Abstract

The multi-Gaussian beam shape is proposed as a model for aperture functions and laser beam profiles that have a nearly flat top but whose sides decrease continuously. Beams and apertures of this type represent a simple, elegant, and intuitive alternative to super-Gaussian beams, which are important in a number of applications such as laser resonator design. Analytical formulas are developed for the propagation of these beams through free space and optical systems representable by ABCD matrices.

© 2001 Optical Society of America

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References

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  1. S. De Silvestri, P. Laporta, V. Magni, O. Svelto, “Unstable laser resonators with super-Gaussian mirrors,” Opt. Lett. 13, 201–203 (1988).
    [CrossRef] [PubMed]
  2. S. De Silvestri, P. Laporta, V. Magni, G. Valentini, G. Cerullo, “Comparative analysis of Nd:YAG unstable resonators with super-Gaussian variable reflectance mirrors,” Opt. Commun. 77, 179–184 (1990).
    [CrossRef]
  3. S. De Silvestri, V. Magni, O. Svelto, G. Valentini, “Lasers with super-Gaussian mirrors,” IEEE J. Quantum Electron. 26, 1500–1509 (1990).
    [CrossRef]
  4. P. A. Bélanger, R. L. Lachance, C. Paré, “Super-Gaussian output from a CO2 laser by using a graded-phase mirror resonator,” Opt. Lett. 17, 739–741 (1991).
    [CrossRef]
  5. A. Parent, M. Morin, P. Lavigne, “Propagation of super-Gaussian field distributions,” Opt. Quantum Electron. 24, 1071–1079 (1992).
    [CrossRef]
  6. M. S. Bowers, “Diffractive analysis of unstable optical resonators with super-Gaussian mirrors,” Opt. Lett. 19, 1319–1321 (1992).
    [CrossRef]
  7. J. Serna, P. M. Mejias, Martinez-Herrero, “Quality changes of Hermite–Gauss mode beams and Gauss Schell-model fields propagating through super-Gaussian apertures,” Opt. Commun. 102, 162–168 (1993).
  8. M. R. Perrone, A. Piegari, S. Scaglione, “On the super-Gaussian unstable resonators for high-gain short-pulse laser media,” IEEE J. Quantum Electron. 29, 1423–1427 (1993).
    [CrossRef]
  9. J. Ojeda-Castañeda, G. Saavedra, E. Lopez-Olazagasti, “Super-Gaussian beams of continuous order as GRIN modes,” Opt. Commun. 102, 21–24 (1993).
    [CrossRef]
  10. R. van Neste, C. Paré, R. L. Lachance, P. A. Bélanger, “Graded-phase mirror resonator with a super-Gaussian output in a CW-CO2 laser,” IEEE J. Quantum Electron. 30, 2663–2669 (1994).
    [CrossRef]
  11. J. J. Wen, M. A. Breazeale, “A diffraction beam field expressed as a superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752–1756 (1988).
    [CrossRef]
  12. D. Ding, X. Liu, “Approximate description for Bessel, Bessel–Gauss, and Gaussian beams with finite aperture,” J. Opt. Soc. Am. A 16, 1286–1293 (1999).
    [CrossRef]
  13. F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
    [CrossRef]
  14. V. Bagini, R. Borghi, F. Gori, A. M. Pacileo, M. Santarsiero, D. Ambrosini, G. S. Spagnolo, “Propagation of axially symmetric flattened Gaussian beams,” J. Opt. Soc. Am. A 13, 1385–1394 (1996).
    [CrossRef]
  15. A. A. Tovar, L. W. Casperson, “Generalized beam matrices: Gaussian beam propagation in misaligned complex optical systems,” J. Opt. Soc. Am. A 12, 1522–1533 (1995).
    [CrossRef]
  16. L. W. Casperson, A. A. Tovar, “Hermite–sinusoidal-Gaussian beams in complex optical systems,” J. Opt. Soc. Am. A 15, 954–961 (1998).
    [CrossRef]
  17. A. A. Tovar, L. W. Casperson, “Production and propagation of Hermite–sinusoidal-Gaussian beams,” J. Opt. Soc. Am. A 15, 2425–2432 (1998).
    [CrossRef]

1999 (1)

1998 (2)

1996 (1)

1995 (1)

1994 (2)

F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
[CrossRef]

R. van Neste, C. Paré, R. L. Lachance, P. A. Bélanger, “Graded-phase mirror resonator with a super-Gaussian output in a CW-CO2 laser,” IEEE J. Quantum Electron. 30, 2663–2669 (1994).
[CrossRef]

1993 (3)

J. Serna, P. M. Mejias, Martinez-Herrero, “Quality changes of Hermite–Gauss mode beams and Gauss Schell-model fields propagating through super-Gaussian apertures,” Opt. Commun. 102, 162–168 (1993).

M. R. Perrone, A. Piegari, S. Scaglione, “On the super-Gaussian unstable resonators for high-gain short-pulse laser media,” IEEE J. Quantum Electron. 29, 1423–1427 (1993).
[CrossRef]

J. Ojeda-Castañeda, G. Saavedra, E. Lopez-Olazagasti, “Super-Gaussian beams of continuous order as GRIN modes,” Opt. Commun. 102, 21–24 (1993).
[CrossRef]

1992 (2)

A. Parent, M. Morin, P. Lavigne, “Propagation of super-Gaussian field distributions,” Opt. Quantum Electron. 24, 1071–1079 (1992).
[CrossRef]

M. S. Bowers, “Diffractive analysis of unstable optical resonators with super-Gaussian mirrors,” Opt. Lett. 19, 1319–1321 (1992).
[CrossRef]

1991 (1)

1990 (2)

S. De Silvestri, P. Laporta, V. Magni, G. Valentini, G. Cerullo, “Comparative analysis of Nd:YAG unstable resonators with super-Gaussian variable reflectance mirrors,” Opt. Commun. 77, 179–184 (1990).
[CrossRef]

S. De Silvestri, V. Magni, O. Svelto, G. Valentini, “Lasers with super-Gaussian mirrors,” IEEE J. Quantum Electron. 26, 1500–1509 (1990).
[CrossRef]

1988 (2)

S. De Silvestri, P. Laporta, V. Magni, O. Svelto, “Unstable laser resonators with super-Gaussian mirrors,” Opt. Lett. 13, 201–203 (1988).
[CrossRef] [PubMed]

J. J. Wen, M. A. Breazeale, “A diffraction beam field expressed as a superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752–1756 (1988).
[CrossRef]

Ambrosini, D.

Bagini, V.

Bélanger, P. A.

R. van Neste, C. Paré, R. L. Lachance, P. A. Bélanger, “Graded-phase mirror resonator with a super-Gaussian output in a CW-CO2 laser,” IEEE J. Quantum Electron. 30, 2663–2669 (1994).
[CrossRef]

P. A. Bélanger, R. L. Lachance, C. Paré, “Super-Gaussian output from a CO2 laser by using a graded-phase mirror resonator,” Opt. Lett. 17, 739–741 (1991).
[CrossRef]

Borghi, R.

Bowers, M. S.

Breazeale, M. A.

J. J. Wen, M. A. Breazeale, “A diffraction beam field expressed as a superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752–1756 (1988).
[CrossRef]

Casperson, L. W.

Cerullo, G.

S. De Silvestri, P. Laporta, V. Magni, G. Valentini, G. Cerullo, “Comparative analysis of Nd:YAG unstable resonators with super-Gaussian variable reflectance mirrors,” Opt. Commun. 77, 179–184 (1990).
[CrossRef]

De Silvestri, S.

S. De Silvestri, V. Magni, O. Svelto, G. Valentini, “Lasers with super-Gaussian mirrors,” IEEE J. Quantum Electron. 26, 1500–1509 (1990).
[CrossRef]

S. De Silvestri, P. Laporta, V. Magni, G. Valentini, G. Cerullo, “Comparative analysis of Nd:YAG unstable resonators with super-Gaussian variable reflectance mirrors,” Opt. Commun. 77, 179–184 (1990).
[CrossRef]

S. De Silvestri, P. Laporta, V. Magni, O. Svelto, “Unstable laser resonators with super-Gaussian mirrors,” Opt. Lett. 13, 201–203 (1988).
[CrossRef] [PubMed]

Ding, D.

Gori, F.

Lachance, R. L.

