Abstract

In contrast to numerical methods for beam shaping, analytical beam shaping consists of two steps: first, finding a purely geometrical distortion between the input plane and the output plane redistributing the intensity of the incoming wave front; and, second, computing a phase-only element realizing this coordinate transform. For the latter the method of stationary phase may be applied. The known classes of possible analytical wave transformation are extended to comprise separable and isotropic super-Gaussian-to-super-Gaussian conversion as well as transformation of Gaussian arrays to super-Gaussian distributions, and vice versa. The resulting optical phase elements contain no spiral phase dislocation and may thus be realized as refractive or diffractive elements. In addition, the outgoing wave front does not contain spiral phase dislocations.

© 1997 Optical Society of America

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References

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  1. J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–305 (1980).
    [CrossRef]
  2. F. Wyrowski, O. Bryngdahl, “Digital holography as part of diffractive optics,” Rep. Prog. Phys. 54, 1481–1571 (1991).
    [CrossRef]
  3. J. Turunen, A. Vasara, J. Westerholm, “Kinoform phase relief synthesis: a stochastic method,” Opt. Eng. 28, 1162–1167 (1989).
    [CrossRef]
  4. M. A. Seldowitz, J. P. Allebach, D. W. Sweeney, “Synthesis of digital holograms by direct binary search,” Appl. Opt. 28, 2788–2798 (1987).
    [CrossRef]
  5. J. R. Leger, D. Chen, Z. Wang, “Diffractive optical element for mode shaping of a ND:YAG laser,” Opt. Lett. 19, 108–110 (1994).
    [CrossRef] [PubMed]
  6. J. R. Leger, D. Chen, G. Mowry, “Design and performance of diffractive optics for custom laser resonators,” Appl. Opt. 34, 2498–2509 (1995).
    [CrossRef] [PubMed]
  7. C. C. Aleksoff, K. K. Ellis, B. D. Neagle, “Holographic conversion of a Gaussian beam to a near-field uniform beam,” Opt. Eng. 30, 537–543 (1991).
    [CrossRef]
  8. M. Duparré, M. A. Golub, B. Lüdge, V. S. Pavelyev, V. A. Soifer, G. V. Uspleniev, S. G. Volotovskii, “Investigation of computer-generated diffractive beam shapers for flattening of single-modal CO2 laser beams,” Appl. Opt. 32, 2489–2497 (1995).
    [CrossRef]
  9. C.-Y. Han, Y. Ishii, K. Murata, “Reshaping collimated laser beams with Gaussian profile to uniform profiles,” Appl. Opt. 22, 3644–3647 (1983).
    [CrossRef] [PubMed]
  10. F. S. Roux, “Intensity distribution transformation for rotationally symmetric beam shaping,” Opt. Eng. 30, 529–536 (1991).
    [CrossRef]
  11. X. Tan, B.-Y Gu, B.-Z Dong, “Diffractive phase elements for beam shaping: a new design method,” Appl. Opt. 34, 1314–1320 (1995).
    [CrossRef] [PubMed]
  12. P. H. Malyak, “Two-mirror unobscured optical system for reshaping the irradiance distribution of a laser beam,” Appl. Opt. 31, 4376–4383 (1992).
    [CrossRef]
  13. T. Dresel, M. Beyerlein, J. Schwider, “Design and fabrication of computer-generated beam-shaping holograms,” Appl. Opt. 35, 4615–4621 (1996).
    [CrossRef] [PubMed]
  14. H. Aagedal, S. Teiwes, F. Wyrowski, “Consequence of illumination wave on optical function of nonperiodic diffractive elements,” Opt. Commun. 109, 22–28 (1994).
    [CrossRef]
  15. A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968), Chap. 7.3, p. 234.
  16. O. Bryngdahl, “Optical map transformations,” Opt. Commun. 10, 164–168 (1974).
    [CrossRef]
  17. O. Bryngdahl, “Geometrical transformations in optics,” J. Opt. Soc. Am. 64, 1092–1099 (1974).
    [CrossRef]
  18. M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), Chap. 6.5, p. 260.
  19. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes. The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, 1992), Chap. 6.2, p. 216.
  20. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes. The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, 1992), Chap. 9.1, p. 350.
  21. H. Aagedal, M. Schmid, T. Beth, S. Teiwes, F. Wyrowski, “Theory of speckles in diffractive optics and its application to beam shaping,” J. Mod. Opt. 43, 1409–1421 (1996).
    [CrossRef]

1996 (2)

H. Aagedal, M. Schmid, T. Beth, S. Teiwes, F. Wyrowski, “Theory of speckles in diffractive optics and its application to beam shaping,” J. Mod. Opt. 43, 1409–1421 (1996).
[CrossRef]

T. Dresel, M. Beyerlein, J. Schwider, “Design and fabrication of computer-generated beam-shaping holograms,” Appl. Opt. 35, 4615–4621 (1996).
[CrossRef] [PubMed]

1995 (3)

X. Tan, B.-Y Gu, B.-Z Dong, “Diffractive phase elements for beam shaping: a new design method,” Appl. Opt. 34, 1314–1320 (1995).
[CrossRef] [PubMed]

J. R. Leger, D. Chen, G. Mowry, “Design and performance of diffractive optics for custom laser resonators,” Appl. Opt. 34, 2498–2509 (1995).
[CrossRef] [PubMed]

M. Duparré, M. A. Golub, B. Lüdge, V. S. Pavelyev, V. A. Soifer, G. V. Uspleniev, S. G. Volotovskii, “Investigation of computer-generated diffractive beam shapers for flattening of single-modal CO2 laser beams,” Appl. Opt. 32, 2489–2497 (1995).
[CrossRef]

1994 (2)

H. Aagedal, S. Teiwes, F. Wyrowski, “Consequence of illumination wave on optical function of nonperiodic diffractive elements,” Opt. Commun. 109, 22–28 (1994).
[CrossRef]

J. R. Leger, D. Chen, Z. Wang, “Diffractive optical element for mode shaping of a ND:YAG laser,” Opt. Lett. 19, 108–110 (1994).
[CrossRef] [PubMed]

1992 (1)

P. H. Malyak, “Two-mirror unobscured optical system for reshaping the irradiance distribution of a laser beam,” Appl. Opt. 31, 4376–4383 (1992).
[CrossRef]

1991 (3)

F. S. Roux, “Intensity distribution transformation for rotationally symmetric beam shaping,” Opt. Eng. 30, 529–536 (1991).
[CrossRef]

F. Wyrowski, O. Bryngdahl, “Digital holography as part of diffractive optics,” Rep. Prog. Phys. 54, 1481–1571 (1991).
[CrossRef]

C. C. Aleksoff, K. K. Ellis, B. D. Neagle, “Holographic conversion of a Gaussian beam to a near-field uniform beam,” Opt. Eng. 30, 537–543 (1991).
[CrossRef]

1989 (1)

J. Turunen, A. Vasara, J. Westerholm, “Kinoform phase relief synthesis: a stochastic method,” Opt. Eng. 28, 1162–1167 (1989).
[CrossRef]

1987 (1)

M. A. Seldowitz, J. P. Allebach, D. W. Sweeney, “Synthesis of digital holograms by direct binary search,” Appl. Opt. 28, 2788–2798 (1987).
[CrossRef]

1983 (1)

1980 (1)

J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–305 (1980).
[CrossRef]

1974 (2)

O. Bryngdahl, “Optical map transformations,” Opt. Commun. 10, 164–168 (1974).
[CrossRef]

O. Bryngdahl, “Geometrical transformations in optics,” J. Opt. Soc. Am. 64, 1092–1099 (1974).
[CrossRef]

Aagedal, H.

