Abstract

The governing equations for the paraxial approximation are deduced from Maxwell’s equations. The plane wave, in the paraxial approximation, is found to be transverse electromagnetic. For a field distribution at the input plane having a general azimuthal variation and a radial variation in the form of a Bessel function of an integer order with a Gaussian envelope of a given waist size, the spreading due to the Fresnel diffraction is determined as the paraxial beam is transported in the axial direction. The effects of Fresnel diffraction are illustrated with examples for a beam transporting unit power. Diffraction patterns of azimuthally symmetrical and dipolar modes are presented.

© 1998 Optical Society of America

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References

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  1. D. Marcuse, Light Transmission Optics (Van NostrandReinhold, New York, 1972), Chap. 6.
  2. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 4.
  3. A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1967), Chap. 6.
  4. R. H. Jordan, D. G. Hall, “Free-space azimuthal paraxial wave equation: the azimuthal Bessel–Gauss beam solution,” Opt. Lett. 19, 427–429 (1994).
    [CrossRef] [PubMed]
  5. D. G. Hall, “Vector-beam solutions of Maxwell’s wave equation,” Opt. Lett. 21, 9–11 (1996).
    [CrossRef] [PubMed]
  6. M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
    [CrossRef]
  7. J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), Chaps. 1 and 6.
  8. N. N. Bogoliubov, Y. A. Mitropolsky, Asymptotic Methods in the Theory of Nonlinear Oscillations (Hindustan, Delhi, India, 1961), Chap. 1.
  9. G. N. Watson, Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, New York, 1966), Chaps. 2 and 13.
  10. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1984), Chap. 8.
  11. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995), pp. 263–285.
  12. F. Gori, “Fresnel transform and sampling theorem,” Opt. Commun. 39, 293–297 (1981).
    [CrossRef]
  13. F. Gori, “Why is Fresnel transform so little known?” in Current Trends in Optics, J. C. Dainty, ed. (Academic, New York, 1994), pp. 139–148.
  14. F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
    [CrossRef]
  15. C. Palma, R. Borghi, G. Cincotti, “Beams originated by J0-correlated Schell-model planar sources,” Opt. Commun. 125, 113–121 (1996).
    [CrossRef]
  16. A. A. Tovar, G. H. Clark, “Concentric-circle-grating, surface-emitting laser beam propagation in complex optical systems,” J. Opt. Soc. Am. A 14, 3333–3340 (1997).
    [CrossRef]

1997 (1)

1996 (2)

C. Palma, R. Borghi, G. Cincotti, “Beams originated by J0-correlated Schell-model planar sources,” Opt. Commun. 125, 113–121 (1996).
[CrossRef]

D. G. Hall, “Vector-beam solutions of Maxwell’s wave equation,” Opt. Lett. 21, 9–11 (1996).
[CrossRef] [PubMed]

1994 (1)

1987 (1)

F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

1981 (1)

F. Gori, “Fresnel transform and sampling theorem,” Opt. Commun. 39, 293–297 (1981).
[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]

Bogoliubov, N. N.

N. N. Bogoliubov, Y. A. Mitropolsky, Asymptotic Methods in the Theory of Nonlinear Oscillations (Hindustan, Delhi, India, 1961), Chap. 1.

Borghi, R.

C. Palma, R. Borghi, G. Cincotti, “Beams originated by J0-correlated Schell-model planar sources,” Opt. Commun. 125, 113–121 (1996).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1984), Chap. 8.

Cincotti, G.

C. Palma, R. Borghi, G. Cincotti, “Beams originated by J0-correlated Schell-model planar sources,” Opt. Commun. 125, 113–121 (1996).
[CrossRef]

Clark, G. H.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 4.

Gori, F.

F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

F. Gori, “Fresnel transform and sampling theorem,” Opt. Commun. 39, 293–297 (1981).
[CrossRef]

F. Gori, “Why is Fresnel transform so little known?” in Current Trends in Optics, J. C. Dainty, ed. (Academic, New York, 1994), pp. 139–148.

Guattari, G.

F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

Hall, D. G.

Jordan, R. H.

Lax, M.

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

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]

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995), pp. 263–285.

Marcuse, D.

D. Marcuse, Light Transmission Optics (Van NostrandReinhold, New York, 1972), Chap. 6.

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]

Mitropolsky, Y. A.

N. N. Bogoliubov, Y. A. Mitropolsky, Asymptotic Methods in the Theory of Nonlinear Oscillations (Hindustan, Delhi, India, 1961), Chap. 1.

Padovani, C.

F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

Palma, C.

C. Palma, R. Borghi, G. Cincotti, “Beams originated by J0-correlated Schell-model planar sources,” Opt. Commun. 125, 113–121 (1996).
[CrossRef]

Stratton, J. A.

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), Chaps. 1 and 6.

Tovar, A. A.

Watson, G. N.

G. N. Watson, Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, New York, 1966), Chaps. 2 and 13.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1984), Chap. 8.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995), pp. 263–285.

Yariv, A.

A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1967), Chap. 6.

