Abstract

It is shown that Hermite–Gaussian beams, Laguerre–Gaussian beams, and certain linear combinations thereof are the only finite-energy coherent beams that propagate, on free propagation, in a shape-invariant manner. All shape-invariant beams have Gouy phase of the universal c arctan(z/zR) form, with quantized values for the prefactor c. It is also shown that, as limiting cases, even two- and three-dimensional nondiffracting beams belong to this class when the Rayleigh distance goes to infinity. The results are deduced from the transport-of-intensity equations, by elementary means as well as by use of the Iwasawa decomposition. A pivotal role in the analysis is the finding that the only possible change in the phase front of a shape-invariant beam from one transverse plane to another is quadratic.

© 2004 Optical Society of America

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References

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  1. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986).
  2. J. Durnin, J. J. Miceli, J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
    [CrossRef] [PubMed]
  3. R. Simon, N. Mukunda, “Iwasawa decomposition in first-order optics: universal treatment of shape-invariant propagation for coherent and partially coherent beams,” J. Opt. Soc. Am. A 15, 2146–2155 (1998).
    [CrossRef]
  4. M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999).
  5. R. W. Boyd, “Intuitive explanation of the phase anomaly of focused light beams,” J. Opt. Soc. Am. 70, 877–880 (1980).
    [CrossRef]
  6. R. Simon, N. Mukunda, “Bargmann invariant and the geometry of the Gouy effect,” Phys. Rev. Lett. 70, 880–883 (1993).
    [CrossRef] [PubMed]
  7. D. Subbarao, “Topological phase in Gaussian beam optics,” Opt. Lett. 20, 2162–2164 (1995).
    [CrossRef] [PubMed]
  8. S. Feng, H. G. Winful, “Physical origin of the Gouy phase shift,” Opt. Lett. 26, 485–487 (2001).
    [CrossRef]
  9. Throughout the paper we set k=1,k being the wave number. This means that in all the subsequent formulas the variable z is intended to be the propagation distance divided by the wave number.
  10. T. E. Gureyev, A. Roberts, K. A. Nugent, “Partially coherent fields, the transport-of-intensity equation, and phase uniqueness,” J. Opt. Soc. Am. A 12, 1942–1946 (1995).
    [CrossRef]
  11. E. Madelung, “Quantum theory in hydrodynamical form,” Z. Phys. 40, 332–336 (1926).
  12. S. A. Akhmanov, A. P. Sukhorukov, V. Kokhlov, Laser Handbook (North-Holland, Amsterdam, 1972).
  13. M. Abramowitz, I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972).
  14. C. Bernardini, F. Gori, M. Santarsiero, “Converting states of a particle under uniform or elastic forces into free-particle states,” Eur. J. Phys. 16, 58–62 (1995).
    [CrossRef]
  15. We follow the standard convention by which ρ is negative for a diverging lens.
  16. G. S. Agarwal, R. Simon, “A simple realization of fractional Fourier transformation and relation to harmonic oscillator Green’s function,” Opt. Commun. 110, 23–26 (1994).
    [CrossRef]
  17. F. Gori, M. Santarsiero, V. Bagini, “Fractional Fourier transform and Fresnel transform,” Atti Fond. Giorgio Ronchi 51, 387–390 (1994).
  18. R. Simon, K. Sundar, N. Mukunda, “Twisted Gaussian Schell-Model beams. I. Symmetry structure and normal-mode spectrum,” J. Opt. Soc. Am. A 10, 2008–2016 (1993).
    [CrossRef]

2001 (1)

1998 (1)

1995 (3)

1994 (2)

G. S. Agarwal, R. Simon, “A simple realization of fractional Fourier transformation and relation to harmonic oscillator Green’s function,” Opt. Commun. 110, 23–26 (1994).
[CrossRef]

F. Gori, M. Santarsiero, V. Bagini, “Fractional Fourier transform and Fresnel transform,” Atti Fond. Giorgio Ronchi 51, 387–390 (1994).

1993 (2)

1987 (1)

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

1980 (1)

1926 (1)

E. Madelung, “Quantum theory in hydrodynamical form,” Z. Phys. 40, 332–336 (1926).

Abramowitz, M.

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

Agarwal, G. S.

G. S. Agarwal, R. Simon, “A simple realization of fractional Fourier transformation and relation to harmonic oscillator Green’s function,” Opt. Commun. 110, 23–26 (1994).
[CrossRef]

Akhmanov, S. A.

S. A. Akhmanov, A. P. Sukhorukov, V. Kokhlov, Laser Handbook (North-Holland, Amsterdam, 1972).

Bagini, V.

F. Gori, M. Santarsiero, V. Bagini, “Fractional Fourier transform and Fresnel transform,” Atti Fond. Giorgio Ronchi 51, 387–390 (1994).

Bernardini, C.

