Abstract

The propagation and focusing properties of a class of Gaussian beams generated by optical resonators with Gaussian reflectivity mirrors are investigated. Attention is concentrated on the following two beams in this class: (a) the annular Gaussian beam (the Gaussian doughnut mode) and (b) the flat-topped Gaussian beam. A class of flat-topped Gaussian beams is introduced. All analysis is limited to a coherent superposition scheme of the lowest-order Gaussian modes (TEM00) that have different parameters.

© 2002 Optical Society of America

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References

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  1. H. Zucker, “Optical resonators with variable reflectivity mirrors,” Bell Syst. Tech. J. 49, 2349–2376 (1970).
    [CrossRef]
  2. A. Yariv, P. Yeh, “Confinement and stability in optical resonators employing mirrors with Gaussian reflectivity tapers,” Opt. Commun. 13, 370–374 (1975).
    [CrossRef]
  3. N. McCarthy, P. Lavigne, “Optical resonators with Gaussian reflectivity mirrors: output beam characteristics,” Appl. Opt. 23, 3845–3850 (1984).
    [CrossRef] [PubMed]
  4. Laser Component GmbH, “Graded reflectivity output couplers with Gaussian profile of reflectivity” (Data Sheet) (Werner-von-Siemens-Str. 15, Olching, Germany, 1994).
  5. Y. Li, “Focusing properties of the Gaussian beam generated by optical resonators with Gaussian reflectivity mirrors,” J. Mod. Opt. 35, 1833–1846 (1988).
    [CrossRef]
  6. S. De Silestri, P. Laporta, V. Magni, O. Svelto, “Solid-state laser unstable resonators with tapered reflectivity mirrors: the super-Gaussian approach,” IEEE J. Quantum Electron. 24, 1172–1177 (1988).
    [CrossRef]
  7. J. Serna, P. M. Mejfas, R. Martinez-Herrero, “Quality changes of beams propagating through super-Gaussian apertures,” in Laser Energy Distribution Profiles: Measurement and Applications, J. M. Darchuk, ed., Proc. SPIE1834, 162–168 (1993).
    [CrossRef]
  8. S.-A. Amarande, “Approximation of supergaussian beams by generalized gaussian beams,” in XI International Symposium on Gas Flow and Chemical Lasers and High-Power Laser Conference, D. R. Hall, H. J. Baker, eds., Proc. SPIE3092, 345–348 (1997).
    [CrossRef]
  9. F. Gori, “Flattened Gaussian beams,” Opt. Commun. 107, 335–341 (1994).
    [CrossRef]
  10. V. Bagini, R. Borghi, F. Gori, M. Pacileo, M. Santarsiero, D. Ambrosini, G. Schirripa Spagnolo, “Propagation of axially symmetric flattened Gaussian beams,” J. Opt. Soc. Am. A 13, 1385–1394 (1996).
    [CrossRef]
  11. B. Lu, B. Zhang, S. Luo, “Far-field intensity distribution, M2 factor, and propagation of flattened Gaussian beams,” Appl. Opt. 38, 4581–4584 (1999).
    [CrossRef]
  12. R. Borghi, “Elegant Laguerre–Gaussian beams as a new tool for describing axisymmetric flattened Gaussian beams,” J. Opt. Soc. Am. A 18, 1627–1632 (2001).
    [CrossRef]
  13. S. A. Self, “Focusing of spherical Gaussian beams,” Appl. Opt. 22, 659–661 (1983).
    [CrossRef]
  14. Y. Li, “Propagation of focal shift through axisymmetrical optical system,” Opt. Commun. 95, 13–17 (1993).
    [CrossRef]
  15. J. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, London, 1980), p. 20, Eq. (2).
  16. A. D. Poularikas, S. Seely, Signals and Systems (PWS-Kent, Boston, Mass.1985), p. 38, Eq. (1.61).

