Abstract

New exact solutions to the paraxial wave equation are obtained in the form of a product of Laguerre polynomials, Bessel functions, and Gaussian functions. In the limit of large Laguerre–Gaussian beam size, the Bessel factor dominates and the solution sets reduce to the modes of closed resonators, hollow metal waveguides, and dielectric waveguides. In the opposite limit the solutions reduce to Laguerre–Gaussian modes of open resonators and graded-index waveguides. These solutions are valid for electromagnetic waves traveling through free space, and they are valid for propagation through circularly symmetric optical systems representable by ABCD matrices as well. An interesting feature of the new solution set is the existence of three mode indices, where only two are required for an orthogonal expansion. As an example, Laguerre–Gaussian beam propagation through an optical system that contains a Bessel-like amplitude filter is discussed.

© 2000 Optical Society of America

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References

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  1. R. H. Dicke, “Molecular amplification and generation systems and methods,” U.S. patent2,851,652 (September9, 1958).
  2. G. D. Boyd, J. P. Gordon, “Confocal multimode resona-tor for millimeter through optical wavelength masers,” Bell Syst. Tech. J. 40, 489–508 (1961).
    [CrossRef]
  3. G. D. Boyd, H. Kogelnik, “Generalized confocal resonator theory,” Bell Syst. Tech. J. 41, 1347–1369 (1962).
    [CrossRef]
  4. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), 798–804.
  5. L. W. Casperson, A. A. Tovar, “Hermite–sinusoidal–Gaussian beam in complex optical systems,” J. Opt. Soc. Am. A 15, 954–961 (1998).
    [CrossRef]
  6. L. W. Casperson, D. G. Hall, A. A. Tovar, “Sinusoidal–Gaussian beams in complex optical systems,” J. Opt. Soc. Am. A 14, 3341–3348 (1997).
    [CrossRef]
  7. J. W. Strutt ( Rayleigh), “On the passage of electric waves through tubes, or the vibrations of dielectric cylinders,” Phil. Mag. Suppl. 43, 125–132 (1897).
    [CrossRef]
  8. R. H. Jordan, D. G. Hall, “Free-space azimuthal paraxial wave equation: the azimuthal Bessel–Gauss beam,” Opt. Lett. 19, 427–429 (1994).
    [CrossRef] [PubMed]
  9. 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]
  10. H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).
    [CrossRef]
  11. H. Kogelnik, T. Li, “Laser beams and resonators,” Appl. Opt. 5, 1550–1567 (1966).
    [CrossRef] [PubMed]
  12. A. A. Tovar, L. W. Casperson, “Generalized beam matrices. II. Mode selection in lasers and periodic misaligned complex optical systems,” J. Opt. Soc. Am. A 13, 90–96 (1997).
    [CrossRef]
  13. I. Kimel, L. R. Elias, “Relations between Hermite and Laguerre Gaussian Modes,” IEEE J. Quantum Electron. 29, 2562–2567 (1993).
    [CrossRef]
  14. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), Eq. (9.1.21).
  15. See, for example, M. R. Spiegel, Schaum’s Outline Series: Mathematical Handbook (McGraw-Hill, New York, 1991), Chap. 24.
  16. L. W. Casperson, “Spatial modulation of Gaussian laser beams,” Opt. Quantum Electron. 10, 483–493 (1978).
    [CrossRef]

1998 (1)

1997 (3)

1994 (1)

1993 (1)

I. Kimel, L. R. Elias, “Relations between Hermite and Laguerre Gaussian Modes,” IEEE J. Quantum Electron. 29, 2562–2567 (1993).
[CrossRef]

1978 (1)

L. W. Casperson, “Spatial modulation of Gaussian laser beams,” Opt. Quantum Electron. 10, 483–493 (1978).
[CrossRef]

1966 (1)

1965 (1)

H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).
[CrossRef]

1962 (1)

G. D. Boyd, H. Kogelnik, “Generalized confocal resonator theory,” Bell Syst. Tech. J. 41, 1347–1369 (1962).
[CrossRef]

1961 (1)

G. D. Boyd, J. P. Gordon, “Confocal multimode resona-tor for millimeter through optical wavelength masers,” Bell Syst. Tech. J. 40, 489–508 (1961).
[CrossRef]

1897 (1)

J. W. Strutt ( Rayleigh), “On the passage of electric waves through tubes, or the vibrations of dielectric cylinders,” Phil. Mag. Suppl. 43, 125–132 (1897).
[CrossRef]

J. W. Strutt ( Rayleigh), “On the passage of electric waves through tubes, or the vibrations of dielectric cylinders,” Phil. Mag. Suppl. 43, 125–132 (1897).
[CrossRef]

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), Eq. (9.1.21).

Boyd, G. D.

G. D. Boyd, H. Kogelnik, “Generalized confocal resonator theory,” Bell Syst. Tech. J. 41, 1347–1369 (1962).
[CrossRef]

G. D. Boyd, J. P. Gordon, “Confocal multimode resona-tor for millimeter through optical wavelength masers,” Bell Syst. Tech. J. 40, 489–508 (1961).
[CrossRef]

Casperson, L. W.

Clark, G. H.

Dicke, R. H.

R. H. Dicke, “Molecular amplification and generation systems and methods,” U.S. patent2,851,652 (September9, 1958).

Elias, L. R.

I. Kimel, L. R. Elias, “Relations between Hermite and Laguerre Gaussian Modes,” IEEE J. Quantum Electron. 29, 2562–2567 (1993).
[CrossRef]

Gordon, J. P.

G. D. Boyd, J. P. Gordon, “Confocal multimode resona-tor for millimeter through optical wavelength masers,” Bell Syst. Tech. J. 40, 489–508 (1961).
[CrossRef]

Hall, D. G.

Jordan, R. H.

Kimel, I.

I. Kimel, L. R. Elias, “Relations between Hermite and Laguerre Gaussian Modes,” IEEE J. Quantum Electron. 29, 2562–2567 (1993).
[CrossRef]

Kogelnik, H.

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

H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).
[CrossRef]

G. D. Boyd, H. Kogelnik, “Generalized confocal resonator theory,” Bell Syst. Tech. J. 41, 1347–1369 (1962).
[CrossRef]

Li, T.

Rayleigh,

J. W. Strutt ( Rayleigh), “On the passage of electric waves through tubes, or the vibrations of dielectric cylinders,” Phil. Mag. Suppl. 43, 125–132 (1897).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), 798–804.

Spiegel, M. R.

See, for example, M. R. Spiegel, Schaum’s Outline Series: Mathematical Handbook (McGraw-Hill, New York, 1991), Chap. 24.

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), Eq. (9.1.21).

Strutt, J. W.

J. W. Strutt ( Rayleigh), “On the passage of electric waves through tubes, or the vibrations of dielectric cylinders,” Phil. Mag. Suppl. 43, 125–132 (1897).
[CrossRef]

Tovar, A. A.

Appl. Opt. (1)

Bell Syst. Tech. J. (3)

G. D. Boyd, J. P. Gordon, “Confocal multimode resona-tor for millimeter through optical wavelength masers,” Bell Syst. Tech. J. 40, 489–508 (1961).
[CrossRef]

G. D. Boyd, H. Kogelnik, “Generalized confocal resonator theory,” Bell Syst. Tech. J. 41, 1347–1369 (1962).
[CrossRef]

H. Kogelnik, “Imaging of optical modes—resonators with internal lenses,” Bell Syst. Tech. J. 44, 455–494 (1965).
[CrossRef]

IEEE J. Quantum Electron. (1)

I. Kimel, L. R. Elias, “Relations between Hermite and Laguerre Gaussian Modes,” IEEE J. Quantum Electron. 29, 2562–2567 (1993).
[CrossRef]

J. Opt. Soc. Am. A (4)

Opt. Lett. (1)

Opt. Quantum Electron. (1)

L. W. Casperson, “Spatial modulation of Gaussian laser beams,” Opt. Quantum Electron. 10, 483–493 (1978).
[CrossRef]

Phil. Mag. Suppl. (1)

J. W. Strutt ( Rayleigh), “On the passage of electric waves through tubes, or the vibrations of dielectric cylinders,” Phil. Mag. Suppl. 43, 125–132 (1897).
[CrossRef]

Other (4)

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), 798–804.

R. H. Dicke, “Molecular amplification and generation systems and methods,” U.S. patent2,851,652 (September9, 1958).

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1970), Eq. (9.1.21).

See, for example, M. R. Spiegel, Schaum’s Outline Series: Mathematical Handbook (McGraw-Hill, New York, 1991), Chap. 24.

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Figures (2)

Fig. 1
Fig. 1

 l=2, p=3, q=0 Laguerre–Bessel–Gaussian mode profiles where the ratio of the width of the Laguerre–Gaussian portion of the beam to the J0 Bessel portion of the beam is (a) 0, (b) 0.5, (c) 1, (d) 2, (e) 10. The first plot is a pure Laguerre–Gaussian mode, and the last plot is primarily a Bessel–Gaussian beam.

Fig. 2
Fig. 2

Example of an optical system containing a Bessel-like transmission filter, as discussed in Section 4. In this system the filter is centered between two identical lenses of focal length f and two lengths of free space, each with distance d.

Equations (65)

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2E+k2E=-2kk E.
E(x, y, z)=ψ(x, y, z)exp-i0zk0(z)dzixiy.
k(x, y, z)=k0(z)-k2x(z)(x2+y2)/2,
k2(x, y, z)k0(z)[k0(z)-k2x(z)(x2+y2)].
2ψx2+2ψy2-2ik0(z) ψz-[k0(z)k2x(z)(x2+y2)]ψ=0,
ψmn(x, y, z)=Hm21/2xW(z)Hn21/2yW(z)×exp-ik0(z)2q(z) (x2+y2)+Sx(z)x+Sy(z)y+P(z),
1q(z)=C(z)+D(z)/q(0)A(z)+B(z)/q(0),
Sx(z)=Sx(0)A(z)+B(z)/q(0),
Sy(z)=Sy(0)A(z)+B(z)/q(0),
W2(z)=W2(0)[A(z)+B(z)/q(0)]2+4iB(z)[A(z)+B(z)/q(0)]/k0(0),
P(z)=P(0)-i ln[A(z)+B(z)/q(0)]-i2(m+n)ln1+4ik0(0)W2(0)×B(z)A(z)+B(z)/q(0)×12k0(0)[Sx2(0)+Sy2(0)]B(z)A(z)+B(z)/q(0).
Melement=A(z)B(z)C(z)D(z).
ψmn,2(x, y)
=Hm21/2xW2Hn21/2yW2×exp-ik022q2 (x2+y2)+Sx2x+Sy2y+P2,
1q2=C+D/q1A+B/q1,
Sx2=Sx1A+B/q1,
Sy2=Sy1A+B/q1,
W22=W12(A+B/q1)2+4iB(A+B/q1)/k01,
P2=P1-i ln(A+B/q1)+i2(m+n)ln1+4ik01W12BA+B/q1-12k01(Sx12+Sy12)BA+B/q1.
uu/q2=ABCD uu/q1.
uu/q3=A2B2C2D2 uu/q2.
uu/q3=A2B2C2D2 ABCD uu/q.
Msystem=A2B2C2D2 ABCD.
Msystem=MnMn-1M3M2M1.
E0,2=AJE0,1,
AJ=exp-i0zk0(z)dz.
ψ2r-2s+l,2p-2r+2s,2(x, y)
=H2r-2s+l21/2xW2H2p-2r+2s21/2yW2×exp-ik022q2 (x2+y2)+Sx2x+Sy2y+P2.
P2=P1-i ln(A+B/q1)+i2(2p+l)ln1+4ik01W12BA+B/q1×-12k01(Sx12+Sy12)BA+B/q1.
ψtotal(x, y)=(-1)p22p+lp!r=0ps=0l/2(-1)sprl2s×ψ2r-2s+l,2p-2r+2s,2(x, y).
21/2rWlLpl2r2W2cos(lϕ)
=(-1)p22p+lp!r=0ps=0l/2(-1)sprl2s×H2r-2s+l21/2xWH2p-2r+2s21/2yW.