J. Opt. Soc. Am. A (1)

Opt. Commun. (3)

F. Gori, G. Guattari, C. Padovani, “Bessel–Gauss beams,” Opt. Commun. 64, 491–495 (1987).
[CrossRef]

C. Palma, R. Borghi, G. Cincotti, “Beams originated by J0-correlated Schell-model planar sources,” Opt. Commun. 125, 113–121 (1996).
[CrossRef]

F. Gori, “Fresnel transform and sampling theorem,” Opt. Commun. 39, 293–297 (1981).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. A (1)

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

Other (9)

J. A. Stratton, Electromagnetic Theory (McGraw-Hill, New York, 1941), Chaps. 1 and 6.

N. N. Bogoliubov, Y. A. Mitropolsky, Asymptotic Methods in the Theory of Nonlinear Oscillations (Hindustan, Delhi, India, 1961), Chap. 1.

G. N. Watson, Theory of Bessel Functions, 2nd ed. (Cambridge U. Press, New York, 1966), Chaps. 2 and 13.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1984), Chap. 8.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, New York, 1995), pp. 263–285.

F. Gori, “Why is Fresnel transform so little known?” in Current Trends in Optics, J. C. Dainty, ed. (Academic, New York, 1994), pp. 139–148.

D. Marcuse, Light Transmission Optics (Van NostrandReinhold, New York, 1972), Chap. 6.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 4.

A. Yariv, Quantum Electronics, 2nd ed. (Wiley, New York, 1967), Chap. 6.

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Figures (2)

Fig. 1
Fig. 1

Transverse distribution of the normalized Poynting vector for the m=0 beam. The physical parameters are P=1 and βw0/2=1. (a) z/b=0, (b) z/b=0.2, (c) z/b=0.4, and (d) z/b=0.6.

Fig. 2
Fig. 2

Same as in Fig. 1, but for the m=1 dipolar beam.

Equations (81)