2001 (1)

1999 (1)

1996 (1)

1994 (1)

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

1993 (1)

Y. Li, “Propagation of focal shift through axisymmetrical optical system,” Opt. Commun. 95, 13–17 (1993).
[CrossRef]

1988 (2)

Y. Li, “Focusing properties of the Gaussian beam generated by optical resonators with Gaussian reflectivity mirrors,” J. Mod. Opt. 35, 1833–1846 (1988).
[CrossRef]

S. De Silestri, P. Laporta, V. Magni, O. Svelto, “Solid-state laser unstable resonators with tapered reflectivity mirrors: the super-Gaussian approach,” IEEE J. Quantum Electron. 24, 1172–1177 (1988).
[CrossRef]

1984 (1)

1983 (1)

S. A. Self, “Focusing of spherical Gaussian beams,” Appl. Opt. 22, 659–661 (1983).
[CrossRef]

1975 (1)

A. Yariv, P. Yeh, “Confinement and stability in optical resonators employing mirrors with Gaussian reflectivity tapers,” Opt. Commun. 13, 370–374 (1975).
[CrossRef]

1970 (1)

H. Zucker, “Optical resonators with variable reflectivity mirrors,” Bell Syst. Tech. J. 49, 2349–2376 (1970).
[CrossRef]

Amarande, S.-A.

S.-A. Amarande, “Approximation of supergaussian beams by generalized gaussian beams,” in XI International Symposium on Gas Flow and Chemical Lasers and High-Power Laser Conference, D. R. Hall, H. J. Baker, eds., Proc. SPIE3092, 345–348 (1997).
[CrossRef]

Ambrosini, D.

Bagini, V.

Borghi, R.

De Silestri, S.

S. De Silestri, P. Laporta, V. Magni, O. Svelto, “Solid-state laser unstable resonators with tapered reflectivity mirrors: the super-Gaussian approach,” IEEE J. Quantum Electron. 24, 1172–1177 (1988).
[CrossRef]

Gori, F.

Gradshteyn, J. S.

J. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, London, 1980), p. 20, Eq. (2).

Laporta, P.

S. De Silestri, P. Laporta, V. Magni, O. Svelto, “Solid-state laser unstable resonators with tapered reflectivity mirrors: the super-Gaussian approach,” IEEE J. Quantum Electron. 24, 1172–1177 (1988).
[CrossRef]

Lavigne, P.

Li, Y.

Y. Li, “Propagation of focal shift through axisymmetrical optical system,” Opt. Commun. 95, 13–17 (1993).
[CrossRef]

Y. Li, “Focusing properties of the Gaussian beam generated by optical resonators with Gaussian reflectivity mirrors,” J. Mod. Opt. 35, 1833–1846 (1988).
[CrossRef]

Lu, B.

Luo, S.

Magni, V.

S. De Silestri, P. Laporta, V. Magni, O. Svelto, “Solid-state laser unstable resonators with tapered reflectivity mirrors: the super-Gaussian approach,” IEEE J. Quantum Electron. 24, 1172–1177 (1988).
[CrossRef]

Martinez-Herrero, R.

J. Serna, P. M. Mejfas, R. Martinez-Herrero, “Quality changes of beams propagating through super-Gaussian apertures,” in Laser Energy Distribution Profiles: Measurement and Applications, J. M. Darchuk, ed., Proc. SPIE1834, 162–168 (1993).
[CrossRef]

McCarthy, N.

Mejfas, P. M.

J. Serna, P. M. Mejfas, R. Martinez-Herrero, “Quality changes of beams propagating through super-Gaussian apertures,” in Laser Energy Distribution Profiles: Measurement and Applications, J. M. Darchuk, ed., Proc. SPIE1834, 162–168 (1993).
[CrossRef]

Pacileo, M.

Poularikas, A. D.

A. D. Poularikas, S. Seely, Signals and Systems (PWS-Kent, Boston, Mass.1985), p. 38, Eq. (1.61).

Ryzhik, I. M.

J. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, London, 1980), p. 20, Eq. (2).

Santarsiero, M.

Seely, S.

A. D. Poularikas, S. Seely, Signals and Systems (PWS-Kent, Boston, Mass.1985), p. 38, Eq. (1.61).

Self, S. A.

S. A. Self, “Focusing of spherical Gaussian beams,” Appl. Opt. 22, 659–661 (1983).
[CrossRef]

Serna, J.