H. Aagedal, M. Schmid, T. Beth, S. Teiwes, F. Wyrowski, “Theory of speckles in diffractive optics and its application to beam shaping,” J. Mod. Opt. 43, 1409–1421 (1996).
[CrossRef]

H. Aagedal, S. Teiwes, F. Wyrowski, “Consequence of illumination wave on optical function of nonperiodic diffractive elements,” Opt. Commun. 109, 22–28 (1994).
[CrossRef]

Abramowitz, M.

M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), Chap. 6.5, p. 260.

Aleksoff, C. C.

C. C. Aleksoff, K. K. Ellis, B. D. Neagle, “Holographic conversion of a Gaussian beam to a near-field uniform beam,” Opt. Eng. 30, 537–543 (1991).
[CrossRef]

Allebach, J. P.

M. A. Seldowitz, J. P. Allebach, D. W. Sweeney, “Synthesis of digital holograms by direct binary search,” Appl. Opt. 28, 2788–2798 (1987).
[CrossRef]

Beth, T.

H. Aagedal, M. Schmid, T. Beth, S. Teiwes, F. Wyrowski, “Theory of speckles in diffractive optics and its application to beam shaping,” J. Mod. Opt. 43, 1409–1421 (1996).
[CrossRef]

Beyerlein, M.

Bryngdahl, O.

F. Wyrowski, O. Bryngdahl, “Digital holography as part of diffractive optics,” Rep. Prog. Phys. 54, 1481–1571 (1991).
[CrossRef]

O. Bryngdahl, “Geometrical transformations in optics,” J. Opt. Soc. Am. 64, 1092–1099 (1974).
[CrossRef]

O. Bryngdahl, “Optical map transformations,” Opt. Commun. 10, 164–168 (1974).
[CrossRef]

Chen, D.

Dong, B.-Z

Dresel, T.

Duparré, M.

M. Duparré, M. A. Golub, B. Lüdge, V. S. Pavelyev, V. A. Soifer, G. V. Uspleniev, S. G. Volotovskii, “Investigation of computer-generated diffractive beam shapers for flattening of single-modal CO2 laser beams,” Appl. Opt. 32, 2489–2497 (1995).
[CrossRef]

Ellis, K. K.

C. C. Aleksoff, K. K. Ellis, B. D. Neagle, “Holographic conversion of a Gaussian beam to a near-field uniform beam,” Opt. Eng. 30, 537–543 (1991).
[CrossRef]

Fienup, J. R.

J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–305 (1980).
[CrossRef]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes. The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, 1992), Chap. 6.2, p. 216.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes. The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, 1992), Chap. 9.1, p. 350.

Golub, M. A.

M. Duparré, M. A. Golub, B. Lüdge, V. S. Pavelyev, V. A. Soifer, G. V. Uspleniev, S. G. Volotovskii, “Investigation of computer-generated diffractive beam shapers for flattening of single-modal CO2 laser beams,” Appl. Opt. 32, 2489–2497 (1995).
[CrossRef]

Gu, B.-Y

Han, C.-Y.

Ishii, Y.

Leger, J. R.

Lüdge, B.

M. Duparré, M. A. Golub, B. Lüdge, V. S. Pavelyev, V. A. Soifer, G. V. Uspleniev, S. G. Volotovskii, “Investigation of computer-generated diffractive beam shapers for flattening of single-modal CO2 laser beams,” Appl. Opt. 32, 2489–2497 (1995).
[CrossRef]

Malyak, P. H.

P. H. Malyak, “Two-mirror unobscured optical system for reshaping the irradiance distribution of a laser beam,” Appl. Opt. 31, 4376–4383 (1992).
[CrossRef]

Mowry, G.

Murata, K.

Neagle, B. D.

C. C. Aleksoff, K. K. Ellis, B. D. Neagle, “Holographic conversion of a Gaussian beam to a near-field uniform beam,” Opt. Eng. 30, 537–543 (1991).
[CrossRef]

Papoulis, A.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968), Chap. 7.3, p. 234.