ψtotal=21/2rWlLPl2r2W2cos(lϕ)exp-ik02r22q2+P2×exp[-i(Sx2x+Sy2y)].
ψtotal=21/2rWlLpl2r2W2exp-ik02r22q2+P2J(r, ϕ),
J(r, ϕ)exp(-i[Sx2r cos(ϕ)+Sy2r sin(ϕ)])×exp[-i(l-q)ϕ]exp(-iqϕ)exp[i(l-q)ϕ]exp(iqϕ),
J(r, ϕ)14πexp(-i[(l-q)ϕ]exp(-i[q(ϕ0+ϕ)+Sx2r cos(ϕ)+Sy2r sin(ϕ)])exp(i[(l-q)ϕ]exp(-i[q(ϕ0-ϕ)+Sx2r cos(ϕ)+Sy2r sin(ϕ)]),
Sx2Ω2sin ϕ0,Sy2±Ω2cos ϕ0,
J(r, ϕ)=14πexp[-i(l-q)ϕ]exp(-i[q(ϕ0+ϕ)+Ω2 r sin(ϕ0+ϕ)])exp[i(l-q)ϕ]exp(-i[q(ϕ0-ϕ)+Ω2 r sin(ϕ0-ϕ)]),
Sx2-Ω2sin ϕ0,Sy2±Ω2cos ϕ0,
J(r, ϕ)14πexp(-i(l-q)ϕ]exp(i[q(ϕ0-ϕ)+Ω2r sin(ϕ0-ϕ)])exp(i(l-q)ϕ]exp(i[q(ϕ0+ϕ)+Ω2r sin(ϕ0+ϕ)]).
J(r, ϕ)12πcos[(l-q)ϕ+q(ϕ0+ϕ)+Ω2r sin(ϕ0+ϕ)]sin[(l-q)ϕ+q(ϕ0+ϕ)+Ω2r sin(ϕ0+ϕ)].
J(r, ϕ)|top12π(cos[(l-q)ϕ]cos[q(ϕ0+ϕ)+Ω2r sin(ϕ0+ϕ)]-sin[(l-q)ϕ]×sin[q(ϕ0+ϕ)+Ω2r sin(ϕ0+ϕ)).
J(r, ϕ)-ππJ(r, ϕ, ϕ0)dϕ0.
J(r, ϕ)|top12π(cos[(l-q)ϕ]-ππcos[q(ϕ0+ϕ)+Ω2r sin(ϕ0+ϕ)]dϕ0-sin[(l-q)ϕ]-ππsin[q(ϕ0+ϕ)+Ω2r sin(ϕ0+ϕ)]dϕ0),
J(r, ϕ)|top=12π(cos[(l-q)ϕ]ϕ-πϕ+πcos[qϕˆ+Ω2r sin(ϕˆ)]dϕˆ-sin[(l-q)ϕ]×ϕ-πϕ+πsin[qϕˆ+Ω2r sin(ϕˆ)]dϕˆ).
J(r, ϕ)|top=12π(cos[(l-q)ϕ]-ππcos[qϕˆ+Ω2r sin(ϕˆ)]dϕˆ-sin[(l-q)ϕ]×-ππsin[qϕˆ+Ω2r sin(ϕˆ)]dϕˆ).
J(r, ϕ)|top1πcos[(l-q)ϕ]×0πcos[qϕˆ+Ω2r sin(ϕˆ)]dϕˆ.
Jn(a)=1π0πcos[nϕˆ+a sin(ϕˆ)]dϕˆ,
J(r)|top=cos[(l-q)ϕ]Jq(Ω2r),
ψtotal=21/2rWlLpl2r2W2cos[(l-q)ϕ]sin[(l-q)ϕ]×Jq(Ω2r)exp-ik02r22q2+P2,
Iq(x)=(-1)qJq(ix)
E2(r, ϕ)=E0,2exp-ik02r22q2+P221/2rW2l×Lpl2r2W22Jq(Ω2r)Iq(Ω2r)Yq(Ω2r)Kq(Ω2r)×cos[(l-q)ϕ]sin[(l-q)ϕ]exp[i(l-q)ϕ]exp[-i(l-q)ϕ]ixiy.
I2(r, ϕ)I0,2=[2r2exp(-r2)L32(2r2)J0(ar)cos(2ϕ)]2,
Ω2sin(ϕ0)=Ω1sin(ϕ0)A+B/q1,
±Ω2cos(ϕ0)=±Ω1cos(ϕ0)A+B/q1.
Ω2=Ω1A+B/q1.
P2=P1-i ln(A+B/q1)+i2(2p+l)ln1+4ik01W12BA+B/q1±12k01Ω12BA+B/q1.
E1(r, ϕ)=E0,1exp-ik01r22q1+P1×21/2rW1lLpl2r2W12cos(lϕ)sin(lϕ)
E2(r, ϕ)=E0,2exp-ik02r22q2+P221/2rW2l×Lpl2r2W22cos(lϕ)sin(lϕ)J0(Ω2r),
T(r)=kAkJ0(λkr),
Ak=2J12(λk)01rT(r)J0(λkr)dr,
T(r)=T0|J0(ar)|.
Ak=2T0J12(λk)01r|J0(ar)|J0(λkr)dr.
Mbeforefilter=1d01 10-1/f1=1-d/fd-1/f1.
Mafterfilter=10-1/f1 1d01=1d-1/f1-d/f.

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