Abstract

The second-order intensity moments and beam-propagation factor (M2 factor) of partially coherent beams have been generalized to include the case of hard-edged diffraction. A laser beam with amplitude modulation and phase fluctuation and a Gaussian Schell-model beam are taken as two typical examples of partially coherent beams. Analytical expressions for the generalized M2 factor are derived.

© 2002 Optical Society of America

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References

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  1. A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE1224, 2–14 (1990).
    [CrossRef]
  2. A. E. Siegman, “How to (maybe) measure laser beam quality,” in DPSS 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.
  3. F. Gori, M. Santarsiero, “The change of width for a partially coherent beam on paraxial propagation,” Opt. Commun. 82, 197–203 (1991).
    [CrossRef]
  4. M. Santarsiero, F. Gori, “Spreading properties of beams radiated by partially coherent Schell-model sources,” J. Opt. Soc. Am. A 16, 106–112 (1999).
    [CrossRef]
  5. R. Martı́nez-Herrero, P. M. Mejias, “Second-order spatial characterization of hard-edge diffracted beams,” Opt. Lett. 18, 1669–1671 (1993).
    [CrossRef] [PubMed]
  6. R. Martı́nez-Herrero, P. M. Mejias, M. Arias, “Parametric characterization of coherent, lowest-order Gaussian beams propagating through hard-edged apertures,” Opt. Lett. 20, 124–126 (1995).
    [CrossRef] [PubMed]
  7. J. Serna, R. Martı́nez-Herrero, P. M. Mejias, “Parametric characterization of general partially coherent beams propagating through ABCD optical systems,” J. Opt. Soc. Am. A 8, 1094–1098 (1991).
    [CrossRef]
  8. H. Weber, “Propagation of high-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24, 1027–1049 (1992).
    [CrossRef]
  9. T. Shirai, T. Asakura, “Spatial coherence of light generated from a partially coherent source and its control using a source filter,” Optik (Stuttgart) 94, 1–15 (1993).
  10. A. Erdelyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Table of Integral Transforms (McGraw-Hill, New York, 1954).
  11. K. R. Manes, W. W. Simmons, “Statistical optics applied to high-power glass lasers,” J. Opt. Soc. Am. A 2, 528–538 (1985).
    [CrossRef]
  12. A. Starikov, E. Wolf, “Coherent-mode representation of Gaussian Schell-model sources and their radiation fields,” J. Opt. Soc. Am. 72, 923–928 (1982).
    [CrossRef]

1999 (1)

1995 (1)

1993 (2)

T. Shirai, T. Asakura, “Spatial coherence of light generated from a partially coherent source and its control using a source filter,” Optik (Stuttgart) 94, 1–15 (1993).

R. Martı́nez-Herrero, P. M. Mejias, “Second-order spatial characterization of hard-edge diffracted beams,” Opt. Lett. 18, 1669–1671 (1993).
[CrossRef] [PubMed]

1992 (1)

H. Weber, “Propagation of high-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24, 1027–1049 (1992).
[CrossRef]

1991 (2)

1985 (1)

1982 (1)

Arias, M.

Asakura, T.

T. Shirai, T. Asakura, “Spatial coherence of light generated from a partially coherent source and its control using a source filter,” Optik (Stuttgart) 94, 1–15 (1993).

Erdelyi, A.

A. Erdelyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Table of Integral Transforms (McGraw-Hill, New York, 1954).

Gori, F.

M. Santarsiero, F. Gori, “Spreading properties of beams radiated by partially coherent Schell-model sources,” J. Opt. Soc. Am. A 16, 106–112 (1999).
[CrossRef]

F. Gori, M. Santarsiero, “The change of width for a partially coherent beam on paraxial propagation,” Opt. Commun. 82, 197–203 (1991).
[CrossRef]

Magnus, W.

A. Erdelyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Table of Integral Transforms (McGraw-Hill, New York, 1954).

Manes, K. R.

Marti´nez-Herrero, R.

Mejias, P. M.

Oberhettinger, F.

A. Erdelyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Table of Integral Transforms (McGraw-Hill, New York, 1954).

Santarsiero, M.

M. Santarsiero, F. Gori, “Spreading properties of beams radiated by partially coherent Schell-model sources,” J. Opt. Soc. Am. A 16, 106–112 (1999).
[CrossRef]

F. Gori, M. Santarsiero, “The change of width for a partially coherent beam on paraxial propagation,” Opt. Commun. 82, 197–203 (1991).
[CrossRef]

Serna, J.

Shirai, T.

T. Shirai, T. Asakura, “Spatial coherence of light generated from a partially coherent source and its control using a source filter,” Optik (Stuttgart) 94, 1–15 (1993).

Siegman, A. E.

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

A. E. Siegman, “How to (maybe) measure laser beam quality,” in DPSS 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.

Simmons, W. W.

Starikov, A.

Tricomi, F. G.

A. Erdelyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Table of Integral Transforms (McGraw-Hill, New York, 1954).

Weber, H.

H. Weber, “Propagation of high-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24, 1027–1049 (1992).
[CrossRef]

Wolf, E.

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Opt. Commun. (1)

F. Gori, M. Santarsiero, “The change of width for a partially coherent beam on paraxial propagation,” Opt. Commun. 82, 197–203 (1991).
[CrossRef]

Opt. Lett. (2)

Opt. Quantum Electron. (1)

H. Weber, “Propagation of high-order intensity moments in quadratic-index media,” Opt. Quantum Electron. 24, 1027–1049 (1992).
[CrossRef]

Optik (Stuttgart) (1)

T. Shirai, T. Asakura, “Spatial coherence of light generated from a partially coherent source and its control using a source filter,” Optik (Stuttgart) 94, 1–15 (1993).

Other (3)

A. Erdelyi, W. Magnus, F. Oberhettinger, F. G. Tricomi, Table of Integral Transforms (McGraw-Hill, New York, 1954).

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

A. E. Siegman, “How to (maybe) measure laser beam quality,” in DPSS 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.

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Figures (3)

Fig. 1
Fig. 1

Generalized M2 factor of laser beams with AMs and PFs as a function of the beam truncation parameter δ.

Fig. 2
Fig. 2

Generalized M2 factor of laser beams with AMs and PFs as a function of phase fluctuation parameters σp2/[(Lp/w0)2].

Fig. 3
Fig. 3

Generalized M2 factor of laser beams with AMs and PFs as a function of amplitude modulation parameters σA2.

Equations (56)