Pavelyev, V. S.

M. Duparré, M. A. Golub, B. Lüdge, V. S. Pavelyev, V. A. Soifer, G. V. Uspleniev, S. G. Volotovskii, “Investigation of computer-generated diffractive beam shapers for flattening of single-modal CO2 laser beams,” Appl. Opt. 32, 2489–2497 (1995).
[CrossRef]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes. The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, 1992), Chap. 6.2, p. 216.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes. The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, 1992), Chap. 9.1, p. 350.

Roux, F. S.

F. S. Roux, “Intensity distribution transformation for rotationally symmetric beam shaping,” Opt. Eng. 30, 529–536 (1991).
[CrossRef]

Schmid, M.

H. Aagedal, M. Schmid, T. Beth, S. Teiwes, F. Wyrowski, “Theory of speckles in diffractive optics and its application to beam shaping,” J. Mod. Opt. 43, 1409–1421 (1996).
[CrossRef]

Schwider, J.

Seldowitz, M. A.

M. A. Seldowitz, J. P. Allebach, D. W. Sweeney, “Synthesis of digital holograms by direct binary search,” Appl. Opt. 28, 2788–2798 (1987).
[CrossRef]

Soifer, V. A.

M. Duparré, M. A. Golub, B. Lüdge, V. S. Pavelyev, V. A. Soifer, G. V. Uspleniev, S. G. Volotovskii, “Investigation of computer-generated diffractive beam shapers for flattening of single-modal CO2 laser beams,” Appl. Opt. 32, 2489–2497 (1995).
[CrossRef]

Stegun, I.

M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), Chap. 6.5, p. 260.

Sweeney, D. W.

M. A. Seldowitz, J. P. Allebach, D. W. Sweeney, “Synthesis of digital holograms by direct binary search,” Appl. Opt. 28, 2788–2798 (1987).
[CrossRef]

Tan, X.

Teiwes, S.

H. Aagedal, M. Schmid, T. Beth, S. Teiwes, F. Wyrowski, “Theory of speckles in diffractive optics and its application to beam shaping,” J. Mod. Opt. 43, 1409–1421 (1996).
[CrossRef]

H. Aagedal, S. Teiwes, F. Wyrowski, “Consequence of illumination wave on optical function of nonperiodic diffractive elements,” Opt. Commun. 109, 22–28 (1994).
[CrossRef]

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes. The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, 1992), Chap. 9.1, p. 350.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes. The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, 1992), Chap. 6.2, p. 216.

Turunen, J.

J. Turunen, A. Vasara, J. Westerholm, “Kinoform phase relief synthesis: a stochastic method,” Opt. Eng. 28, 1162–1167 (1989).
[CrossRef]

Uspleniev, G. V.

M. Duparré, M. A. Golub, B. Lüdge, V. S. Pavelyev, V. A. Soifer, G. V. Uspleniev, S. G. Volotovskii, “Investigation of computer-generated diffractive beam shapers for flattening of single-modal CO2 laser beams,” Appl. Opt. 32, 2489–2497 (1995).
[CrossRef]

Vasara, A.

J. Turunen, A. Vasara, J. Westerholm, “Kinoform phase relief synthesis: a stochastic method,” Opt. Eng. 28, 1162–1167 (1989).
[CrossRef]

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes. The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, 1992), Chap. 9.1, p. 350.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes. The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, 1992), Chap. 6.2, p. 216.

Volotovskii, S. G.

M. Duparré, M. A. Golub, B. Lüdge, V. S. Pavelyev, V. A. Soifer, G. V. Uspleniev, S. G. Volotovskii, “Investigation of computer-generated diffractive beam shapers for flattening of single-modal CO2 laser beams,” Appl. Opt. 32, 2489–2497 (1995).
[CrossRef]

Wang, Z.

Westerholm, J.

