Abstract

A detailed study of the coherent and the incoherent combinations of two-dimensional off-axis Hermite–Gaussian beams with rectangular symmetry is made. The closed-form propagation formulas of the resulting beam are derived, and the resulting beam quality in terms of the M 2 factor and power in the bucket is discussed and compared for the coherent and the incoherent combinations. In addition, it is shown that the resulting astigmatic beam can be symmetrized in the sense of the second-moment definition of beam width. However, the symmetrizing transformation of the resulting astigmatic beams is incomplete, because there exist different irradiance profiles.

© 2000 Optical Society of America

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References

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  1. R. A. Chodzko, J. M. Bernard, H. Mirels, “Coherent combination of multiline lasers,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE1224, 239–253 (1990).
    [CrossRef]
  2. B. Ozygus, H. Laabs, “Gain-induced coupling of solid-state lasers,” Chaos, Solitons, Fractals 4, 1559–1572 (1994).
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    [CrossRef] [PubMed]
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  8. G. L. Schuster, J. R. Andrews, “Coherent beam combining: optical loss effects on power scaling,” Appl. Opt. 34, 6801–6805 (1995).
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  9. A. E. Siegman, “How to (maybe) measure laser beam quality,” in Diode Pumped Solid State Lasers: Applications and Issues, M. W. Dowley, ed., Vol. 17 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1998), pp. 184–199.
  10. A. E. Siegman, “New development in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE1224, 2–14 (1990).
    [CrossRef]
  11. S. A. Collins, “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. 60, 1168–1177 (1970).
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  13. H. Weber, “Some historical and technical aspects of beam quality,” Opt. Quant. Electron. 24, 861–864 (1992).
    [CrossRef]
  14. D. Zhao, Q. Lin, S. Wang, “Symmetrizing transformation of general astigmatic Gaussian beams,” Opt. Quantum Electron. 26, 903–910 (1994).
  15. B. Eppich, A. T. Friberg, C. Gao, H. Weber, “Twist of coherent field and beam quality,” in Third International Workshop on Laser Beam and Optics Characterization, M. Morin, A. Giesen, eds., Proc. SPIE2870, 260–267 (1996).
    [CrossRef]
  16. P. A. Belanger, “Beam propagation and the ABCD ray matrix,” Opt. Lett. 16, 196–198 (1991).
    [CrossRef]

1998 (2)

1997 (1)

1995 (1)

1994 (2)

B. Ozygus, H. Laabs, “Gain-induced coupling of solid-state lasers,” Chaos, Solitons, Fractals 4, 1559–1572 (1994).

D. Zhao, Q. Lin, S. Wang, “Symmetrizing transformation of general astigmatic Gaussian beams,” Opt. Quantum Electron. 26, 903–910 (1994).

1993 (1)

1992 (1)

H. Weber, “Some historical and technical aspects of beam quality,” Opt. Quant. Electron. 24, 861–864 (1992).
[CrossRef]

1991 (2)

1989 (1)

1970 (1)

Andrews, J. R.

Belanger, P. A.

Bernard, J. M.

R. A. Chodzko, J. M. Bernard, H. Mirels, “Coherent combination of multiline lasers,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE1224, 239–253 (1990).
[CrossRef]

Bretenaker, F.

Brunel, M.

Capjack, C. E.

Chodzko, R. A.

R. A. Chodzko, J. M. Bernard, H. Mirels, “Coherent combination of multiline lasers,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE1224, 239–253 (1990).
[CrossRef]

Collins, S. A.

Eppich, B.

B. Eppich, A. T. Friberg, C. Gao, H. Weber, “Twist of coherent field and beam quality,” in Third International Workshop on Laser Beam and Optics Characterization, M. Morin, A. Giesen, eds., Proc. SPIE2870, 260–267 (1996).
[CrossRef]

Floch, A. L.

Friberg, A. T.

B. Eppich, A. T. Friberg, C. Gao, H. Weber, “Twist of coherent field and beam quality,” in Third International Workshop on Laser Beam and Optics Characterization, M. Morin, A. Giesen, eds., Proc. SPIE2870, 260–267 (1996).
[CrossRef]

Gao, C.

B. Eppich, A. T. Friberg, C. Gao, H. Weber, “Twist of coherent field and beam quality,” in Third International Workshop on Laser Beam and Optics Characterization, M. Morin, A. Giesen, eds., Proc. SPIE2870, 260–267 (1996).
[CrossRef]

Laabs, H.

B. Ozygus, H. Laabs, “Gain-induced coupling of solid-state lasers,” Chaos, Solitons, Fractals 4, 1559–1572 (1994).

Lin, Q.