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x2=1I-+x2W(x, x)dx,
u2=1k2I-+2W(x1, x2)x1x2x1=x2=xdx,
I=-W(x, x)dx
W(x1, x2)=E(x1)E*(x2),
x2=1I-aax2W(x, x)dx,
u2=1k2I-aa2W(x1, x2)x1x2x1=x2=xdx+4W(a, a)k2Ia+4W(-a, -a)k2Ia,
xu=12ikI-aax1W(x1, x2)x2-x2W(x1, x2)x1x1=x2=xdx,
I=-aaW(x, x)dx
QG=x2u2-xu2,
MG2=2k(QG)1/2=2k(x2u2-xu2)1/2.
Wo(x1, x2)=k2πB-aa-aaWi(x1, x2)×exp-ik2B [A(x12-x22)-2(x1x1-x2x2)+D(x12-x22)]dx1dx2,
Wi(x1, x2)=kB2π Wi(Bx1, Bx2)×exp-ikAB2 (x12-x22),
Wo(x1, x2)=-aa-aaWi(x1, x2)exp[ik(x1x1-x2x2)]×exp-ikD2B (x12-x22)dx1dx2.
x2o=1I-x2Wo(x, x)dx,
I=-Wo(x, x)dx.
x2o=1I-x2-aa-aaWi(x1, x2)×exp[ikx(x1-x2)]dx1dx2dx.
x2o=1k2I-aa2Wi(x1, x2)x1x2x1=x2=xdx+4Wi(a, a)k2Ia+4Wi(-a,-a)k2Ia,
I=-aaWi(x, x)dx.
x2o=A2I-aax2Wi(x, x)dx+B2k2I-aa2Wi(x1, x2)x1x2x1=x2=xdx+4B2Wi(a, a)k2Ia+4B2Wi(-a, -a)k2Ia+2AB2ikI-aax1Wi(x1, x2)x2-x2Wi(x1, x2)x1x1=x2=xdx,
x2o=A2x2i+B2u2i+2ABxui,
Wo(u1, u2)=--Wo(x1, x2)exp [ik(x1u1-x2u2)]dx1dx2,
-+exp-x22a2dx=2πa,
AD-BC=1,
Wo(u1, u2)=1D-aa-aaWi(x1, x2)×exp-ik2D [C(x12-x22)-2(u1x1-u2x2)-B(u12-u22)]dx1dx2.
Wi(x1, x2)=DWi(Dx1, Dx2)exp-ikCD2 (x12-x22),
Wo(u1, u2)=-aa-aaWi(x1, x2)exp[ik(u1x1-u2x2)]×expikB2D (u12-u22)dx1dx2.
u2o=1I-u2Wo(u, u)du,
I=-Wo(u, u)du.
u2o=1I-u2-aa-aaWi(x1, x2)×exp[iku(x1-x2)]dx1dx2du.
u2o=C2x2i+D2u2i+2CDxui.
xuo=1I-DB x2-aa-aaWi(x1, x2)×exp[ikx(x1-x2)]dx1dx2dx+12ikI-x1-aa-aaWi(x1, x2)×(-ikx2)exp[ik(x1x1-x2x2)]dx1dx2-x2-aa-aaWi(x1, x2)(ikx1)×exp[ik(x1x1-x2x2)]dx1dx2x1=x2=xdx.
xuo=Dx2oB-12ikI-aax1Wi(x1, x2)x2-x2Wi(x1, x2)x1x1=x2=xdx.
xuo=DB x2o-xui-AB x2i.
xuo=ACx2i+BDu2i+(AD+BC)xui.
MGo2=MGi2.
W(x1, x2, 0)=I0exp-σp2Lp2 (x1-x2)2+σA2exp-1LA2+σp2Lp2(x1-x2)2,
Io=exp-x12+x22w02,
x2=w0214-Eδ+σA2Eδ43 δ2-1exp(2δ2),
u2=1(kw0)21+2 σp2(Lp/w0)2+4Eδ4δ2-1+4σA2Eδ exp(2δ2)4δ2+2(LA/w0)2-1,
xu=0,
δ=a/w0
E=exp(-2δ2)/[2πerf(2δ)+4δσA2]
erf(s)=2π0sexp(-t2)dt.
MG2=214-Eδ+σA2Eδ43 δ2-1exp(2δ2)×1+2 σp2(Lp/w0)2+4Eδ4δ2-1+4σA2Eδ ×exp(2δ2)4δ2+2(LA/w0)2-11/2.
σA2=0,
σpLp2=12σ02,
W(x1, x2)=exp-x12+x22w02exp-(x1-x2)22σ02,
β=1+w02σ02-1/2,
MG2=2[1/4β2+(4/δ2-1/β2-1)Eδ-4E2δ2(4/δ2-1)]1/2,
E=exp(-2δ2)/2πerf(2δ).
limδ Eδ=limδδexp(2δ2)2πerf(2δ)=limδδexp(2δ2)2π.
limδ Eδ=0.
MG2=limδ 2[1/4β2+(4/δ2-1/β2-1)Eδ-4E2δ2(4/δ2-1)]1/2=1/β,
MG2=2[1/4+(4/δ2-2)Eδ-4E2δ2(4/δ2-1)]1/2.
k2QG=14-q2+4L212-qq,
q=2Eδ,L=δ.

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