J. Serna, P. M. Mejfas, R. Martinez-Herrero, “Quality changes of beams propagating through super-Gaussian apertures,” in Laser Energy Distribution Profiles: Measurement and Applications, J. M. Darchuk, ed., Proc. SPIE1834, 162–168 (1993).
[CrossRef]

Spagnolo, G. Schirripa

Svelto, O.

S. De Silestri, P. Laporta, V. Magni, O. Svelto, “Solid-state laser unstable resonators with tapered reflectivity mirrors: the super-Gaussian approach,” IEEE J. Quantum Electron. 24, 1172–1177 (1988).
[CrossRef]

Yariv, A.

A. Yariv, P. Yeh, “Confinement and stability in optical resonators employing mirrors with Gaussian reflectivity tapers,” Opt. Commun. 13, 370–374 (1975).
[CrossRef]

Yeh, P.

A. Yariv, P. Yeh, “Confinement and stability in optical resonators employing mirrors with Gaussian reflectivity tapers,” Opt. Commun. 13, 370–374 (1975).
[CrossRef]

Zhang, B.

Zucker, H.

H. Zucker, “Optical resonators with variable reflectivity mirrors,” Bell Syst. Tech. J. 49, 2349–2376 (1970).
[CrossRef]

Appl. Opt. (3)

Bell Syst. Tech. J. (1)

H. Zucker, “Optical resonators with variable reflectivity mirrors,” Bell Syst. Tech. J. 49, 2349–2376 (1970).
[CrossRef]

IEEE J. Quantum Electron. (1)

S. De Silestri, P. Laporta, V. Magni, O. Svelto, “Solid-state laser unstable resonators with tapered reflectivity mirrors: the super-Gaussian approach,” IEEE J. Quantum Electron. 24, 1172–1177 (1988).
[CrossRef]

J. Mod. Opt. (1)

Y. Li, “Focusing properties of the Gaussian beam generated by optical resonators with Gaussian reflectivity mirrors,” J. Mod. Opt. 35, 1833–1846 (1988).
[CrossRef]

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

Opt. Commun. (3)

A. Yariv, P. Yeh, “Confinement and stability in optical resonators employing mirrors with Gaussian reflectivity tapers,” Opt. Commun. 13, 370–374 (1975).
[CrossRef]

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

Y. Li, “Propagation of focal shift through axisymmetrical optical system,” Opt. Commun. 95, 13–17 (1993).
[CrossRef]

Other (5)

J. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, London, 1980), p. 20, Eq. (2).

A. D. Poularikas, S. Seely, Signals and Systems (PWS-Kent, Boston, Mass.1985), p. 38, Eq. (1.61).

Laser Component GmbH, “Graded reflectivity output couplers with Gaussian profile of reflectivity” (Data Sheet) (Werner-von-Siemens-Str. 15, Olching, Germany, 1994).

J. Serna, P. M. Mejfas, R. Martinez-Herrero, “Quality changes of beams propagating through super-Gaussian apertures,” in Laser Energy Distribution Profiles: Measurement and Applications, J. M. Darchuk, ed., Proc. SPIE1834, 162–168 (1993).
[CrossRef]

S.-A. Amarande, “Approximation of supergaussian beams by generalized gaussian beams,” in XI International Symposium on Gas Flow and Chemical Lasers and High-Power Laser Conference, D. R. Hall, H. J. Baker, eds., Proc. SPIE3092, 345–348 (1997).
[CrossRef]

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Figures (12)

Fig. 1
Fig. 1

(a) Gaussian beam irradiance profile exp(-2r2/w02) and (b) the transmission factor 1-K exp(-r2/wc2) on the right-hand side of Eq. (1).

Fig. 2
Fig. 2

Beam irradiance profiles. (a) Gaussian beams with a central hole (Gaussian doughnut mode) generated by optical resonators with K=1 and β=1, 2, 4, and 8. (b) Flat-topped Gaussian beam generated by optical resonators with (K=0.5, β=1), (K=0.667, β=0.5), , and (K=0.909, β=0.1).

Fig. 3
Fig. 3

Geometry of Gaussian beams propagating in the +z direction. These beams are the zeroth, first, second, etc. components of the beams generated by the Gaussian mirror resonators.

Fig. 4
Fig. 4

Changes in the transverse irradiance patterns during free-space propagation of a beam with a central hole in the plane of z=0, assuming that K=1 and β=1.

