Abstract

We explain a technique that extracts both the structure and the modal weights of spatial modes of lasers by analyzing the spatial coherence of the beam. This is the first time, to our knowledge, that an experimental method is being used to measure arbitrary forms of the spatial modes. We applied this method to an edge-emitting Fabry–Perot semiconductor laser with a stripe width of 5 µm and extracted fundamental and first-order lateral modes with relative power weights of 96.2% and 3.8%. There was a single transverse mode.

© 2000 Optical Society of America

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References

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1999 (1)

1998 (1)

1996 (1)

1995 (3)

A. Cutolo, T. Isernia, I. Izzo, R. Pierri, L. Zeni, “Transverse mode analysis of a laser beam by near- and far-field intensity measurements,” Appl. Opt. 34, 7974–7978 (1995).
[CrossRef] [PubMed]

A. Liesenhoff, F. Rühl, “An interferometric method of laser beam analysis,” Rev. Sci. Instrum. 38, 4059–4065 (1995).
[CrossRef]

L. J. Pelz, B. L. Anderson, “Practical use of the spatial coherence function for determining laser transverse mode structure,” Opt. Eng. 34, 3323–3328 (1995).
[CrossRef]

1993 (3)

1992 (1)

D. Wright, P. Greve, J. Fleischer, L. Austin, “Laser beam width, divergence and beam propagation factor–an international standardization approach,” Opt. Quantum Electron. 24, S993–S1000 (1992).
[CrossRef]

1990 (1)

1989 (1)

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

1985 (1)

A. A. Maciejewski, C. A. Klein, “Obstacle avoidance for kinematically redundant manipulators in dynamically varying environments,” Int. J. Robotics Res. 4, 109–117 (1985).
[CrossRef]

1984 (1)

1982 (1)

1980 (1)

P. Spano, “Connection between spatial coherence and modal structure in optical fibers and semiconductor lasers,” Opt. Commun. 33, 265–270 (1980).
[CrossRef]

1975 (1)

W. P. Dumke, “The angular beam divergence in double-heterojunction lasers with very thin active regions,” IEEE J. Quantum Electron. 11, 400–402 (1975).
[CrossRef]

1966 (1)

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

1961 (1)

A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).
[CrossRef]

Agarwal, G. S.

Anderson, B. L.

L. J. Pelz, B. L. Anderson, “Practical use of the spatial coherence function for determining laser transverse mode structure,” Opt. Eng. 34, 3323–3328 (1995).
[CrossRef]

Austin, L.

D. Wright, P. Greve, J. Fleischer, L. Austin, “Laser beam width, divergence and beam propagation factor–an international standardization approach,” Opt. Quantum Electron. 24, S993–S1000 (1992).
[CrossRef]

Borghi, R.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1980).

Cutolo, A.

Dumke, W. P.

W. P. Dumke, “The angular beam divergence in double-heterojunction lasers with very thin active regions,” IEEE J. Quantum Electron. 11, 400–402 (1975).
[CrossRef]

Durst, F.

Fleischer, J.

D. Wright, P. Greve, J. Fleischer, L. Austin, “Laser beam width, divergence and beam propagation factor–an international standardization approach,” Opt. Quantum Electron. 24, S993–S1000 (1992).
[CrossRef]

Fox, A. G.

A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).
[CrossRef]

Friberg, A. T.

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

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

Gori, F.

Greve, P.

D. Wright, P. Greve, J. Fleischer, L. Austin, “Laser beam width, divergence and beam propagation factor–an international standardization approach,” Opt. Quantum Electron. 24, S993–S1000 (1992).
[CrossRef]

Guattari, G.

Iaconis, C.

Isernia, T.

Izzo, I.

Klein, C. A.

A. A. Maciejewski, C. A. Klein, “Obstacle avoidance for kinematically redundant manipulators in dynamically varying environments,” Int. J. Robotics Res. 4, 109–117 (1985).
[CrossRef]

Kogelnik, H.

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Li, T.

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).
[CrossRef]

Liesenhoff, A.

A. Liesenhoff, F. Rühl, “An interferometric method of laser beam analysis,” Rev. Sci. Instrum. 38, 4059–4065 (1995).
[CrossRef]

Maciejewski, A. A.

A. A. Maciejewski, C. A. Klein, “Obstacle avoidance for kinematically redundant manipulators in dynamically varying environments,” Int. J. Robotics Res. 4, 109–117 (1985).
[CrossRef]

Naqwi, A.

Pelz, L. J.

L. J. Pelz, B. L. Anderson, “Practical use of the spatial coherence function for determining laser transverse mode structure,” Opt. Eng. 34, 3323–3328 (1995).
[CrossRef]

Pierri, R.

Rühl, F.

A. Liesenhoff, F. Rühl, “An interferometric method of laser beam analysis,” Rev. Sci. Instrum. 38, 4059–4065 (1995).
[CrossRef]

Santarsiero, M.

Siegman, A. E.

A. E. Siegman, S. W. Townsend, “Output beam propagation and beam quality from a multimode stable-cavity laser,” IEEE J. Quantum Electron. 29, 1212–1217 (1993).
[CrossRef]

Spano, P.

P. Spano, “Connection between spatial coherence and modal structure in optical fibers and semiconductor lasers,” Opt. Commun. 33, 265–270 (1980).
[CrossRef]

Tervonen, E.

