Abstract

We model the spatial coherence of broad-area laser diodes (BALDs) by representing the mutual intensity as superpositions of individually fully coherent but mutually uncorrelated fields. Consideration of spectroscopic modal structure measurements and intensity-based mode recovery shows that the standard Mercer-type coherent-mode expansion can lead to unsatisfactory results for real BALDs. However, we show that a so-called shifted elementary-field method provides a sufficiently accurate tool for spatial coherence and propagation modeling even if the modal structure of the BALD is severely distorted.

© 2013 Optical Society of America

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  4. J. Turunen, E. Tervonen, and A. T. Friberg, “Coherence-theoretic algorithm to determine the transverse mode structure of lasers,” Opt. Lett. 14, 627–629 (1989).
    [CrossRef]
  5. E. Tervonen, J. Turunen, and A. T. Friberg, “Transverse mode structure determination from spatial coherence measurements: experimental results,” Appl. Phys. B 49, 409–414 (1989).
    [CrossRef]
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  7. M. Santarsiero, F. Gori, R. Borghi, and G. Guattari, “Evaluation of the modal structure for light beams with Hermite–Gaussian modes,” Appl. Opt. 38, 5272–5281 (1999).
    [CrossRef]
  8. R. Borghi, G. Piquero, and M. Santarsiero, “Use of biorthogonal functions for the modal decomposition of multimode beams,” Opt. Commun. 194, 235–242 (2001).
    [CrossRef]
  9. R. Borghi, G. Guattari, F. de la Torre, F. Gori, and M. Santarsiero, “Evaluation of the spatial coherence of a light beam through transverse intensity measurements,” J. Opt. Soc. Am. A 20, 1763–1770 (2003).
    [CrossRef]
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    [CrossRef]
  11. N. Stelmakh, “Harnessing multimode broad-area laser-diode emission into a single-lobe diffraction-limited spot,” IEEE Photon. Technol. Lett. 19, 1392–1394 (2007).
    [CrossRef]
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    [CrossRef]
  13. F. Gori, “Directionality and partial coherence,” Opt. Acta 27, 1025–1034 (1980).
    [CrossRef]
  14. P. Vahimaa and J. Turunen, “Finite-elementary-source model for partially coherent radiation,” Opt. Express 14, 1376–1381 (2006).
    [CrossRef]
  15. J. Tervo, J. Turunen, P. Vahimaa, and F. Wyrowski, “Shifted-elementary-mode representation for partially coherent vectorial fields,” J. Opt. Soc. Am. A 27, 2004–2014 (2010).
    [CrossRef]
  16. J. Turunen, “Elementary-field representations in partially coherent optics,” J. Mod. Opt. 58, 509–527 (2011).
    [CrossRef]
  17. A. Greynolds, “Propagation of generally astigmatic Gaussian beams along skew ray paths,” Proc. SPIE 560, 33–50 (1985).
    [CrossRef]
  18. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).
  19. P. Spano, “Connection between spatial coherence and mode structure in optical fibers and semiconductor lasers,” Opt. Commun. 33, 265–270 (1980).
    [CrossRef]
  20. F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
    [CrossRef]
  21. J. W. Goodman, Statistical Optics (Wiley, 1985).
  22. J. Turunen and P. Vahimaa, “Independent-elementary-field model for three-dimensional spatially partially coherent sources,” Opt. Express 16, 6433–6442 (2008).
    [CrossRef]

2011

J. Turunen, “Elementary-field representations in partially coherent optics,” J. Mod. Opt. 58, 509–527 (2011).
[CrossRef]

2010

2008

2007

N. Stelmakh, “Harnessing multimode broad-area laser-diode emission into a single-lobe diffraction-limited spot,” IEEE Photon. Technol. Lett. 19, 1392–1394 (2007).
[CrossRef]

2006

N. Stelmakh and M. Flowers, “Measurement of spatial modes of broad-area diode lasers with 1 GHz resolution grating spectrometer,” IEEE Photon. Technol. Lett. 18, 1618–1620 (2006).
[CrossRef]

P. Vahimaa and J. Turunen, “Finite-elementary-source model for partially coherent radiation,” Opt. Express 14, 1376–1381 (2006).
[CrossRef]

2005

2003

2001

R. Borghi, G. Piquero, and M. Santarsiero, “Use of biorthogonal functions for the modal decomposition of multimode beams,” Opt. Commun. 194, 235–242 (2001).
[CrossRef]

1999

1998

1989

J. Turunen, E. Tervonen, and A. T. Friberg, “Coherence-theoretic algorithm to determine the transverse mode structure of lasers,” Opt. Lett. 14, 627–629 (1989).
[CrossRef]

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

1985

A. Greynolds, “Propagation of generally astigmatic Gaussian beams along skew ray paths,” Proc. SPIE 560, 33–50 (1985).
[CrossRef]

1984

1982

1980

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

F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[CrossRef]

F. Gori, “Directionality and partial coherence,” Opt. Acta 27, 1025–1034 (1980).
[CrossRef]

1978

F. Gori and C. Palma, “Partially coherent sources which give rise to highly directional fields,” Opt. Commun. 27, 185–188 (1978).
[CrossRef]

Borghi, R.

de la Torre, F.