C. Bernardini, F. Gori, M. Santarsiero, “Converting states of a particle under uniform or elastic forces into free-particle states,” Eur. J. Phys. 16, 58–62 (1995).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999).

Boyd, R. W.

Durnin, J.

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

Eberly, J. H.

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

Feng, S.

Gori, F.

C. Bernardini, F. Gori, M. Santarsiero, “Converting states of a particle under uniform or elastic forces into free-particle states,” Eur. J. Phys. 16, 58–62 (1995).
[CrossRef]

F. Gori, M. Santarsiero, V. Bagini, “Fractional Fourier transform and Fresnel transform,” Atti Fond. Giorgio Ronchi 51, 387–390 (1994).

Gureyev, T. E.

Kokhlov, V.

S. A. Akhmanov, A. P. Sukhorukov, V. Kokhlov, Laser Handbook (North-Holland, Amsterdam, 1972).

Madelung, E.

E. Madelung, “Quantum theory in hydrodynamical form,” Z. Phys. 40, 332–336 (1926).

Miceli, J. J.

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

Mukunda, N.

Nugent, K. A.

Roberts, A.

Santarsiero, M.

C. Bernardini, F. Gori, M. Santarsiero, “Converting states of a particle under uniform or elastic forces into free-particle states,” Eur. J. Phys. 16, 58–62 (1995).
[CrossRef]

F. Gori, M. Santarsiero, V. Bagini, “Fractional Fourier transform and Fresnel transform,” Atti Fond. Giorgio Ronchi 51, 387–390 (1994).

Siegman, A. E.

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

Simon, R.

R. Simon, N. Mukunda, “Iwasawa decomposition in first-order optics: universal treatment of shape-invariant propagation for coherent and partially coherent beams,” J. Opt. Soc. Am. A 15, 2146–2155 (1998).
[CrossRef]

G. S. Agarwal, R. Simon, “A simple realization of fractional Fourier transformation and relation to harmonic oscillator Green’s function,” Opt. Commun. 110, 23–26 (1994).
[CrossRef]

R. Simon, K. Sundar, N. Mukunda, “Twisted Gaussian Schell-Model beams. I. Symmetry structure and normal-mode spectrum,” J. Opt. Soc. Am. A 10, 2008–2016 (1993).
[CrossRef]

R. Simon, N. Mukunda, “Bargmann invariant and the geometry of the Gouy effect,” Phys. Rev. Lett. 70, 880–883 (1993).
[CrossRef] [PubMed]

Stegun, I.

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

Subbarao, D.

Sukhorukov, A. P.

S. A. Akhmanov, A. P. Sukhorukov, V. Kokhlov, Laser Handbook (North-Holland, Amsterdam, 1972).

Sundar, K.

Winful, H. G.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999).

Atti Fond. Giorgio Ronchi (1)

F. Gori, M. Santarsiero, V. Bagini, “Fractional Fourier transform and Fresnel transform,” Atti Fond. Giorgio Ronchi 51, 387–390 (1994).

Eur. J. Phys. (1)

C. Bernardini, F. Gori, M. Santarsiero, “Converting states of a particle under uniform or elastic forces into free-particle states,” Eur. J. Phys. 16, 58–62 (1995).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Commun. (1)

G. S. Agarwal, R. Simon, “A simple realization of fractional Fourier transformation and relation to harmonic oscillator Green’s function,” Opt. Commun. 110, 23–26 (1994).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. Lett. (2)

R. Simon, N. Mukunda, “Bargmann invariant and the geometry of the Gouy effect,” Phys. Rev. Lett. 70, 880–883 (1993).
[CrossRef] [PubMed]

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

Z. Phys. (1)

E. Madelung, “Quantum theory in hydrodynamical form,” Z. Phys. 40, 332–336 (1926).

Other (6)

S. A. Akhmanov, A. P. Sukhorukov, V. Kokhlov, Laser Handbook (North-Holland, Amsterdam, 1972).

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

We follow the standard convention by which ρ is negative for a diverging lens.

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

M. Born, E. Wolf, Principles of Optics, 7th ed. (Cambridge U. Press, Cambridge, UK, 1999).

Throughout the paper we set k=1,k being the wave number. This means that in all the subsequent formulas the variable z is intended to be the propagation distance divided by the wave number.