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

Townsend, S. W.

A. E. Siegman, S. W. Townsend, “Output beam propagation and beam quality from a multimode stable-cavity laser,” IEEE J. Quantum Electron. 29, 1212–1217 (1993).
[CrossRef]

Turunen, J.

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

Walmsley, I. A.

Wolf, E.

Wright, D.

D. Wright, P. Greve, J. Fleischer, L. Austin, “Laser beam width, divergence and beam propagation factor–an international standardization approach,” Opt. Quantum Electron. 24, S993–S1000 (1992).
[CrossRef]

Zeng, X.

Zeni, L.

Appl. Opt. (4)

Appl. Phys. B (1)

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

Bell Syst. Tech. J. (1)

A. G. Fox, T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40, 453–488 (1961).
[CrossRef]

IEEE J. Quantum Electron. (2)

A. E. Siegman, S. W. Townsend, “Output beam propagation and beam quality from a multimode stable-cavity laser,” IEEE J. Quantum Electron. 29, 1212–1217 (1993).
[CrossRef]

W. P. Dumke, “The angular beam divergence in double-heterojunction lasers with very thin active regions,” IEEE J. Quantum Electron. 11, 400–402 (1975).
[CrossRef]

Int. J. Robotics Res. (1)

A. A. Maciejewski, C. A. Klein, “Obstacle avoidance for kinematically redundant manipulators in dynamically varying environments,” Int. J. Robotics Res. 4, 109–117 (1985).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Opt. Commun. (1)

P. Spano, “Connection between spatial coherence and modal structure in optical fibers and semiconductor lasers,” Opt. Commun. 33, 265–270 (1980).
[CrossRef]

Opt. Eng. (1)

L. J. Pelz, B. L. Anderson, “Practical use of the spatial coherence function for determining laser transverse mode structure,” Opt. Eng. 34, 3323–3328 (1995).
[CrossRef]

Opt. Lett. (2)

Opt. Quantum Electron. (1)

D. Wright, P. Greve, J. Fleischer, L. Austin, “Laser beam width, divergence and beam propagation factor–an international standardization approach,” Opt. Quantum Electron. 24, S993–S1000 (1992).
[CrossRef]

Proc. IEEE (1)

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Rev. Sci. Instrum. (1)

A. Liesenhoff, F. Rühl, “An interferometric method of laser beam analysis,” Rev. Sci. Instrum. 38, 4059–4065 (1995).
[CrossRef]

Other (2)

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1980).

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

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Figures (7)

Fig. 1
Fig. 1

Intensity profiles of a Gaussian beam (solid curve) and a beam with 95% power in the fundamental mode and 5% in the first-order Hermite–Gaussian mode (dashed curve).

Fig. 2
Fig. 2

Twin-fiber interferometer sampling a beam on the left, with the detector at the top right. A polarization controller (PC) is on one arm of the interferometer and a piezoelectric stretcher (PZS) is on the other arm.

Fig. 3
Fig. 3

Input ends of the fibers are attached to metal plates that are mounted on computer-controlled xyz translation stages.

Fig. 4
Fig. 4

For each row of data in the correlation matrix, one input is kept at a constant x 1, and the other input steps through the range of positions for x 2.

Fig. 5
Fig. 5

Single transverse spatial mode extracted by the Jacobian method from measured data.

Fig. 6
Fig. 6

Lateral spatial modes extracted by the Jacobian method from measured data indicated by +’s. Exact Hermite–Gaussian modes are shown for comparison with a dashed curve. The actual physical modes resemble, but are not exactly, Hermite–Gaussian.

Fig. 7
Fig. 7

Noise simulations with 50 realizations superimposed. Background noise is 0% in (c) and (e), ±0.5% in (a), and ±1% in (b), (d), and (f). Intensity-dependent noise is 0% in (a) and (b), ±2.5% in (c) and (d), and ±5% in (e) and (f).

Tables (1)

Tables Icon

Table 1 Relative Weight of the First-Order Mode as Intensity-Dependent Noise and Background Noise are Varieda

Equations (18)

Equations on this page are rendered with MathJax. Learn more.

ψmnx, y, z=ϕmx, zϕny, zexpjαmnx, y, z,
Γ12τ=EP1, t+τE*P2, t,
Γ12τJ12 exp-j2πν0τ,
J12=m λmψm*x1ψmx2,
J12=J12r expjβ12.
J12r=m λmϕmx1ϕmx2,
- J12rϕmx2dx=λmϕmx1
ID=K1Ex1, t+τ+K2Ex2, t×K1Ex1, t+τ+K2Ex2, t*,
ID=I1+I2+2|K1K2J12r|cos2πν0τ-γ12,
|J12r|=Imax-Imin/4|K1K2|,
R=ΦΛΦT,
JΔv=r=fv+Δv-fv,
Δv=J+r+I-J+Jz,
Δv=Jo+ro+JmI-Jo+Jo+rm-JmJo+ro.
0=v1+Δv1Tv2+Δv2=v1Tv2+Δv1Tv2+v1TΔv2+Δv1TΔv2.
JoΔv=v2Tv1T0v3T0v1T0v3Tv2TΔv1Δv2Δv3=-v1Tv2-v1Tv3-v2Tv3=ro.
Ri, j=k=1n vkivkj.
Jmh, k-1q+i=vkj,Jmh, k-1q+j=vki.

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