Flowers, M.

N. Stelmakh and M. Flowers, “Measurement of spatial modes of broad-area diode lasers with 1 GHz resolution grating spectrometer,” IEEE Photon. Technol. Lett. 18, 1618–1620 (2006).
[CrossRef]

Friberg, A. T.

Goodman, J. W.

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

Gori, F.

Greynolds, A.

A. Greynolds, “Propagation of generally astigmatic Gaussian beams along skew ray paths,” Proc. SPIE 560, 33–50 (1985).
[CrossRef]

Guattari, G.

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Palma, C.

F. Gori and C. Palma, “Partially coherent sources which give rise to highly directional fields,” Opt. Commun. 27, 185–188 (1978).
[CrossRef]

Piquero, G.

R. Borghi, G. Piquero, and M. Santarsiero, “Use of biorthogonal functions for the modal decomposition of multimode beams,” Opt. Commun. 194, 235–242 (2001).
[CrossRef]

Saastamoinen, T.

Santarsiero, M.

Setälä, T.

Spano, P.

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

Stelmakh, N.

N. Stelmakh, “Harnessing multimode broad-area laser-diode emission into a single-lobe diffraction-limited spot,” IEEE Photon. Technol. Lett. 19, 1392–1394 (2007).
[CrossRef]

N. Stelmakh and M. Flowers, “Measurement of spatial modes of broad-area diode lasers with 1 GHz resolution grating spectrometer,” IEEE Photon. Technol. Lett. 18, 1618–1620 (2006).
[CrossRef]

Tervo, J.

Tervonen, E.

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

J. Turunen, E. Tervonen, and A. T. Friberg, “Coherence-theoretic algorithm to determine the transverse mode structure of lasers,” Opt. Lett. 14, 627–629 (1989).
[CrossRef]

Turunen, J.

Vahimaa, P.

Wolf, E.

Wyrowski, F.

Appl. Opt.

Appl. Phys. B

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

IEEE Photon. Technol. Lett.

N. Stelmakh and M. Flowers, “Measurement of spatial modes of broad-area diode lasers with 1 GHz resolution grating spectrometer,” IEEE Photon. Technol. Lett. 18, 1618–1620 (2006).
[CrossRef]

N. Stelmakh, “Harnessing multimode broad-area laser-diode emission into a single-lobe diffraction-limited spot,” IEEE Photon. Technol. Lett. 19, 1392–1394 (2007).
[CrossRef]

J. Mod. Opt.

J. Turunen, “Elementary-field representations in partially coherent optics,” J. Mod. Opt. 58, 509–527 (2011).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Acta

F. Gori, “Directionality and partial coherence,” Opt. Acta 27, 1025–1034 (1980).
[CrossRef]

Opt. Commun.

F. Gori and C. Palma, “Partially coherent sources which give rise to highly directional fields,” Opt. Commun. 27, 185–188 (1978).
[CrossRef]

R. Borghi, G. Piquero, and M. Santarsiero, “Use of biorthogonal functions for the modal decomposition of multimode beams,” Opt. Commun. 194, 235–242 (2001).
[CrossRef]

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

F. Gori, “Collett–Wolf sources and multimode lasers,” Opt. Commun. 34, 301–305 (1980).
[CrossRef]

Opt. Express

Opt. Lett.

Proc. SPIE

A. Greynolds, “Propagation of generally astigmatic Gaussian beams along skew ray paths,” Proc. SPIE 560, 33–50 (1985).
[CrossRef]

Other

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

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

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Figures (9)

Fig. 1.
Fig. 1.

Cavity of an ideal BALD with exit aperture dimensions Lx×Ly (where LxLy) and cavity length Lz.

Fig. 2.
Fig. 2.

Partial view of the measured resonator mode structure in the x direction at the image plane of the source (top) and at the Fourier plane (bottom). The graphs on the right show the total intensity distributions at these planes. The detailed mode structure and the intensity profiles depend on the drive current.