D. Zhao, Q. Lin, S. Wang, “Symmetrizing transformation of general astigmatic Gaussian beams,” Opt. Quantum Electron. 26, 903–910 (1994).

Marty, J.

Mirels, H.

R. A. Chodzko, J. M. Bernard, H. Mirels, “Coherent combination of multiline lasers,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE1224, 239–253 (1990).
[CrossRef]

Molva, E.

Ozygus, B.

B. Ozygus, H. Laabs, “Gain-induced coupling of solid-state lasers,” Chaos, Solitons, Fractals 4, 1559–1572 (1994).

Palma, C.

Schuster, G. L.

Seguin, H. J. J.

Siegman, A. E.

A. E. Siegman, “How to (maybe) measure laser beam quality,” in Diode Pumped Solid State Lasers: Applications and Issues, M. W. Dowley, ed., Vol. 17 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1998), pp. 184–199.

A. E. Siegman, “New development in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE1224, 2–14 (1990).
[CrossRef]

Strohschein, J. D.

Wang, S.

D. Zhao, Q. Lin, S. Wang, “Symmetrizing transformation of general astigmatic Gaussian beams,” Opt. Quantum Electron. 26, 903–910 (1994).

Weber, H.

H. Weber, “Some historical and technical aspects of beam quality,” Opt. Quant. Electron. 24, 861–864 (1992).
[CrossRef]

B. Eppich, A. T. Friberg, C. Gao, H. Weber, “Twist of coherent field and beam quality,” in Third International Workshop on Laser Beam and Optics Characterization, M. Morin, A. Giesen, eds., Proc. SPIE2870, 260–267 (1996).
[CrossRef]

Zhao, D.

D. Zhao, Q. Lin, S. Wang, “Symmetrizing transformation of general astigmatic Gaussian beams,” Opt. Quantum Electron. 26, 903–910 (1994).

Appl. Opt. (4)

Chaos, Solitons, Fractals (1)

B. Ozygus, H. Laabs, “Gain-induced coupling of solid-state lasers,” Chaos, Solitons, Fractals 4, 1559–1572 (1994).

J. Opt. Soc. Am. (1)

Opt. Lett. (4)

Opt. Quant. Electron. (1)

H. Weber, “Some historical and technical aspects of beam quality,” Opt. Quant. Electron. 24, 861–864 (1992).
[CrossRef]

Opt. Quantum Electron. (1)

D. Zhao, Q. Lin, S. Wang, “Symmetrizing transformation of general astigmatic Gaussian beams,” Opt. Quantum Electron. 26, 903–910 (1994).

Other (4)

B. Eppich, A. T. Friberg, C. Gao, H. Weber, “Twist of coherent field and beam quality,” in Third International Workshop on Laser Beam and Optics Characterization, M. Morin, A. Giesen, eds., Proc. SPIE2870, 260–267 (1996).
[CrossRef]

R. A. Chodzko, J. M. Bernard, H. Mirels, “Coherent combination of multiline lasers,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE1224, 239–253 (1990).
[CrossRef]

A. E. Siegman, “How to (maybe) measure laser beam quality,” in Diode Pumped Solid State Lasers: Applications and Issues, M. W. Dowley, ed., Vol. 17 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1998), pp. 184–199.

A. E. Siegman, “New development in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE1224, 2–14 (1990).
[CrossRef]

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Figures (14)

Fig. 1
Fig. 1

2D model of the P × Q off-axis Hermite–Gaussian beam array.

Fig. 2
Fig. 2

Relative irradiance distribution of the resulting beam focused by a lens for the incoherent combination. (a) I(x′, y′, Δz)/I(0, 0, Δz) versus x′ and y′, (b) I(x′, y′, Δz)/I(0, 0, Δz) versus x′, (c) I(x′, y′, Δz)/I(0, 0, Δz) versus y′. The calculation parameters are P = Q = 5, N f = 5, M = 4, xd = yd = 3.0, and Δz = -0.9.

Fig. 3
Fig. 3

Relative irradiance distribution of the resulting beam focused by a lens for the coherent combination. (a) I(x′, y′, Δz)/I(0, 0, Δz) versus x′ and y′, (b) I(x′, y′, Δz)/I(0, 0, Δz) versus x′, (c) I(x′, y′, Δz)/I(0, 0, Δz) versus y′. The calculation parameters are the same as those in Fig. 2.

Fig. 4
Fig. 4

Relative irradiance distribution of the resulting beam focused by a lens for the incoherent combination. (a) I(x′, y′, Δz)/I(0, 0, Δz) versus x′ and y′, (b) I(x′, y′, Δz)/I(0, 0, Δz) versus x′, (c) I(x′, y′, Δz)/I(0, 0, Δz) versus y′. The calculation parameters are the same as those in Fig. 2, except that Δz = -0.014.