J. Turunen, A. Vasara, J. Westerholm, “Kinoform phase relief synthesis: a stochastic method,” Opt. Eng. 28, 1162–1167 (1989).
[CrossRef]

Wyrowski, F.

H. Aagedal, M. Schmid, T. Beth, S. Teiwes, F. Wyrowski, “Theory of speckles in diffractive optics and its application to beam shaping,” J. Mod. Opt. 43, 1409–1421 (1996).
[CrossRef]

H. Aagedal, S. Teiwes, F. Wyrowski, “Consequence of illumination wave on optical function of nonperiodic diffractive elements,” Opt. Commun. 109, 22–28 (1994).
[CrossRef]

F. Wyrowski, O. Bryngdahl, “Digital holography as part of diffractive optics,” Rep. Prog. Phys. 54, 1481–1571 (1991).
[CrossRef]

Appl. Opt. (7)

M. A. Seldowitz, J. P. Allebach, D. W. Sweeney, “Synthesis of digital holograms by direct binary search,” Appl. Opt. 28, 2788–2798 (1987).
[CrossRef]

M. Duparré, M. A. Golub, B. Lüdge, V. S. Pavelyev, V. A. Soifer, G. V. Uspleniev, S. G. Volotovskii, “Investigation of computer-generated diffractive beam shapers for flattening of single-modal CO2 laser beams,” Appl. Opt. 32, 2489–2497 (1995).
[CrossRef]

P. H. Malyak, “Two-mirror unobscured optical system for reshaping the irradiance distribution of a laser beam,” Appl. Opt. 31, 4376–4383 (1992).
[CrossRef]

C.-Y. Han, Y. Ishii, K. Murata, “Reshaping collimated laser beams with Gaussian profile to uniform profiles,” Appl. Opt. 22, 3644–3647 (1983).
[CrossRef] [PubMed]

X. Tan, B.-Y Gu, B.-Z Dong, “Diffractive phase elements for beam shaping: a new design method,” Appl. Opt. 34, 1314–1320 (1995).
[CrossRef] [PubMed]

J. R. Leger, D. Chen, G. Mowry, “Design and performance of diffractive optics for custom laser resonators,” Appl. Opt. 34, 2498–2509 (1995).
[CrossRef] [PubMed]

T. Dresel, M. Beyerlein, J. Schwider, “Design and fabrication of computer-generated beam-shaping holograms,” Appl. Opt. 35, 4615–4621 (1996).
[CrossRef] [PubMed]

J. Mod. Opt. (1)

H. Aagedal, M. Schmid, T. Beth, S. Teiwes, F. Wyrowski, “Theory of speckles in diffractive optics and its application to beam shaping,” J. Mod. Opt. 43, 1409–1421 (1996).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Commun. (2)

O. Bryngdahl, “Optical map transformations,” Opt. Commun. 10, 164–168 (1974).
[CrossRef]

H. Aagedal, S. Teiwes, F. Wyrowski, “Consequence of illumination wave on optical function of nonperiodic diffractive elements,” Opt. Commun. 109, 22–28 (1994).
[CrossRef]

Opt. Eng. (4)

F. S. Roux, “Intensity distribution transformation for rotationally symmetric beam shaping,” Opt. Eng. 30, 529–536 (1991).
[CrossRef]

C. C. Aleksoff, K. K. Ellis, B. D. Neagle, “Holographic conversion of a Gaussian beam to a near-field uniform beam,” Opt. Eng. 30, 537–543 (1991).
[CrossRef]

J. R. Fienup, “Iterative method applied to image reconstruction and to computer-generated holograms,” Opt. Eng. 19, 297–305 (1980).
[CrossRef]

J. Turunen, A. Vasara, J. Westerholm, “Kinoform phase relief synthesis: a stochastic method,” Opt. Eng. 28, 1162–1167 (1989).
[CrossRef]

Opt. Lett. (1)

Rep. Prog. Phys. (1)

F. Wyrowski, O. Bryngdahl, “Digital holography as part of diffractive optics,” Rep. Prog. Phys. 54, 1481–1571 (1991).
[CrossRef]

Other (4)

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968), Chap. 7.3, p. 234.