Fig. 3.
Fig. 3.

Top: coefficients Im obtained with the recovery method. Bottom: measured intensity at the source plane (solid red), reconstructed intensity distribution with negative coefficients included (dashed blue), and reconstructed intensity with only 12 lowest-order coefficients included (dash–dotted black).

Fig. 4.
Fig. 4.

Intensity (left) and degree of coherence (right) at different distances from a 15-mode cavity with equal modal weights for fields given by the coherent-mode expansion (solid black), the forced Schell model (dashed red), and the forced quasi-homogeneous model (dash–dotted blue).

Fig. 5.
Fig. 5.

Same as in Fig. 4 at distance z=0.5mm, but with Gaussian weighting Im=I0exp(m2/152).

Fig. 6.
Fig. 6.

Measured (solid red) and simulated (dashed blue) intensities for three propagation distances: 0 mm (top), 20 mm (center), and 60 mm (bottom). The driving current of the source is 500 mA, which is under the lasing threshold. The curves for the single elementary field (dash–dotted black) are scaled to help the comparison.

Fig. 7.
Fig. 7.

Same as Fig. 6 but with driving current 1100 mA leading to lasing.

Fig. 8.
Fig. 8.

Measured (left) and simulated (right) intensity propagation, with driving current 500 mA.

Fig. 9.
Fig. 9.

Same as Fig. 8 but with driving current 1100 mA.

Equations (32)

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J(S)(x1,x2)=mImvm(S)*(x1)vm(S)(x2),
I(S)(x)=J(S)(x,x)=mIm|vm(S)(x)|2,
J(x1,x2,z)=mImvm*(x1,z)vm(x2,z),
vm(x,z)=am(kx)exp[i(kxx+kzz)]dkx,
am(kx)=12πvm(S)(x)exp(ikxx)dx
J(S)(x1,x2)=p(x)e(S)*(x1x)e(S)(x2x)dx,
I(S)(x)=p(x)|e(S)(xx)|2dx,
J(x1,x2,z)=A(kx1,kx2)exp[i(kx2x2kx1x1)]exp[i(kz2kz1*)z]dkx1dkx2,
A(kx1,kx2)=1(2π)2J(S)(x1,x2)exp[i(kx1x1kx2x2)]dx1dx2.
e(S)(x)=f(kx)exp(ikxx)dkx
p(x)=g(Δkx)exp(iΔkxx)dΔkx,
A(kx1,kx2)=2πg(Δkx)f*(kx1)f(kx2).
J(x1,x2,z)=p(x)e*(x1x,z)e(x2x,z)dx,
e(x,z)=f(kx)exp[i(kxx+kzz)]dkx
J(F)(u1,u2)=(2π)2(k0/F)g(k0Δu/F)f*(k0u1/F)f(k0u2/F).
I(F)(u)=(2π)2(k0/F)g(0)|f(k0u/F)|2.
e(S)(x)=C[I(F)(u)]1/2exp(ik0xu/F)du,
J(x1,x2,z)=p(x¯)p(0)e*(x12Δx,z)e(x+12Δx,z)dx,
I(I)(ξ)=J(I)(ξ,ξ)=1|M|p(ξ/M)p(0)|e(S)(x)|2dx.
vm(S)(x)=(2/Lxsin(πmx/Lx)when|x|Lx/2andmiseven2/Lxcos(πmx/Lx)when|x|Lx/2andmisodd0otherwise
am(kx)=(im(1)m/22Lxsin(kxLx/2)π2m2kx2Lx2whenmiseven,m(1)(m1)/22Lxcos(kxLx/2)π2m2kx2Lx2whenmisodd.
Fn(x)=2(1)ncos(2πnx/Lx),
Lx/2Lx/2Fn(x)|vm(S)(x)|2dx=δmn.
In=2(1)nLx/2Lx/2cos(2πnx/Lx)I(S)(x)dx.
A(kx1,kx2)=mImam*(kx1)am(kx2),
γ(kx1,kx2)=A(kkx1,kx2)A(kx1,kx1)A(kx2,kx2).
γ(kx1,kx2)=γ(k¯xΔkx/2,k¯x+Δkx/2).
γ(kx1,kx2)=γ(Δkx/2,Δkx/2).
A(kx1,kx2)=γ(Δkx/2,Δkx/2)A(kx1,kx1)A(kkx2,kx2).
g(Δkx)=γ(Δkx/2,Δkx/2),
f(kx)=A(kx,kx).
μ(x1,x2,z)=J(x1,x2,z)I(x1,z)I(x2,z)

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