Abstract

Improved beam propagation method (BPM) equations are derived for the general case of arbitrary refractive-index spatial distributions. It is shown that in the paraxial approximation the discrete equations admit an analytical solution for the propagation of a paraxial spherical wave, which converges to the analytical solution of the paraxial Helmholtz equation. The generalized Kirchhoff–Fresnel diffraction integral between the object and the image planes can be derived, with its coefficients expressed in terms of the standard ABCD matrix. This result allows the substitution, in the case of an unaberrated system, of the many numerical steps with a single analytical step. We compared the predictions of the standard and improved BPM equations by considering the cases of a Maxwell fish-eye and of a Luneburg lens.

© 1998 Optical Society of America

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References

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  1. M. D. Feit, J. A. Fleck, “Light propagation in graded-index optical fibers,” Appl. Opt. 17, 3990–3998 (1978).
    [CrossRef] [PubMed]
  2. J. van der Donk, P. E. Lagasse, “Analysis of geodesic lenses by beam propagation method,” Electron. Lett. 16, 292–294 (1980).
    [CrossRef]
  3. J. Van Roey, J. van der Donk, P. E. Lagasse, “Beam-propagation method: analysis and assessment,” J. Opt. Soc. Am. 71, 803–810 (1981).
    [CrossRef]
  4. J. J. Gribble, J. M. Arnold, “Beam-propagation method ray equation,” Opt. Lett. 13, 611–613 (1988).
    [CrossRef] [PubMed]
  5. G. N. Lawrence, S.-H. Hwang, “Beam propagation in gradient refractive index media,” Appl. Opt. 31, 5201–5210 (1992).
    [CrossRef] [PubMed]
  6. A. Di Sebastiano, G. Pozzi, “Improved beam-propagation method equations for the analysis of integrated optics lenses,” Opt. Lett. 17, 472–474 (1992).
    [CrossRef] [PubMed]
  7. C. E. Pearson, Numerical Methods in Engineering and Science (Van Nostrand Reinhold, New York, 1986), Chap. 6.
  8. G. Pozzi, “Multislice approach to lens analysis,” Adv. Imaging Electron Phys. 93, 173–217 (1995).
    [CrossRef]
  9. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1987), Chap. 4.
  10. A. R. Mickelson, Physical Optics (Van Nostrand Reinhold, New York, 1992), Chap. 7.
    [CrossRef]
  11. L. W. Casperson, “Beam propagation in tapered quadratic index waveguides: numerical solutions,” J. Lightwave Technol. LT-3, 256–263 (1985).
    [CrossRef]
  12. L. W. Casperson, “Beam propagation in periodic quadratic-index waveguides,” Appl. Opt. 24, 4395–4403 (1985).
    [CrossRef] [PubMed]
  13. A. A. Tovar, L. W. Casperson, “Beam propagation in parabolically tapered graded-index waveguides,” Appl. Opt. 33, 7733–7739 (1994).
    [CrossRef] [PubMed]
  14. J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978), Chap. 9.
  15. Ref. 14, Chap. 10.
  16. G. Dattoli, J. C. Gallardo, A. Torre, “An algebraic view to the operatorial ordering and its application to optics,” Riv. Nuovo Cimento 11, 1–79 (1988).
    [CrossRef]
  17. S. A. Collins, “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. 60, 1168–1177 (1970).
    [CrossRef]
  18. A. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986), Chap. 20.
  19. K. Tanaka, “Paraxial theory of rotationally distributed-index media by means of Gaussian constants,” Appl. Opt. 23, 1700–1706 (1984).
    [CrossRef] [PubMed]
  20. J. N. McMullin, “The ABCD matrix in arbitrarily tapered quadratic-index waveguides,” Appl. Opt. 25, 2184–2187 (1986).
    [CrossRef] [PubMed]
  21. mathematica V. 2.2 (Wolfram Research, Champaign, Ill., 1994).
  22. Ref. 7, Chap. 7.

1995

G. Pozzi, “Multislice approach to lens analysis,” Adv. Imaging Electron Phys. 93, 173–217 (1995).
[CrossRef]

1994

1992

1988

J. J. Gribble, J. M. Arnold, “Beam-propagation method ray equation,” Opt. Lett. 13, 611–613 (1988).
[CrossRef] [PubMed]

G. Dattoli, J. C. Gallardo, A. Torre, “An algebraic view to the operatorial ordering and its application to optics,” Riv. Nuovo Cimento 11, 1–79 (1988).
[CrossRef]

1986

1985

L. W. Casperson, “Beam propagation in tapered quadratic index waveguides: numerical solutions,” J. Lightwave Technol. LT-3, 256–263 (1985).
[CrossRef]

L. W. Casperson, “Beam propagation in periodic quadratic-index waveguides,” Appl. Opt. 24, 4395–4403 (1985).
[CrossRef] [PubMed]

1984

1981

1980

J. van der Donk, P. E. Lagasse, “Analysis of geodesic lenses by beam propagation method,” Electron. Lett. 16, 292–294 (1980).
[CrossRef]

1978

1970

Arnold, J. M.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1987), Chap. 4.

Casperson, L. W.

Collins, S. A.

Dattoli, G.

G. Dattoli, J. C. Gallardo, A. Torre, “An algebraic view to the operatorial ordering and its application to optics,” Riv. Nuovo Cimento 11, 1–79 (1988).
[CrossRef]

Di Sebastiano, A.

Feit, M. D.

Fleck, J. A.

Gallardo, J. C.

G. Dattoli, J. C. Gallardo, A. Torre, “An algebraic view to the operatorial ordering and its application to optics,” Riv. Nuovo Cimento 11, 1–79 (1988).
[CrossRef]

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978), Chap. 9.

Gribble, J. J.

Hwang, S.-H.

Lagasse, P. E.

J. Van Roey, J. van der Donk, P. E. Lagasse, “Beam-propagation method: analysis and assessment,” J. Opt. Soc. Am. 71, 803–810 (1981).
[CrossRef]

J. van der Donk, P. E. