Abstract

A unified operator approach is described for deriving Hermite–Gaussian and Laguerre–Gaussian laser beams by using as a starting point a plane-wave-spectrum representation of the electromagnetic field. We show that by using the plane-wave representation of the fundamental Gaussian mode as a seed function, all higher-order beam modes can be derived by acting with differential operators on this fundamental solution. The approach presented can be easily generalized to nonparaxial situations and to include vector effects of the electromagnetic field.

© 2004 Optical Society of America

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References

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  1. H. Kogelnik, “On the propagation of Gaussian beams of light through lenslike media including those with a loss or gain variation,” Appl. Opt. 4, 1562–1569 (1965).
    [CrossRef]
  2. H. Kogelnik, T. Li, “Laser beams and resonators,” Appl. Opt. 5, 1550–1567 (1966).
    [CrossRef] [PubMed]
  3. A. E. Siegman, Lasers (University Science, Books, Mill Valley, Calif., 1986), pp. 642–652.
  4. R. Menzel, Photonics (Springer, New York, 2001), pp. 362–370.
  5. J. T. Verdeyen, Laser Electronics (Prentice Hall, Englewood Cliffs, N.J., 1981), pp. 53–69.
  6. A. Maitland, M. H. Dunn, Laser Physics (North-Holland, Amsterdam, 1969), pp. 153–175.
  7. O. Svelto, Principles of Lasers (Plenum, New York, 1998), pp. 148–159.
  8. K. J. Kuhn, Laser Engineering (Prentice Hall, Englewood Cliffs, N.J., 1998), pp. 84–88.
  9. K. Shimoda, Introduction to Laser Physics (Springer, New York, 1991).
  10. D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
    [CrossRef] [PubMed]
  11. A. T. O’Neil, M. J. Padgett, “Axial and lateral trapping efficiency of Laguerre–Gaussian modes in inverted optical tweezers,” Opt. Commun. 193, 45–50 (2001).
    [CrossRef]
  12. K. T. Gahagan, G. A. Schwartzlander, “Optical vortex trapping of particles,” Opt. Lett. 21, 827–829 (1996).
    [CrossRef] [PubMed]
  13. A. T. O’Neil, M. J. Padgett, “Three-dimensional optical confinement of micron-sized metal particles and the decoupling of the spin and orbital angular momentum within an optical spanner,” Opt. Commun. 185, 139–143 (2000).
    [CrossRef]
  14. M. Padgett, L. Allen, “Optical tweezers and spanners,” Phys. World 10, 35–38 (1997).
  15. M. Padgett, L. Allen, “The angular momentum of light: optical spanners and the rotational frequency shift,” Opt. Quantum Electron. 31, 1–12 (1999).
    [CrossRef]
  16. E. Zauderer, “Complex argument Hermite–Gaussian and Laguerre–Gaussian beams,” J. Opt. Soc. Am. A 3, 465–469 (1986).
    [CrossRef]
  17. A. E. Siegman, “Hermite–Gaussian functions of complex argument as optical-beam eigenfunctions,” J. Opt. Soc. Am. 63, 1093–1094 (1973).
    [CrossRef]
  18. A. Wünsche, “Generalized Gaussian beam solutions of paraxial optics and their connection to a hidden symmetry,” J. Opt. Soc. Am. A 6, 1320–1329 (1989).
    [CrossRef]
  19. D. R. Rhodes, “On the stored energy of planar apertures,” IEEE Trans. Antennas Propag. 14, 676–683 (1966).
    [CrossRef]
  20. D. R. Rhodes, “On a fundamental principle in the theory of planar antennas,” Proc. IEEE 52, 1013–1021 (1964).
    [CrossRef]
  21. W. H. Carter, “Electromagnetic beam fields,” J. Mod. Opt. 21, 871–92 (1974).
  22. J. Durnin, J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
    [CrossRef] [PubMed]
  23. D. G. Hall, “Vector-beam solutions of Maxwells wave equation,” Opt. Lett. 21, 9–11 (1996).
    [CrossRef] [PubMed]
  24. H. C. Kim, Y. H. Lee, “Hermite–Gaussian and Laguerre–Gaussian beams beyond the paraxial approximation,” Opt. Commun. 169, 9–16 (1999).
    [CrossRef]
  25. M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
    [CrossRef]
  26. D. N. Pattanayak, G. P. Agrawal, “Representation of vector electromagnetic beams,” Phys. Rev. A 22, 1159–1164 (1980).
    [CrossRef]
  27. C. G. Chen, P. T. Konkola, J. Ferrera, R. K. Heilmann, M. L. Schattenburg, “Analyses of vector Gaussian beam propagation and the validity of paraxial and spherical approximations,” J. Opt. Soc. Am. A 19, 404–412 (2002).
    [CrossRef]

2003 (1)

D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
[CrossRef] [PubMed]

2002 (1)

2001 (1)

A. T. O’Neil, M. J. Padgett, “Axial and lateral trapping efficiency of Laguerre–Gaussian modes in inverted optical tweezers,” Opt. Commun. 193, 45–50 (2001).
[CrossRef]

2000 (1)

A. T. O’Neil, M. J. Padgett, “Three-dimensional optical confinement of micron-sized metal particles and the decoupling of the spin and orbital angular momentum within an optical spanner,” Opt. Commun. 185, 139–143 (2000).
[CrossRef]

1999 (2)

M. Padgett, L. Allen, “The angular momentum of light: optical spanners and the rotational frequency shift,” Opt. Quantum Electron. 31, 1–12 (1999).
[CrossRef]

H. C. Kim, Y. H. Lee, “Hermite–Gaussian and Laguerre–Gaussian beams beyond the paraxial approximation,” Opt. Commun. 169, 9–16 (1999).
[CrossRef]

1997 (1)

M. Padgett, L. Allen, “Optical tweezers and spanners,” Phys. World 10, 35–38 (1997).

1996 (2)

1989 (1)

1987 (1)

J. Durnin, J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

1986 (1)

1980 (1)

D. N. Pattanayak, G. P. Agrawal, “Representation of vector electromagnetic beams,” Phys. Rev. A 22, 1159–1164 (1980).
[CrossRef]

1975 (1)

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

1974 (1)

W. H. Carter, “Electromagnetic beam fields,” J. Mod. Opt. 21, 871–92 (1974).

1973 (1)

1966 (2)

H. Kogelnik, T. Li, “Laser beams and resonators,” Appl. Opt. 5, 1550–1567 (1966).
[CrossRef] [PubMed]

D. R. Rhodes, “On the stored energy of planar apertures,” IEEE Trans. Antennas Propag. 14, 676–683 (1966).
[CrossRef]

1965 (1)

1964 (1)

D. R. Rhodes, “On a fundamental principle in the theory of planar antennas,” Proc. IEEE 52, 1013–1021 (1964).
[CrossRef]

Agrawal, G. P.

D. N. Pattanayak, G. P. Agrawal, “Representation of vector electromagnetic beams,” Phys. Rev. A 22, 1159–1164 (1980).
[CrossRef]

Allen, L.

M. Padgett, L. Allen, “The angular momentum of light: optical spanners and the rotational frequency shift,” Opt. Quantum Electron. 31, 1–12 (1999).
[CrossRef]

M. Padgett, L. Allen, “Optical tweezers and spanners,” Phys. World 10, 35–38 (1997).

Carter, W. H.

W. H. Carter, “Electromagnetic beam fields,” J. Mod. Opt. 21, 871–92 (1974).

Chen, C. G.

Dunn, M. H.

A. Maitland, M. H. Dunn, Laser Physics (North-Holland, Amsterdam, 1969), pp. 153–175.

Durnin, J.

J. Durnin, J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

Eberly, J. H.

J. Durnin, J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

Ferrera, J.

Gahagan, K. T.

Grier, D. G.

D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
[CrossRef] [PubMed]

Hall, D. G.

Heilmann, R. K.

Kim, H. C.

H. C. Kim, Y. H. Lee, “Hermite–Gaussian and Laguerre–Gaussian beams beyond the paraxial approximation,” Opt. Commun. 169, 9–16 (1999).
[CrossRef]

Kogelnik, H.

Konkola, P. T.

Kuhn, K. J.

K. J. Kuhn, Laser Engineering (Prentice Hall, Englewood Cliffs, N.J., 1998), pp. 84–88.

Lax, M.

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Lee, Y. H.

H. C. Kim, Y. H. Lee, “Hermite–Gaussian and Laguerre–Gaussian beams beyond the paraxial approximation,” Opt. Commun. 169, 9–16 (1999).
[CrossRef]

Li, T.

Louisell, W. H.

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Maitland, A.

A. Maitland, M. H. Dunn, Laser Physics (North-Holland, Amsterdam, 1969), pp. 153–175.

McKnight, W. B.

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

Menzel, R.

R. Menzel, Photonics (Springer, New York, 2001), pp. 362–370.

Miceli, J.

J. Durnin, J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

O’Neil, A. T.

A. T. O’Neil, M. J. Padgett, “Axial and lateral trapping efficiency of Laguerre–Gaussian modes in inverted optical tweezers,” Opt. Commun. 193, 45–50 (2001).
[CrossRef]

A. T. O’Neil, M. J. Padgett, “Three-dimensional optical confinement of micron-sized metal particles and the decoupling of the spin and orbital angular momentum within an optical spanner,” Opt. Commun. 185, 139–143 (2000).
[CrossRef]

Padgett, M.

M. Padgett, L. Allen, “The angular momentum of light: optical spanners and the rotational frequency shift,” Opt. Quantum Electron. 31, 1–12 (1999).
[CrossRef]

M. Padgett, L. Allen, “Optical tweezers and spanners,” Phys. World 10, 35–38 (1997).

Padgett, M. J.

A. T. O’Neil, M. J. Padgett, “Axial and lateral trapping efficiency of Laguerre–Gaussian modes in inverted optical tweezers,” Opt. Commun. 193, 45–50 (2001).
[CrossRef]

A. T. O’Neil, M. J. Padgett, “Three-dimensional optical confinement of micron-sized metal particles and the decoupling of the spin and orbital angular momentum within an optical spanner,” Opt. Commun. 185, 139–143 (2000).
[CrossRef]

Pattanayak, D. N.

D. N. Pattanayak, G. P. Agrawal, “Representation of vector electromagnetic beams,” Phys. Rev. A 22, 1159–1164 (1980).
[CrossRef]

Rhodes, D. R.

D. R. Rhodes, “On the stored energy of planar apertures,” IEEE Trans. Antennas Propag. 14, 676–683 (1966).
[CrossRef]

D. R. Rhodes, “On a fundamental principle in the theory of planar antennas,” Proc. IEEE 52, 1013–1021 (1964).
[CrossRef]

Schattenburg, M. L.

Schwartzlander, G. A.

Shimoda, K.

K. Shimoda, Introduction to Laser Physics (Springer, New York, 1991).

Siegman, A. E.

Svelto, O.

O. Svelto, Principles of Lasers (Plenum, New York, 1998), pp. 148–159.

Verdeyen, J. T.

J. T. Verdeyen, Laser Electronics (Prentice Hall, Englewood Cliffs, N.J., 1981), pp. 53–69.

Wünsche, A.

Zauderer, E.

Appl. Opt. (2)

IEEE Trans. Antennas Propag. (1)

D. R. Rhodes, “On the stored energy of planar apertures,” IEEE Trans. Antennas Propag. 14, 676–683 (1966).
[CrossRef]

J. Mod. Opt. (1)

W. H. Carter, “Electromagnetic beam fields,” J. Mod. Opt. 21, 871–92 (1974).

J. Opt. Soc. Am. (1)

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

Nature (1)

D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
[CrossRef] [PubMed]

Opt. Commun. (3)

A. T. O’Neil, M. J. Padgett, “Axial and lateral trapping efficiency of Laguerre–Gaussian modes in inverted optical tweezers,” Opt. Commun. 193, 45–50 (2001).
[CrossRef]

A. T. O’Neil, M. J. Padgett, “Three-dimensional optical confinement of micron-sized metal particles and the decoupling of the spin and orbital angular momentum within an optical spanner,” Opt. Commun. 185, 139–143 (2000).
[CrossRef]

H. C. Kim, Y. H. Lee, “Hermite–Gaussian and Laguerre–Gaussian beams beyond the paraxial approximation,” Opt. Commun. 169, 9–16 (1999).
[CrossRef]

Opt. Lett. (2)

Opt. Quantum Electron. (1)

M. Padgett, L. Allen, “The angular momentum of light: optical spanners and the rotational frequency shift,” Opt. Quantum Electron. 31, 1–12 (1999).
[CrossRef]

Phys. Rev. A (2)

M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
[CrossRef]

D. N. Pattanayak, G. P. Agrawal, “Representation of vector electromagnetic beams,” Phys. Rev. A 22, 1159–1164 (1980).
[CrossRef]

Phys. Rev. Lett. (1)

J. Durnin, J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
[CrossRef] [PubMed]

Phys. World (1)

M. Padgett, L. Allen, “Optical tweezers and spanners,” Phys. World 10, 35–38 (1997).

Proc. IEEE (1)

D. R. Rhodes, “On a fundamental principle in the theory of planar antennas,” Proc. IEEE 52, 1013–1021 (1964).
[CrossRef]

Other (7)

A. E. Siegman, Lasers (University Science, Books, Mill Valley, Calif., 1986), pp. 642–652.

R. Menzel, Photonics (Springer, New York, 2001), pp. 362–370.

J. T. Verdeyen, Laser Electronics (Prentice Hall, Englewood Cliffs, N.J., 1981), pp. 53–69.

A. Maitland, M. H. Dunn, Laser Physics (North-Holland, Amsterdam, 1969), pp. 153–175.

O. Svelto, Principles of Lasers (Plenum, New York, 1998), pp. 148–159.

K. J. Kuhn, Laser Engineering (Prentice Hall, Englewood Cliffs, N.J., 1998), pp. 84–88.

K. Shimoda, Introduction to Laser Physics (Springer, New York, 1991).

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Equations (45)

