Abstract

The concepts of shape-invariance error (SIE) and shape-invariance range (SIR) have recently been introduced to specify in a quantitative way the shape changes suffered by a beam on propagation. Here such parameters are evaluated for the case of a fundamental Gaussian beam in the presence of a quartic aberration of its wave front. Numerical results are presented for the case of a collimated aberrated beam. Generalization to the case of noncollimated beams is also given.

© 2001 Optical Society of America

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References

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  1. F. Gori, S. Vicalvi, M. Santarsiero, R. Borghi, “Shape invariance range of a light beam,” Opt. Lett. 21, 1205–1207 (1996).
    [Crossref] [PubMed]
  2. S. Vicalvi, R. Borghi, M. Santarsiero, F. Gori, “Shape-invariance error for axially symmetric light beams,” IEEE J. Quantum Electron. 34, 2109–2116 (1998).
    [Crossref]
  3. P.-A. Bélanger“Beam propagation and the ABCD matrices,” Opt. Lett. 16, 196–198 (1991).
    [Crossref]
  4. A. E. Siegman, “Defining the effective radius of curvature for a nonideal optical beam,” IEEE J. Quantum Electron. 27, 1146–1148 (1991).
    [Crossref]
  5. W. T. Welford, Aberration of the Symmetric Optical System (Academic, New York, 1974).
  6. V. N. Mahajan, Aberration Theory Made Simple, Vol. TT06 of SPIE Tutorial Text Series (SPIE, Bellingham, Wash., 1991).
  7. C. A. Klein, “Optical distortion coefficients of high-power laser windows,” Opt. Eng. 20, 343–360 (1990).
    [Crossref]
  8. A. E. Siegman, “Analysis of laser beam quality degradation caused by quartic phase distortion,” Appl. Opt. 32, 5893–5901 (1993).
    [Crossref] [PubMed]
  9. A. E. Siegman, “New developments in laser resonators,” in Optical Resonators, D. A. Holmes, ed., Proc. SPIE1224, 2–14 (1990).
    [Crossref]
  10. R. Martı́nez-Herrero, P. M. Mejı́as, G. Piquero, “Quality improvement of partially coherent symmetric-intensity beams caused by quartic phase distortion,” Opt. Lett. 17, 1650–1652 (1992).
    [Crossref]
  11. G. Piquero, P. M. Mejı́as, R. Martı́nez-Herrero, “Sharpness changes of Gaussian beams induced by spherically aberrated lenses,” Opt. Commun. 107, 179–183 (1994).
    [Crossref]
  12. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 17.
  13. M. Abramowitz, I. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972), Chap. 22.
  14. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995), Chap. 5.
  15. F. Gori, “Why is the Fresnel transform so little known?” in Current Trends in Optics, L. C. Dainty, ed. (Academic, London, 1994), Chap. 10.

1998 (1)

S. Vicalvi, R. Borghi, M. Santarsiero, F. Gori, “Shape-invariance error for axially symmetric light beams,” IEEE J. Quantum Electron. 34, 2109–2116 (1998).
[Crossref]

1996 (1)

1994 (1)

G. Piquero, P. M. Mejı́as, R. Martı́nez-Herrero, “Sharpness changes of Gaussian beams induced by spherically aberrated lenses,” Opt. Commun. 107, 179–183 (1994).
[Crossref]

1993 (1)

1992 (1)

1991 (2)

P.-A. Bélanger“Beam propagation and the ABCD matrices,” Opt. Lett. 16, 196–198 (1991).
[Crossref]

A. E. Siegman, “Defining the effective radius of curvature for a nonideal optical beam,” IEEE J. Quantum Electron. 27, 1146–1148 (1991).
[Crossref]

1990 (1)

C. A. Klein, “Optical distortion coefficients of high-power laser windows,” Opt. Eng. 20, 343–360 (1990).
[Crossref]

Bélanger, P.-A.

Borghi, R.

S. Vicalvi, R. Borghi, M. Santarsiero, F. Gori, “Shape-invariance error for axially symmetric light beams,” IEEE J. Quantum Electron. 34, 2109–2116 (1998).
[Crossref]

F. Gori, S. Vicalvi, M. Santarsiero, R. Borghi, “Shape invariance range of a light beam,” Opt. Lett. 21, 1205–1207 (1996).
[Crossref] [PubMed]

Gori, F.