M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), Chap. 6.5, p. 260.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes. The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, 1992), Chap. 6.2, p. 216.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes. The Art of Scientific Computing, 2nd ed. (Cambridge U. Press, Cambridge, 1992), Chap. 9.1, p. 350.

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Figures (5)

Fig. 1
Fig. 1

Distortion transforming a Gaussian beam to a uniform distribution.

Fig. 2
Fig. 2

In the top row (a) an elliptical standard Gaussian beam is transformed to (c) a separable super-Gaussian distribution of order n=12 with (b) an analytically specified phase-only element. The bottom row shows an isotropic transformation of (d) a circular super-Gaussian of order n1=4.5 to (f) a standard Gaussian with (e) a phase element.

Fig. 3
Fig. 3

Tranforming (a) an array of overlapping elliptical Gaussian beams to (c) a standard Gaussian distribution with (b) an analytically specified phase-only element.

Fig. 4
Fig. 4

Transforming (a) two completely separated Gaussian beams with (b) a phase-only element to (c) one Gaussian beam. The energy of the leftmost Gaussian beam in the input wave is mapped to (d) the left-hand half of the Gaussian distribution in the output wave.

Fig. 5
Fig. 5

Transforming (a) a Gaussian beam to (c) an array of Gaussian beams with (b) an analytically specified phase-only element.

Equations (26)

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

|f1(x)|2dx=|f2(h)|2dh.
-x|f1(t)|2dt=-h(x)|f2(t)|2dt.
h(x)=F2-1[F1(x)]
ϕ(x)=2πh(x)dx=2πF2-1[F1(x)]dx
B|f1(x)|2dx=h(B)|f2(x)|2dxforallB.
F2[h(B)]=F1(B).
erfn(x)=nΓ(1/n) 0x exp(-tn)dt.
P(a, x)=1Γ(a) 0x exp(-t)ta-1dt.
|fk(x, y)|=ckuk(x)dkvk(y),
ck=nk21/nk-1Γ(1/nk)αk1/2uk(x)=exp-xαknk,
dk=mk21/mk-1Γ(1/mk)βk1/2vk(y)=exp-yβkmk
Fk(x)=-x|ckuk(ξ)|2dξ=12 1+erfnk21/nkαkx.
ϕ1(n1, n2, α1, α2; x)
=2πF2-1[F1(x)]dx=2π α221/n2 erfn2-1erfn121/n1α1xdx.
ϕ1(2, , α1, α2; x)=2πα1α2 exp-2α12x2
+2πα2x erf22α1x.
ϕ(x, y)=ϕ1(n1, n2, α1, α2; x)×ϕ1(m1, m2, β1, β2; y).
|fk(ρ)|=ck exp-ραknk,
withck=21/nkαk nkΓ(2/nk)1/2
Fk(r)=B(r)|fk(x)|2dx=0r|fk(ρ)|2ρdρ
=P2nk, 2(r)nkαknk=erfnk/222/nkαk2r2.
F1(r)=F2[h(r)].
ϕ(ρ)=2πF2-1[F1(ρ)]dρ=2π α221/n2 (erfn2/2-1{erfn1/2[22/n1(r/α1)2]})1/2dρ.
|f1(x)|=cν=0n-1cν exp-x-bνaν2,
F1(x)=-x|f1(ξ)|2dξ=c2-xνcν exp-ξ-bνaν22dξ=c2 π2 ν,μ cνcμaνaμaν2+aμ2 exp-(bν-bμ)2aν2+aμ2×erf21+(aν2+aμ2)x-aμ2bν-aν2bμaνaμ(aν2+aμ2)1/2.
c=πν,μ cνcμaνaμaν2+aμ2 exp-(bν-bμ)2aν2+aμ2-1/2.

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