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Ekx2+ky2k2dkxdky(2π)2 A(kx, ky)×exp[ikxx+ikyy+i(k2-kx2-ky2)1/2z],
2Ex2+2Ey2+2Ez2+k2E=0
divergence-dkx2π-dky2π (kx2+ky2)|A|2,
transversalextension
-dx-dy(x2+y2)|E|2
=-dkx2π-dky2πAx2+Ay2.
-dkx2π kx2|A|2 -dkx2πAkx2,
-dξ|f(ξ)|2-dξ|g(ξ)|2-dξf(ξ)¯g(ξ)2
-dkx2π kx2|A|2 -dkx2πAkx2
-dkx(2π)212kxA¯ Akx+kxA A¯kx2=-dkx(2π)2kx2|A|2kx2=18π2-dkx|A|22=A4(8π2)2,
A(kx, ky)exp-w024 (kx2+ky2),
E0-dkx2π-dky2πexpikxx+ikyy+ikz-i kx2+ky22k z-w024 (kx2+ky2).
E0=1w02+2iz/kexpikz-x2+y2w02+2iz/k,
zR=kw022=πw02λ
ζ=z/zR,
11+iζ=1-iζ1+ζ2=exp(-i arctan ζ)1+ζ2,
E0=1w01+ζ2expikz-1-iζw02(1+ζ2) ρ2-i arctan ζ,
E˜m,nH=-dkx2π-dky2π (ikx)m(iky)nexp S
S(kx, ky, x, y, z)=ikxx+ikyy+ikz-w024 (1+iζ)×(kx2+ky2).
ikxexp S=xexp S
E˜m,nH=m+nxmyn E0.
Hp(x)=(-1)pexp(x2) dpdxpexp(-x2),
E˜m,nH=1w(ζ)(m+n)/2+1 Hmxw0(1+iζ)1/2×Hnyw0(1+iζ)1/2×expikz-ρ2w02(1+iζ)-iψ˜m,n,
ψ˜m,n=1+m+n2arctan ζ.
Em,nH-dkx2π-dky2πikx+1ukxm×iky+1ukynexp S.
1ukxexp S=ixu-w022u (1+iζ)kxexp S,
Em,nH=fm+nx+ixufmy+iyufnE0,
f=1-w022iu (1+iζ)
x+αxpexp-αx22xexpαx22p=exp-αx22pxpexpαx22
Em,nHfm+nexp-ix22ufxexpix22ufm×exp-iy22ufyexpiy22ufnE0=fm+nexp-iρ22ufmxmnynexpiρ22ufE0.
iρ22uf-ρ2w02(1+iζ)=-ρ22iu-w02(1+iζ)-ρ2w02(1+iζ),
f=1-w022iu (1+iζ)=1-iζ2.
Em,nHfm+nexp-ir22ufmxmnynexpiρ22ufE0=fm+nπw0wexp-ir22ufmxmnyn×expikz-2ρ2w02(1+ζ2)-iψ0=(-f)m+nπw0w2w02(1+ζ2)m+n/2Hm2xwHn2yw×expikz-ρ2w02(1+iζ)-iψ0=(-1)m+n2(m+n)/2πw0m+n+1w Hm2xwHn2yw×expikz-ρ2w02(1+iζ)-iψm,n,
Em,nH=1w(ζ) Hm2xw(ζ)Hn2yw(ζ)×expikz-ρ2w02(1+iζ)-iψm,n.
kx±iky=kx2+ky2exp(iα),
E˜m,nL-dkx2π-dky2π (kx+iky)m(kx-iky)n+mexp S.
E˜m,nL(x+iy)m(x-iy)n+mE0,
E˜m,nLω¯mωn+mE0.
Lmn(r)=exp(r)r-nm!dmdrm [exp(-r)rn+m],
ω¯mωn+mE0=ω¯m-ω¯w02(1+iζ)n+mE0=(-ω)-n-mm!ωω¯w02(1+iζ)nωw02(1+iζ)m×Lmnωω¯w02(1+iζ)E0.
E˜m,nL=exp(-inϕ)w(ζ)n+m+1 ρnLmnρ2w02(1+iζ)×expikz-ρ2w02(1+iζ)-iψ˜m,nL,
Em,nL-dkx2π-dky2πi(kx+iky)+1ukx+i kymi(kx-iky)+1ukx-i kyn+mexp S
Em,nLf2m+n2ω¯+iωufm2ω+iω¯ufn+mE0=(2f)2m+nexp-iωω¯2ufω¯m-2ω¯w2n+mexpiωω¯2ufE0=(2f)2m+n(-ω)-n-mm!2ωω¯w2n2ωw2mLmn2ωω¯w2E0=(-1)n+m2m+nm! exp(-inϕ)w2(n+m) ρnLmn2ρ2w2×expikz-ρ2w02(1+iζ)-iψm,nL,
Em,nL=exp(-inϕ)w(ζ)ρw(ζ)nLmn2ρ2w2(ζ)×expikz-ρ2w02(1+iζ)-iψn,mL.
Ekx2+ky2k2dkxdky(2π)2 (k×xˆ)×exp[ikxx+ikyy+i(k2-kx2-ky2)1/2z],

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