S. Vicalvi, R. Borghi, M. Santarsiero, F. Gori, “Shape-invariance error for axially symmetric light beams,” IEEE J. Quantum Electron. 34, 2109–2116 (1998).
[Crossref]

F. Gori, S. Vicalvi, M. Santarsiero, R. Borghi, “Shape invariance range of a light beam,” Opt. Lett. 21, 1205–1207 (1996).
[Crossref] [PubMed]

F. Gori, “Why is the Fresnel transform so little known?” in Current Trends in Optics, L. C. Dainty, ed. (Academic, London, 1994), Chap. 10.

Klein, C. A.

C. A. Klein, “Optical distortion coefficients of high-power laser windows,” Opt. Eng. 20, 343–360 (1990).
[Crossref]

Mahajan, V. N.

V. N. Mahajan, Aberration Theory Made Simple, Vol. TT06 of SPIE Tutorial Text Series (SPIE, Bellingham, Wash., 1991).

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995), Chap. 5.

Marti´nez-Herrero, R.

G. Piquero, P. M. Mejı́as, R. Martı́nez-Herrero, “Sharpness changes of Gaussian beams induced by spherically aberrated lenses,” Opt. Commun. 107, 179–183 (1994).
[Crossref]

R. Martı́nez-Herrero, P. M. Mejı́as, G. Piquero, “Quality improvement of partially coherent symmetric-intensity beams caused by quartic phase distortion,” Opt. Lett. 17, 1650–1652 (1992).
[Crossref]

Meji´as, P. M.

G. Piquero, P. M. Mejı́as, R. Martı́nez-Herrero, “Sharpness changes of Gaussian beams induced by spherically aberrated lenses,” Opt. Commun. 107, 179–183 (1994).
[Crossref]

R. Martı́nez-Herrero, P. M. Mejı́as, G. Piquero, “Quality improvement of partially coherent symmetric-intensity beams caused by quartic phase distortion,” Opt. Lett. 17, 1650–1652 (1992).
[Crossref]

Piquero, G.

G. Piquero, P. M. Mejı́as, R. Martı́nez-Herrero, “Sharpness changes of Gaussian beams induced by spherically aberrated lenses,” Opt. Commun. 107, 179–183 (1994).
[Crossref]

R. Martı́nez-Herrero, P. M. Mejı́as, G. Piquero, “Quality improvement of partially coherent symmetric-intensity beams caused by quartic phase distortion,” Opt. Lett. 17, 1650–1652 (1992).
[Crossref]

Santarsiero, M.

S. Vicalvi, R. Borghi, M. Santarsiero, F. Gori, “Shape-invariance error for axially symmetric light beams,” IEEE J. Quantum Electron. 34, 2109–2116 (1998).
[Crossref]

F. Gori, S. Vicalvi, M. Santarsiero, R. Borghi, “Shape invariance range of a light beam,” Opt. Lett. 21, 1205–1207 (1996).
[Crossref] [PubMed]

Siegman, A. E.

A. E. Siegman, “Analysis of laser beam quality degradation caused by quartic phase distortion,” Appl. Opt. 32, 5893–5901 (1993).
[Crossref] [PubMed]

A. E. Siegman, “Defining the effective radius of curvature for a nonideal optical beam,” IEEE J. Quantum Electron. 27, 1146–1148 (1991).
[Crossref]

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

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 17.

Vicalvi, S.

S. Vicalvi, R. Borghi, M. Santarsiero, F. Gori, “Shape-invariance error for axially symmetric light beams,” IEEE J. Quantum Electron. 34, 2109–2116 (1998).
[Crossref]

F. Gori, S. Vicalvi, M. Santarsiero, R. Borghi, “Shape invariance range of a light beam,” Opt. Lett. 21, 1205–1207 (1996).
[Crossref] [PubMed]

Welford, W. T.

W. T. Welford, Aberration of the Symmetric Optical System (Academic, New York, 1974).

Wolf, E.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995), Chap. 5.

Appl. Opt. (1)

IEEE J. Quantum Electron. (2)

S. Vicalvi, R. Borghi, M. Santarsiero, F. Gori, “Shape-invariance error for axially symmetric light beams,” IEEE J. Quantum Electron. 34, 2109–2116 (1998).
[Crossref]

A. E. Siegman, “Defining the effective radius of curvature for a nonideal optical beam,” IEEE J. Quantum Electron. 27, 1146–1148 (1991).
[Crossref]

Opt. Commun. (1)

G. Piquero, P. M. Mejı́as, R. Martı́nez-Herrero, “Sharpness changes of Gaussian beams induced by spherically aberrated lenses,” Opt. Commun. 107, 179–183 (1994).
[Crossref]

Opt. Eng. (1)

C. A. Klein, “Optical distortion coefficients of high-power laser windows,” Opt. Eng. 20, 343–360 (1990).
[Crossref]

Opt. Lett. (3)

Other (7)

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 17.

M. Abramowitz, I. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972), Chap. 22.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995), Chap. 5.

F. Gori, “Why is the Fresnel transform so little known?” in Current Trends in Optics, L. C. Dainty, ed. (Academic, London, 1994), Chap. 10.

W. T. Welford, Aberration of the Symmetric Optical System (Academic, New York, 1974).

V. N. Mahajan, Aberration Theory Made Simple, Vol. TT06 of SPIE Tutorial Text Series (SPIE, Bellingham, Wash., 1991).

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

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Figures (4)

Fig. 1
Fig. 1

SIE plotted versus the normalized propagation distance z/L for values of the modulus of η from 0.0 to 2.0 in steps of 0.1.

Fig. 2
Fig. 2

SIR, divided by L, plotted versus |η| for different values of the SIE.

Fig. 3
Fig. 3

SIR, divided by L¯, plotted versus |η| for different values of the SIE.

Fig. 4
Fig. 4

Transverse intensity profile of the reference field of the aberrated beam for different values of the propagation distance and of the aberration parameter η together with the intensity profile of the unaberrated Gaussian beam (dotted curves). The values of the parameters are (a) z/L=0.2 and η=0.1, (b) z/L=0.4 and η=0.1, (c) z/L=0.2 and η=0.2, (d) z/L=0.4 and η=0.2, corresponding to =2.8%, 4.9%, 6.0%, 9.6%, respectively.

Equations (40)

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σz2=σ021+M2zkσ022,
M2=2πσ0σ,
M2=2πN0|V0(r)|2r3dr0V0(r)r2rdr1/2,
N=2π0|V0(r)|2rdr.
Rz=-kσz22π0r ϕz(r)r|Vz(r)|2rdr-1,
Rz=z1+kσ02M2z2.
w¯0=2σ02M21/2.
Vz(r)=exp(ikz)exp(-iΦz)expik2Rz r2×n=0exp(-2inΦz)cnψn(r; w¯z),
ψn(r; w)=2πw21/2Ln2r2w2exp-r2w2,
n=0, 1 ,,
w¯z=w¯01+zL¯21/2,
Rz=z1+L¯z2,
Φz=arctanzL¯.
L¯=kw¯022,
cn=2π0V0(r)ψn(r; w¯0)rdr,n=0, 1 , .
(z)=1-2π0V0(r)[VzR(r)]*rdr1/2.
(z)1-n=0|cn|2exp(2inΦz)1/2.
V0(r)=A exp-r2w02+iβr4.
R0=k4βw02,
V0(r)=A exp-r2w02+iβr4-2iβw02r2.
σ02=w022
M2=(1+2β2w08)1/2,
η=βw04
w¯0=w0(1+2η2)1/4.
cn(η)=Jn(η)(1+2η2)1/4.
Jn(η)=0exp[-α(η)x+iγ(η)x2]Ln(x)dx,
α(η)=121+1+2iη(1+2η2)1/2,
γ(η)=η4(1+2η2).
(z)=1-n=0|cn(η)|2exp2in×arctanzL (1+2η2)1/21/2,
L=kw022
α(-η)=α*(η),
γ(-η)=-γ(η),
cn(-η)=cn*(η).
L¯=kw¯022=L(1+2η2)1/2.
V0(r)=V0(r)expikr22R0,
Vz(r)=-iλzexp(ikz)V0(r)expik2z (r-r)2dr,
Vz(r)=-iλzexp(ikz)V0(r)×expikr22R0expik2z (r-r)2dr.
Vz(r)=zzexp[ik(z-z)]expikz2R0z r2Vzzz r,
z=z1+zR0.
(z)=(z).

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