (R = ∞), BS beam splitter, PF pump light filter, ND neutral density filter, L lens, M2: M2 meter.

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Fig. 2
Fig. 2 Illustration of the inner product measurement scheme using a 2f-setup. The correlation of an incoming beam with the hologram pattern (H) results in a correlation signal I at the optical axis in the back focal plane of a lens L.

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Fig. 3
Fig. 3 Modal decomposition into adapted and non-adapted basis sets regarding scale. (a) Modal decomposition into LGp,±4 modes of adapted basis scale w0. (b) Decomposition into LGp,±4 modes with scale 0.75 w0, (c) 2w0, and (d) 3w0. Inset in (b) depicts the measured beam intensity.

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Fig. 4
Fig. 4 Influence of basis set scale on mode spectrum. (a) Relative power ρ2 of mode LG0,±4, measured (me) and simulated (sim), as a function of normalised beam radius w/w0. (b) Simulated power spectrum of LGp,±4 modes (p = 0 ... 20) as a function of nor-malised beam radius w/w0. Inset in (a) depicts corresponding beam intensity.

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Fig. 5
Fig. 5 Reconstruction of the beam by modal decomposition into LGp,l modes of previously determined scale. (a) Modal power spectrum (total power normalised to one). (b) Modal phases. (c) Measured intensity (Me). (d) Reconstructed intensity (Re).

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Fig. 6
Fig. 6 Modal decomposition after determination of correct basis set scale of (a) a Laguerre-Gaussian LG1,0 beam, and (b) of a superposition of an 8-petal beam and a LG1,0 beam. Insets depict corresponding beam intensities.

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Tables (1)

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Table 1 Diameter and M2 of measured and reconstructed intensity.

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Equations (6)

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U ( r ) = ∑ n = 1 n max c n ψ n ( r ) .
〈 ψ n | ψ m 〉 = ∬ ℝ 2 d 2 r ψ n * ( r ) ψ m ( r ) = δ n m ,
c n = ρ n exp ( i Δ ϕ n ) = 〈 ψ n | U 〉 .
LG p l ( r ; w 0 ) = 2 p ! π w 0 2 ( p + | l | ) ! ( 2 r w 0 ) | l | L p | l | ( 2 r 2 w 0 2 ) exp ( − r 2 w 0 2 ) exp ( i l ϕ )
U ( r ) = ∑ p l c p l a LG p l ( r ; w a ) = ∑ p l c p l b LG p l ( r ; w b )
w 0 = w / M 2 ,

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