R. van Neste, C. Paré, R. L. Lachance, P. A. Bélanger, “Graded-phase mirror resonator with a super-Gaussian output in a CW-CO2 laser,” IEEE J. Quantum Electron. 30, 2663–2669 (1994).
[CrossRef]

P. A. Bélanger, R. L. Lachance, C. Paré, “Super-Gaussian output from a CO2 laser by using a graded-phase mirror resonator,” Opt. Lett. 17, 739–741 (1991).
[CrossRef]

Laporta, P.

S. De Silvestri, P. Laporta, V. Magni, G. Valentini, G. Cerullo, “Comparative analysis of Nd:YAG unstable resonators with super-Gaussian variable reflectance mirrors,” Opt. Commun. 77, 179–184 (1990).
[CrossRef]

S. De Silvestri, P. Laporta, V. Magni, O. Svelto, “Unstable laser resonators with super-Gaussian mirrors,” Opt. Lett. 13, 201–203 (1988).
[CrossRef] [PubMed]

Lavigne, P.

A. Parent, M. Morin, P. Lavigne, “Propagation of super-Gaussian field distributions,” Opt. Quantum Electron. 24, 1071–1079 (1992).
[CrossRef]

Liu, X.

Lopez-Olazagasti, E.

J. Ojeda-Castañeda, G. Saavedra, E. Lopez-Olazagasti, “Super-Gaussian beams of continuous order as GRIN modes,” Opt. Commun. 102, 21–24 (1993).
[CrossRef]

Magni, V.

S. De Silvestri, P. Laporta, V. Magni, G. Valentini, G. Cerullo, “Comparative analysis of Nd:YAG unstable resonators with super-Gaussian variable reflectance mirrors,” Opt. Commun. 77, 179–184 (1990).
[CrossRef]

S. De Silvestri, V. Magni, O. Svelto, G. Valentini, “Lasers with super-Gaussian mirrors,” IEEE J. Quantum Electron. 26, 1500–1509 (1990).
[CrossRef]

S. De Silvestri, P. Laporta, V. Magni, O. Svelto, “Unstable laser resonators with super-Gaussian mirrors,” Opt. Lett. 13, 201–203 (1988).
[CrossRef] [PubMed]

Martinez-Herrero,

J. Serna, P. M. Mejias, Martinez-Herrero, “Quality changes of Hermite–Gauss mode beams and Gauss Schell-model fields propagating through super-Gaussian apertures,” Opt. Commun. 102, 162–168 (1993).

Mejias, P. M.

J. Serna, P. M. Mejias, Martinez-Herrero, “Quality changes of Hermite–Gauss mode beams and Gauss Schell-model fields propagating through super-Gaussian apertures,” Opt. Commun. 102, 162–168 (1993).

Morin, M.

A. Parent, M. Morin, P. Lavigne, “Propagation of super-Gaussian field distributions,” Opt. Quantum Electron. 24, 1071–1079 (1992).
[CrossRef]

Ojeda-Castañeda, J.

J. Ojeda-Castañeda, G. Saavedra, E. Lopez-Olazagasti, “Super-Gaussian beams of continuous order as GRIN modes,” Opt. Commun. 102, 21–24 (1993).
[CrossRef]

Pacileo, A. M.

Paré, C.

R. van Neste, C. Paré, R. L. Lachance, P. A. Bélanger, “Graded-phase mirror resonator with a super-Gaussian output in a CW-CO2 laser,” IEEE J. Quantum Electron. 30, 2663–2669 (1994).
[CrossRef]

P. A. Bélanger, R. L. Lachance, C. Paré, “Super-Gaussian output from a CO2 laser by using a graded-phase mirror resonator,” Opt. Lett. 17, 739–741 (1991).
[CrossRef]

Parent, A.

A. Parent, M. Morin, P. Lavigne, “Propagation of super-Gaussian field distributions,” Opt. Quantum Electron. 24, 1071–1079 (1992).
[CrossRef]

Perrone, M. R.

M. R. Perrone, A. Piegari, S. Scaglione, “On the super-Gaussian unstable resonators for high-gain short-pulse laser media,” IEEE J. Quantum Electron. 29, 1423–1427 (1993).
[CrossRef]

Piegari, A.

M. R. Perrone, A. Piegari, S. Scaglione, “On the super-Gaussian unstable resonators for high-gain short-pulse laser media,” IEEE J. Quantum Electron. 29, 1423–1427 (1993).
[CrossRef]

Saavedra, G.