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×E=ikH,
×H=-ikE,
·H=0,
·E=0,
E=ikA-(·A)ik-×F,
H=×A+ikF-(·F)ik
(2+k2)(A, F)=0.
Et=zˆ×tFz,
Ez=0,
Ht=-1ikt zFz,
Hz=-1ik2z2+k2Fz,
Et=-1ikt zAz,
Ez=-1ik2z2+k2Az,
Ht=-zˆ×tAz,
Hz=0,
Fz(z0; x1, y1, z2)=Fz0(z0; x1, y1, z2)+δ2Fz2(z0; x1, y1, z2),
2z02+k2Fz0=0,
2z02+k2Fz2=-2x12+2y12+2 2z0z2Fz0.
Fz0=F0(x1, y1; z2)exp(ikz0),
2x12+2y12+2ik z2F0(x1, y1; z2)=0.
Et=δzˆ×(1tF0)exp(ikz0),
Ez=0,
Ht=-δ1tF0+δ2 1ikF0z2exp(ikz0),
Hz=-δ22 F0z2exp(ikz0),
1t=xˆ x1+yˆ y1.
Ht=zˆ×Et=-(tF0)exp(ikz),
2x2+2y2+2ik zF0(x, y; z)=0.
F0(x, y; z)=1(2π)2--F¯0(ξx, ξy; z)×exp[i(ξxx+ξyy)]dξxdξy,
F¯0(ξx, ξy; z)=--F0(x, y; z)×exp[-i(ξxx+ξyy)]dxdy.
z+i(ξx2+ξy2)2kF¯0(ξx, ξy; z)=0,
F¯0(ξx, ξy; z)=F¯0(ξx, ξy; z=0)exp-i(ξx2+ξy2)2kz.
x=ρ cos ϕ,y=ρ sin ϕ,
ξx=η cos Θ,ξy=η sin Θ
F0(ρ, ϕ; z)=1(2π)20dηη02πdΘF¯0(η, Θ; z)×exp[iηρ cos(Θ-ϕ)],
F¯0(η, Θ; z)=0dρρ02πdϕF0(ρ, ϕ; z)×exp[-iηρ cos(ϕ-Θ)].
Fz0(ρ, ϕ; z=0)=F0(ρ, ϕ; z=0)=N cos mϕJm(βρ)exp(-ρ2/w02),
F¯0(η, Θ; z=0)=N0dρρJm(βρ)exp(-ρ2/w02)×02πcos mϕ exp[-iηρ cos(ϕ-Θ)]dϕ.
F¯0(η, Θ; z=0)
=N2πi-m cos mΘ
×0dρρ exp(-ρ2/w02)Jm(βρ)Jm(ηρ).
F¯0(η, Θ; z=0)=Nπ(-1)-mw02 exp(-β˜2)×cosmΘexp(-η2w02/4)Jm(iβ˜w0η),
β˜=βw0/2.
F0(ρ, ϕ; z)
=N4π(-1)mw02 exp(-β˜2)
×0dηη exp-η2w024(1+iz¯)Jm(iβ˜w0η)×02πdΘ cos mΘ exp[iηρ cos(Θ-ϕ)],
z˜=z/b,
b=12kw02.
F0(ρ, ϕ, z)=N(1+iz˜)cos mϕJmβρ(1+iz˜)×exp-(ρ˜2+iz˜β˜2)(1+iz˜),
ρ˜=ρ/w0.
h(x, y, z)=-ik2πzexpik2z(x2+y2),
--h(x, y, z)dxdy=1.
F0(x, y, z)=--dxpdypF0(xp, yp; z=0)×h(x-xp, y-yp; z).
xp=ρp cos ϕp,yp=ρp sin ϕp,
F0(ρ, ϕ, z)=-ik2πzexpikρ22z002πdρpρpdϕp×F0(ρp, ϕp; z=0)expikρp22z×exp-ikρρpzcos(ϕp-ϕ).
F0(ρ, ϕ, z)=-Nik2πzexpikρ22z×0dρpρp expikρp22z1+i zb×Jm(βρp)02πdϕp cos mϕp×exp-ikρρpzcos(ϕp-ϕ).
F0(ρ, ϕ, z)=-Nikzi-m cos mϕ expikρ22z×0dρpρp expikρp22z1+i zb×Jm(βρp)Jmkρρpz.
Hρ=-Eϕ=-ρF0(ρ, ϕ, z)exp(ikz)=-N(1+iz˜)2cos mϕ exp-(ρ˜2+iz˜β˜2)(1+iz˜)exp(ikz)×βJmβρ(1+iz˜)-2ρw02Jmβρ(1+iz˜),
Hϕ=Eρ=-1ρϕF0(ρ, ϕ, z)exp(ikz)=N(1+iz˜)sin mϕ mρJmβρ(1+iz˜)×exp-(ρ˜2+iz˜β˜2)(1+iz˜)exp(ikz).
S=zˆSz=zˆ c2(EρHϕ*-EϕHρ*).
Sz=c2N2β2(1+z˜2)2exp-2(ρ˜2+z˜2β˜2)(1+z˜2)×cos2mϕJmβρ(1+iz˜)-2ρβw02Jmβρ(1+iz˜)2+sin2 mϕm(1+iz˜)βρJmβρ(1+iz˜)2.
P=0S(ρ, z)ρdρ,
S(ρ, z)=02πSz(ρ, ϕ, z)dϕ=c2N2πβ2(1+z˜2)2exp-2z˜2β˜2(1+z˜2)exp-2ρ˜2(1+z˜2)×Jmβρ(1+iz˜)-2ρβw02Jmβρ(1+iz˜)2+m(1+iz˜)βρJmβρ(1+iz˜)2.
P=c2N2πβ2(1+z˜2)2exp-2z˜2β˜2(1+z˜2)×J1-1β˜w0(J2+J2*)+1β˜2w02J3,
J1=0dρρ exp-2ρ˜2(1+z˜2)Jmβρ(1+iz˜)2+mβρ(1+iz˜)Jmβρ(1+iz˜)2,
J2=0dρρ exp-2ρ˜2(1+z˜2)ρJmβρ(1+iz˜)Jmβρ1-iz˜,
J3=0dρρ exp-2ρ˜2(1+z˜2)ρ2Jmβρ(1+iz˜)×Jmβρ(1-iz˜).
J3=12w02(1+z˜2)J4+β˜w02[(1-iz˜)J2+(1+iz˜)J2*],
J4=0dρρ exp-2ρ˜2(1+z˜2)×Jmβρ(1+iz˜)Jmβρ(1-iz˜).
J2=β˜w02[-(1-iz˜)J4+(1+iz˜)J1].
P=c4N2πβ2(1+z˜2)exp-2z˜2β˜2(1+z˜2)1β˜2(1+β˜2)J4+J1.
J4=14w02(1+z˜2)exp(-β˜2)exp2z˜2β˜2(1+z˜2)Im(β˜2),
2Jm(z)=Jm-1(z)-Jm+1(z),
2 mzJm(z)=Jm-1(z)+Jm+1(z)
J1=0dρρ exp-2ρ˜2(1+z˜2)×12Jm-1βρ(1+iz˜)Jm-1βρ(1-iz˜)+Jm+1βρ(1+iz˜)Jm+1βρ(1-iz˜).
J1=14w02(1+z˜2)exp(-β˜2)exp2z˜2β˜2(1+z˜2)Im(β˜2).
P=c4N2 π exp(-β˜2)[(1+β˜2)Im(β˜2)+β˜2Im(β˜2)].
Sz π2β2=P exp(β˜2)[(1+β˜2)Im(β˜2)+β˜2Im(β˜2)]-1×(1+z˜2)-2 exp-2(ρ˜2+z˜2β˜2)(1+z˜2)×cos2 mϕJm2β˜ρ˜(1+iz˜)-ρ˜β˜Jm2β˜ρ˜(1+iz˜)2+sin2 mϕm(1+iz˜)2β˜ρ˜Jm2β˜ρ˜(1+iz˜)2.
Hρ=-Eϕ=-Q cos ϕ,
Hϕ=Eρ=Q sin ϕ,
Q=Nβ2(1+iz˜)2exp-iz˜β˜2(1+iz˜)exp(ikz).
E=yˆQ,H=-xˆQ.

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