Fig. 5
Fig. 5

Relative irradiance distribution of the resulting beam focused by a lens for the coherent combination. (a) I(x′, y′, Δz)/I(0, 0, Δz) versus x′ and y′, (b) I(x′, y′, Δz)/I(0, 0, Δz) versus x′, (c) I(x′, y′, Δz)/I(0, 0, Δz) versus y′. The calculation parameters are the same as those in Fig. 4.

Fig. 6
Fig. 6

Relative irradiance distribution of the resulting beam focused by a lens for the incoherent combination. (a) I(x′, y′, Δz)/I(0, 0, Δz) versus x′ and y′, (b) I(x′, y′, Δz)/I(0, 0, Δz) versus x′, (c) I(x′, y′, Δz)/I(0, 0, Δz) versus y′. The calculation parameters are the same as those in Fig. 2, except that Δz = 0.

Fig. 7
Fig. 7

Relative irradiance distribution of the resulting beam focused by a lens for the coherent combination. (a) I(x′, y′, Δz)/I(0, 0, Δz) versus x′ and y′, (b) I(x′, y′, Δz)/I(0, 0, Δz) versus x′, (c) I(x′, y′, Δz)/I(0, 0, Δz) versus y′. The calculation parameters are the same as those in Fig. 6.

Fig. 8
Fig. 8

(a) M x 2 factor of the resulting beam as a function of normalized distance xd. (b) M y 2 factor of the resulting beam as a function of normalized distance yd. The calculation parameters are P = Q = 5, M = 4. Solid curve, coherent combination; dashed curve, incoherent combination.

Fig. 9
Fig. 9

(a) M x 2 factor of the resulting beam versus beam number P, M = 3, xd = yd = 2.0. (b) M x 2 factor of the resulting beam versus beam number P, M = 4, xd = yd = 2.0. (c) M y 2 factor of the resulting beam versus beam number Q, M = 0, xd = yd = 2.0. Solid curve, coherent combination; dashed curve, incoherent combination.

Fig. 10
Fig. 10

PIB curves are plotted against normalized bucket’s size x2 for the coherent (solid curve) and the incoherent (dashed curve) cases; the aspect ratio is x′/y′ = 1; w 0 = 1 mm and N f = 5. (a) M = 3, xd = yd = 3, (b) M = 4, xd = yd = 3.

Fig. 11
Fig. 11

Beam widths W z (z) (solid curve) and W x (z) (dashed curve) of the resulting beam vary with the propagation distance z. The calculation parameters are P = Q = 5, w 0 = 1 mm, M = 3, xd = yd = 3, f x = 1000 mm: (a) For the incoherent case, f y = 911 mm; (b) for the coherent case, f y = 1123 mm.

Fig. 12
Fig. 12

Relative irradiance distribution of the resulting beam as the place of z = f x = 1000 mm for the incoherent combination. (a) I(x′, y′, Δz)/I(0, 0, Δz) versus x′ and y′, (b) I(x′, y′, Δz)/I(0, 0, Δz) versus x′, (c) I(x′, y′, Δz)/I(0, 0, Δz) versus y′. The calculation parameters are P = Q = 5, w 0 = 1.0 mm, M = 3, xd = yd = 3.0, and f y = 911 mm.

Fig. 13
Fig. 13

Relative irradiance distribution of the resulting beam at the place of z = f x = 1000 mm for the coherent combination. (a) I(x′, y′, Δz)/I(0, 0, Δz) versus x′ and y′, (b) I(x′, y′, Δz)/I(0, 0, Δz) versus x′, (c) I(x′, y′, Δz)/I(0, 0, Δz) versus y′. The calculation parameters are the same as those in Fig. 12, except that f y = 1123 mm.

Fig. 14
Fig. 14

Relative irradiance distribution of the resulting beam at the place of z = 695 mm for the coherent combination. (a) I(x′, y′, Δz)/I(0, 0, Δz) versus x′ and y′, (b) I(x′, y′, Δz)/I(0, 0, Δz) versus x′, (c) I(x′, y′, Δz)/I(0, 0, Δz) versus y′. The calculation parameters are the same as those in Fig. 13.