J. Ojeda-Castañeda, G. Saavedra, E. Lopez-Olazagasti, “Super-Gaussian beams of continuous order as GRIN modes,” Opt. Commun. 102, 21–24 (1993).
[CrossRef]

Santarsiero, M.

Scaglione, S.

M. R. Perrone, A. Piegari, S. Scaglione, “On the super-Gaussian unstable resonators for high-gain short-pulse laser media,” IEEE J. Quantum Electron. 29, 1423–1427 (1993).
[CrossRef]

Serna, J.

J. Serna, P. M. Mejias, Martinez-Herrero, “Quality changes of Hermite–Gauss mode beams and Gauss Schell-model fields propagating through super-Gaussian apertures,” Opt. Commun. 102, 162–168 (1993).

Spagnolo, G. S.

Svelto, O.

S. De Silvestri, V. Magni, O. Svelto, G. Valentini, “Lasers with super-Gaussian mirrors,” IEEE J. Quantum Electron. 26, 1500–1509 (1990).
[CrossRef]

S. De Silvestri, P. Laporta, V. Magni, O. Svelto, “Unstable laser resonators with super-Gaussian mirrors,” Opt. Lett. 13, 201–203 (1988).
[CrossRef] [PubMed]

Tovar, A. A.

Valentini, G.

S. De Silvestri, P. Laporta, V. Magni, G. Valentini, G. Cerullo, “Comparative analysis of Nd:YAG unstable resonators with super-Gaussian variable reflectance mirrors,” Opt. Commun. 77, 179–184 (1990).
[CrossRef]

S. De Silvestri, V. Magni, O. Svelto, G. Valentini, “Lasers with super-Gaussian mirrors,” IEEE J. Quantum Electron. 26, 1500–1509 (1990).
[CrossRef]

van Neste, R.

R. van Neste, C. Paré, R. L. Lachance, P. A. Bélanger, “Graded-phase mirror resonator with a super-Gaussian output in a CW-CO2 laser,” IEEE J. Quantum Electron. 30, 2663–2669 (1994).
[CrossRef]

Wen, J. J.

J. J. Wen, M. A. Breazeale, “A diffraction beam field expressed as a superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752–1756 (1988).
[CrossRef]

IEEE J. Quantum Electron. (3)

S. De Silvestri, V. Magni, O. Svelto, G. Valentini, “Lasers with super-Gaussian mirrors,” IEEE J. Quantum Electron. 26, 1500–1509 (1990).
[CrossRef]

M. R. Perrone, A. Piegari, S. Scaglione, “On the super-Gaussian unstable resonators for high-gain short-pulse laser media,” IEEE J. Quantum Electron. 29, 1423–1427 (1993).
[CrossRef]

R. van Neste, C. Paré, R. L. Lachance, P. A. Bélanger, “Graded-phase mirror resonator with a super-Gaussian output in a CW-CO2 laser,” IEEE J. Quantum Electron. 30, 2663–2669 (1994).
[CrossRef]

J. Acoust. Soc. Am. (1)

J. J. Wen, M. A. Breazeale, “A diffraction beam field expressed as a superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752–1756 (1988).
[CrossRef]

J. Opt. Soc. Am. A (5)

Opt. Commun. (4)

F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
[CrossRef]

J. Ojeda-Castañeda, G. Saavedra, E. Lopez-Olazagasti, “Super-Gaussian beams of continuous order as GRIN modes,” Opt. Commun. 102, 21–24 (1993).
[CrossRef]

J. Serna, P. M. Mejias, Martinez-Herrero, “Quality changes of Hermite–Gauss mode beams and Gauss Schell-model fields propagating through super-Gaussian apertures,” Opt. Commun. 102, 162–168 (1993).

S. De Silvestri, P. Laporta, V. Magni, G. Valentini, G. Cerullo, “Comparative analysis of Nd:YAG unstable resonators with super-Gaussian variable reflectance mirrors,” Opt. Commun. 77, 179–184 (1990).
[CrossRef]

Opt. Lett. (3)

Opt. Quantum Electron. (1)

A. Parent, M. Morin, P. Lavigne, “Propagation of super-Gaussian field distributions,” Opt. Quantum Electron. 24, 1071–1079 (1992).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

An N=2 multi-Gaussian function is made up of a sum of 2N+1 off-axis Gaussian function components.