Fig. 5
Fig. 5

Changes in the transverse irradiance patterns during free-space propagation of a flat-topped Gaussian beam, assuming that K=0.909 and β=0.1.

Fig. 6
Fig. 6

Contour line diagrams to show the numerical properties of the G function, defined by Eq. (15) and calculated assuming that (a) K=1.0 and β=1.0 for a Gaussian beam with a central hole and (b) K=0.909 and β=0.1 for a flat-topped Gaussian beam. Both (a) and (b) show that the contour lines approach each other and finally converge to a constant E when z0.5zR. This means they revert to a beam with a bell-shaped profile resembling the incident Gaussian beam.

Fig. 7
Fig. 7

Gaussian beams focused by thin lenses, showing the notation used in object and image space for the zeroth- and mth-order component beams.

Fig. 8
Fig. 8

Focusing of a Gaussian beam with a central hole. Axial irradiance distributions in the image space of a thin lens with a focusing geometry of the Fresnel number Nw=1/π, K=1.0 and β=1.0. The plane of the input beam waists is located in object space at (a) s=0, 0.1f, and 0.2 f and (b) s=f, 2 f, and 3f.

Fig. 9
Fig. 9

Focusing of a Gaussian beam with a central hole. Transverse irradiance distributions in the image space of a thin lens with Fresnel number Nw=1/π, K=1.0, and β=1.0. The plane of input beam waists is located in object space at s=0 (immediately in front of the lens).

Fig. 10
Fig. 10

Focusing of a Gaussian beam with a central hole. Transverse irradiance distributions in the image space of a thin lens with Fresnel number Nw=1/π, K=1.0, and β=1.0. The plane of the input beam waists is located in object space at s=2 f.

Fig. 11
Fig. 11

Axial irradiance pattern, plotted under the conditions Nw=1/π, K=1.0, β=1.0, and s=3f, with a dip that indicates a rapid change in transverse irradiance profiles. Conventional Gaussian beam axial irradiance distributions are on both sides of the dip.

Fig. 12
Fig. 12

Flat-topped Gaussian profiles FM(r), given by Eq. (36) for the values of M given on each curve.

Tables (2)

Tables Icon

Table 1 Values of K1, K2, K3  in Eq. (36) When β1=1, β2=2, , β10=10 and Assuming K0=-1

Tables Icon

Table 2 Values of K0, K1, K2  in Table 1 Modified by a Factor M+1, γp=(M+1) Kp

Equations (59)