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Equations (82)

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x2u+2izu=0.
u(x; z)=A(x; z)exp[iϕ(x; z)]
2zϕ=-(xϕ)2+D(I),
zI=-x[Ixϕ].
D(I)=I-1/2x2(I1/2).
I(x; z)=αI0(αx),
-+I0(x)xdx=0.
x[g(x; z)I0(αx)]=0,
g(x; z)=αx+αxϕ.
g(x; z)I0(αx)=f(z),
ϕ(x; z)=-12α(z)α(z)x2+ϕ0(z),
-+I(x; z)xdx=0,
-+I(x; z)xϕ(x; z)dx=0.
ϕ(x; z)=-12α(z)α(z)x2+ϕ0(z)+ψ(x; z),
-+I0(αx)xψ(x; z)dx=0.
I0(αx)xψ(x; z)=f(z)α(z),
zϕ=-12x2αα+ϕ0,
xϕ=-ααx,
D(I)=α2A0(t)A0(t)t=αx,
1α2αα2-ααx2+2ϕ0α2=A0(t)A0(t)t=αx.
αα2-αα=±α4/zR2,
ϕ0=ηα22,
α(z)=11±(z/zR)2,
α(z)=11+(z/zR)2.
ϕ(x; z)=-12ρ(z)x2+ϕ0(z),
ρ(z)=-z[1+(zR/z)2].
ϕ0(z)=ηzR2arctanzzR.
ϕ(x; z)=12x2z+zR2/z+ηzR2arctanzzR.
A0(t)=(t2/zR2+η)A0(t),
F0(ξ)=ξ24+zRη2F0(ξ).
F0,even(ξ)=exp-ξ24MηzR4+14,12,ξ22,
F0,odd(ξ)=ξ expξ24M-ηzR4+34,32, -ξ22,
η=-2zRn+12,n=0, 1, 2,.
A0(x)=exp[-x2/(2zR)]Hn(x/zR).
ϕ(x; z)=12x2z+zR2/z-n+12arctanzzR.
ϕ0(z)=ηz2,
A0(t)+kt2A0(t)=0,
2u+2izu=0,
2zϕ=-(ϕ)2+D(I),
zI=-·(Iϕ),
I(r; z)=α2I0(αr),
I(r; 0)ϕ(r; 0)=×Q(r),
ψ(r; z)=ψ[α(z)r; 0].
·[I0(αr)ψ(αr; 0)]=0.
·[g(r; z)I0(αr)]=0,
g(r; z)=αr+αϕ(r; z).
·[αI0(αr)ψ(αr; 0)]+·(I0(αr)[αr+αϕ1(r; z)])=0.
·[αI0(αr)ψ(αr; 0)]=0,
I0(αr)[αr+αϕ1(r; z)]=0.
ϕ1(r; z)=-12α(z)α(z)r2+ϕ0(z),
zϕ=-12ααr2+ααψ(αr; 0)·r+ϕ0,ϕ=-ααr+ψ(αr; 0),D(I)=α22A0(t)A0(t)t=αr,
1α2αα2-ααr2+2ϕ0α2+[ψ(αr; 0)]2=2A0(αr)A0(αr),
ϕ(r; z)=ψ(α(z)r; 0)-12ρ(z)r2+ϕ0(z),
u(r; z)=α(z)exp[iϕ0(z)]exp-i12ρ(z)r2u[α(z)r; 0].
2A0(t)=t2zR2+η+[ψ(t;0)]2A0(t).
2A0(t)=t2zR2+η+s2t2A0(t).
1tddttdGdt=t2zR2-2zR(2l+s+1)+s2t2G(t),
A0(r)=Ar2sLl(s)r2zRexp-r22zR,
2A0(t)={η+[ψ(t;0)]2}A0(t),
A0(r)=AJs(ktr),
F(z)=1z01,
F(z)=L(ρ)M(α)R(θ),
L(ρ)=10-ρ-11,
M(α)=α-100α,
R(θ)=cos θzR sin θ-zR-1 sin θcos θ.
ρ(z)-1=-z[1+(zR/z)2],
α(z)=[1+(z/zR)2]-1/2,
θ(z)=arctan(z/zR).
Uˆz=Uˆρ(z)Uˆα(z)Uˆθ(z)zR.
Uˆ[α(z)]-1Uˆ-ρ(z)Uˆz=Uˆθ(z)zR.
Uˆθ(z)zR=exp[-iθ(z)Hˆ],
Hˆ=2w02x2+y22-w022x2+y22.
Ψm,n(r)=A2πw021/2exp-x2+y2w02×Hm2xw0Hn2yw0;
HˆΨm,n(r)=(m+n+1)Ψm,n(r),  m,n=0, 1, 2.
Uˆθ(z)zRΨm,n(r)=exp[-i(m+n+1)θ(z)]Ψm,n(r).
u(r; z)=α(z)exp[iϕ0(z)]exp-i12ρ(z)r2u(α(z)r; 0).
u(r; z)Uˆzu(r; 0)=exp[iϕ0(z)]Uˆρ(z)Uˆα(z)u(r; 0).
Uˆ[α(z)]-1Uˆ-ρ(z)Uˆzu(r; 0)=exp[iϕ0(z)]u(r; 0).
Uˆθ(z)zRu(r; 0)=exp[iϕ0(z)]u(r; 0).
u(r; 0)=Ψm,n(r),m,n=0, 1, 2,,
ϕ0(z)=-(m+n+1)arctan(z/zR),
ϕ(r; z)=12r2z+zR2/z-(n+m+1)arctanzzR.

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