Equations (46)

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Epqx, y, 0=Hm2x-pxdw0×exp-x-pxd2+y-qyd2w02, p-P-12, P-12,  q-Q-12, Q-12,
ABCD
Epqx1, y1, z=iλB  Epqx, y, 0×exp-ik2BAx2+y2-2xx1+yy1+Dx12+y12dxdy,
k=2π/λ.
Epqx1, y1, z=1A+B/q0A/Bq0-1A/Bq0+1m/2×Hm2x1-x1dwxz×exp-ik2q1x1-x1d2+y1-y1d2×exp-ik1xx1+1yy1+iϕx+ϕy,
1q1z=C+D/q0A+B/q0,
q0=iπw02λ,
x1d=Apxd,  1x=Cpxd,  ϕx=kAC2 p2xd2;
y1d=Aqyd,  1y=Cqyd,  ϕy=kAC2 q2yd2;
w1z=w0A2-B2/q021/2.
Epqx, y, z=1A+B/q0exp-ik2q1zx-x1d2+y-y1d2-ik1xx+1yy+iϕx+ϕy,
x1d=Apxd,  1x=Cpxd,  ϕx=kAC2 p2xd2;
y1d=Aqyd,  1y=Cqyd,  ϕy=kAC2 q2yd2.
Ix, y, z=pq Epqx, y, zEpq*x, y, z,
Ix, y, z=Ex, y, zE*x, y, z,
Ex, y, z=pq Epqx, y, z.
ABCD=1z0110-1f1=-Δzf1+Δz-1f1,
Δz=z-f/f.
Epqx, y, Δz=iπNf1+Δz1-iπNf×-1+Δz1+iπNf1+Δz1-iπNfm/2×Hm2 πNf1+Δz2+πNfΔz21/2×x+Δzpxd×exp-iπNf1-iπNf1+Δz1-iπNf× x+Δzpxd2+y+Δzqyd2+i2πNfpxdx+qydy×expiπNfΔzpxd2+qyd2,
Nf=w02/λf
x=xw0,  xd=xdw0; y=yw0,  yd=ydw0.
Ix, y, Δz=0=PQπ2Nf2Hm22 πNfx×exp-2π2Nf2x2+y2
Ix, y, Δz=0=π2Nf2Hm22 πNfx×exp-2π2Nf2x2+y2 ×1-cos2πNfPxdx1-cos2πNfxdx1-cos2πNfQydy1-cos2πNfydy
Ix, y, 0=pq Ipqx, y, 0=pq Hm22x-pxdw0×exp-2x-pxd2+y-qyd2w02,
σx2=w0242m+1+P2-112 xd2,
σy2=w024+Q2-112 yd2,
σsx2=2m+14π2w02,
σsy2=14π2w02.
Mx2=4πσx2σsx21/2=2m+11+P2-132m+1xdw021/2=2m+11+P2-132m+1 xd21/2,
My2=4πσy2σsy21/2=1+Q2-13ydw021/2=1+Q2-13 yd21/2.
Ix, y, 0=Ex, y, 0E*x, y, 0,
Ex, y, 0=pq Epqx, y, 0=pq Hm2 x-pxdw0×exp-x-pxd2+y-qyd2w02.
Mx2=3+xd2+3-xd2exp-xd2/21+1-xd2exp-xd2/2½ exp-xd2/2H42 xd/2+62-exp-xd2/2H22 xd/21/2,
My2=1-yd2exp-yd2+2-yd4exp-yd2/2+yd2+11/21+exp-yd2/2
Mx2=33+4 exp-xd2/2+2 exp-2xd2+8xd21-exp-2xd2-4xd4 exp-xd2/23+4 exp-xd2/2+2 exp-2xd2-4xd2exp-xd2/2+2 exp-2xd2×12exp-2xd2H42 xd+exp-xd2/2H422 xd+93-exp-2xd2H22xd-2 expxd2/2H222 xd1/2,
My2=9+24yd2+83+4yd2-4yd4exp-yd2/2+161-yd2exp-yd2+43-2yd2-16yd4exp-2yd2+161-2yd4exp-5yd2/2+41-4yd2exp-4yd21/23+exp-2yd2+4 exp-yd2/2-1
PIB=-a/w0a/w0-b/w0b/w0 Ix, y, 0dxdy-- Ix, y, 0dxdy,
Wj2=W0j2Aj2+BjZ0j2 j=x, y, unless otherwise stated
Z0j=πW0j2Mj2λ.
W0j=4 j2Ix, y, 0dxdy Ix, y, 0dxdy,
Wxz=Wyz.
AjBjCjDj=1-zfjz-1fj1
αz2+βz+γ=0,
α=W0x21fx2+1Z0x2-W0y21fy2+1Z0y2, β=2W0y2fy2-W0x2fx2, γ=W0x2-W0y2.
fy=z1±1W0yW0x21-zfx2+z2W0x2Z0x2-W0y2Z0y21/2.
fy=fx1±λfxπ1W0y2Mx4W0x2-My4W0y21/2.

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