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IS(r)=|A|2 exp-2 r2w021-K exp-2 r2wc2.
β=(w0/wc)2.
IS(r)=|A|2 exp-2 r2w021-K exp-2β r2w02.
VS(r)=A exp-r2w02-iΘ1-K exp-2β r2w021/2.
VS(r)=m=1νm(r, 0),
νm(r, 0)=Am exp-(2mβ+1) r2w02
=Am exp-r2(w0)m2.
αm=-(2m-3)(2m-5)  (3)(1)m! K2m(m2).
(w0)m=w0(2mβ+1)1/2(m=0, 1, 2, ).
zR=π w02λ,
(zR)m=π (w0)m2λ=zR2mβ+1(m=0, 1, 2, ).
νm(r, z)=Am (w0)mwm(z) exp-i[kz-Φm(z)]-r21wm2(z)+ik2Rm(z),
wm(z)=(w0)m1+z2(zR)m21/2,
Φm(z)=arctanz(zR)m,
Rm(z)=z+(zR)m2z
V(r, z)=A exp(-ikz)m=0αm (w0)mwm(z)×expiΦm(z)-kr22Rm(z)-r2wm2(z).
I(r, z)=|A|2m=0αm (w0)mwm(z)×expiΦm(z)-kr22Rm(z)-r2wm2(z)2.
I(r, z)=|A|21+(z/zR)2 exp-2 r2w02(z)G(r, z).
G(r, z)=1+m=0αm w0(z)wm(z) expiΔΦm(z)-kr22ΔRm(z)-r2Δwm2(z)2,
G(r, z)Ewhenz0.5zR.
Eθ=00=2πdθr=0r= |A|21+(z/zR)2 exp-2 r2w02(z)rdr=θ=0θ=2πdθr=0r=IS(r, z)rdr.
E=1-K1+β.
sm=f+sm-f(1-sm/f )2+π2(Nw)m2(smsandm=0, 1, 2 ).
(Nw)m=(w0)m2λf.
(w0)m=(MT)m(w0)m,
(zR)m=(MT)m2(zR)m,
(MT)m=(w0)m(w0)m=1[(1-s/f)2+π2(Nw)m2]1/2.
ςm=sm-s=s-f(1-s/f)2+π2(Nw)m2-s-f(1-s/f)2+π2Nw2.
ςm=4mβ(mβ+1)π2Nw2(s-f)[(1-s/f)2+π2Nw2][(2mβ+1)2(1-s/f)2+π2Nw2].
μm(r, z)=Am¯ (w0)mwm(z-ςm)×exp-i[k(z-ςm)-Φm(z-ςm)]-r2[wm(z-ςm)]2+ikr22Rm(z-ςm),
Am¯=αA exp-ik(s+sm)-arctan s(zR)m-arctan sm(zR)m,
αm=αm(MT)m=αm1-sf2+πNw2mβ+121/2.
I(r, z)=|A|2m=0αm (w0)mwm(z-ςm)×expiΦm(z-ςm)+ΔΨm-kr22Rm(z-ςm)-r2[wm(z-ςm)]22,
ΔΨm=arctan s(zR)m-arctan szR+arctan sm(zR)m-arctan szR.
ΔΨm=arctan 2mβ(πNw)(s/f)(2mβ+1)(s/f)2+π2Nw2-arctan 2mβ(πNw)(1-s/f)(2mβ+1)(1-s/f)2+π2Nw2.
I(0, z)|A|2=m=0αm (w0)mwm(z-ςm)×exp{i[Φm(z-ςm)+ΔΨm]}2
=m=0 αm1+bm2z-ςmf21/2×expiarctanbmz-ςmf+ΔΨm2,
bm=2mβ+1πNw(1-s/f)2+πNw2mβ+1.
SG(r)=exp(-rη+2),
FG(r)=exp(-r2)n=0η r2nn!(η=0, 1, 2 ),
Fr=Fθ=0,2Fr2=2Frθ=2Fθ2=0, ,NFrN=NFrN-1θ==NFrN=0,
d2Fdr2=0,d4Fdr4=0, 
FM(r)=A exp-r2w021-p=1MKp exp-βp r2w02.
dNdrN exp(βr2)=(2βr)N exp(βr2)×p=0M 1(4βr2)p N!p!(N-2p)!.
dNdrN exp(βr2)r=0=N!M!βM.
M=N/2whenNiseven(N-1)/2whenNisodd.
(1+β1)(1+β2)(1+βM)(1+β1)2(1+β2)2(1+βM)2·········(1+β1)M(1+β2)M(1+βM)MK1K2···KM=11···1.
(1+β1)(1+β2)(1+βM)(1+β1)2(1+β2)2(1+βM)2·········(1+β1)M(1+β2)M(1+βM)M=p=1M(1+βp)H(βp, βq),
H(βp, βq)=p=1,q=1qpM-1(βq-βp).
Kp=11+βp H(βp, 1)H(βp, βq).
FM(r)=AM+1 p=1M+1γp exp-p r2w02,
FM(r)1-[1-exp(-r2/w02)]M+1.
w0(z)wm(z)=1+(z/zR)21+(2mβ+1)2(z/zR)21/2,
ΔΦm(z)=Φm(z)-Φ0(z)=arctan 2mβ(z/zR)1+(2mβ+1)(z/zR)2;
kr22ΔRm(z)=kr22 1Rm(z)-1R0(z)=(2mβ+1)(z/zR)21+(2mβ+1)(z/zR)2 r2w02,
r2Δwm2(z)=r21wm2(z)-1w02(z)=2mβ+11+(2mβ+1)2(z/zR)2-11+(z/zR)2 r2w02.
(Nw)m=(w0)m2λf=Nw2mβ+1.
(MT)m=1[(1-s/f)2+π2(Nw)m2]1/2
=2mβ+1[(1-s/f)2(2mβ+1)2+π2Nw2]1/2.

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