Fig. 2
Fig. 2

(a) Different-order super-Gaussian functions and the circ function, (b) different-order multi-Gaussian functions and the circ function. Although multi-Gaussian functions are very similar to super-Gaussian functions, they are much easier to deal with analytically.

Fig. 3
Fig. 3

Intensity distribution of a multi-Gaussian beam as it propagates through free space.

Equations (56)

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

fN(x)=k=1NAkexp(-Bkx2).
fN(x)=exp(-x2)k=0Nx2kk!.
MG(x)=m=-NNexp-x-mww2m=-NNexp(-m2).
E(x)=E0m=-NNexp-x-mww2m=-NNexp(-m2).
w=WN+1-lnm=-NNexp(-m2)1/2,
E2(x)=E0,2exp-iπn02λq2 x2+S2x+P2,
1q=1R-i λπn0w2.
S=2πn0λ-daq+da
1q2=C+D/q1A+B/q1,
S2=S1A+B/q1,
P2=P1-i2ln(A+B/q1)-12β01S12BA+B/q1
w2=w1[(A+B/R1)2+(B/z01)2]1/2,
R2=(A+B/R1)2+(B/z01)2(A+B/R1)(C+D/R1)+(BD/z012),
da2=Ada1+Bda1,
da2=Cda1+Dda1,
uu/q2=ABCD uu/q1.
uu/q3=A2B2C2D2 uu/q2.
uu/q3=A2B2C2D2 ABCD uu/q1.
Msystem=A2B2C2D2 ABCD.
Msystem=MnMn-1M3M2M1.
E0,2=AJE0,1,
AJ=exp-i0zk0(z)dz.
E2(x)=E0,2exp-x-da2w22expda2w22×exp(-iP2)exp-iβ0x22R2×exp-iβ0da2-da2R2x,
E2=E0,2exp-x-da2w22expda2w22-da1w12×exp[-i(P2-P1)]exp-iβ0x22R2×exp-iβ0da2-da2R2x.
da1=mw1,
da1=0,
R1-1=0,
w1=W1N+1-lnm=-NNexp(-m2)1/2-1,
E2(x)=E0,2expda2w22-m2exp-x-da2w22×exp[-i(P2-P1)]exp-iβ0x22R2×exp-iβ0da2-da2R2x.
EMG,2(x)=E0,2m=-NNE2(x)/m=-NN E1(0),
w2=w1[A2+(B/z01)2]1/2,
R2=A2+(B/z01)2AC+BD/z012,
da2=mw1A,
da2=mw1C,
P2=P1-i2ln(A-iB/z01)+m2B/z01A-iB/z01.
EMG,2(x)=E0,2m=-NNfmexp-x-mw1Aw1(A2+B2/z012)1/22exp-i β02AC+BD/z012A2+B2/z012 x-mw1B/z012AC+BD/z0122m=-NNexp(-m2),
fmexpi2tan-1BAz01(A2+B2/z012)exp-im2BC/z01AC+BD/z012.
 
Mfreespace=1z01.
EMG,2(x)=E0,2m=-NNfmexp-x-mw1w1(1+z2/z012)1/22exp-izz01x-mw1w1(1+z2/z012)1/22m=-NNexp(-m2),
fm=exp[(i/2)tan-1(z/z01)](1+z2/z012)1/4.
zm=f1+(f/z01)2
wmin=w1(f/z01)21+(f/z01)22+(z/z01)21/2.
w1=W1N+1-lnm=-NNexp(-m2)1/2.
wmin=WminN+1-lnm=-NNexp(-m2)1/2.
Wmin=W1(f/z01)21+(f/z01)22+(z/z01)21/2.
θMGθGaussian=N+1-lnm=-NNexp(-m2)1/2.
M=100-iλπ-1wga-210i2mwga-101,
N+1-lnm=-NNexp(-m2)1/2<W0λ.
NmaxW0λ-1.
MGN(x)=m=-NNexp-x-mww2m=-NNexp(-m2).
g(x)=exp-x-Nww2m=-NNexp(-m2).
g(W)=e-1.
exp-W-Nww2m=-NNexp(-m2)=1e.
W=wN+1-lnm=-NNexp(-m2)1/2.
w=WN+1-lnm=-NNexp(-m2)1/2.

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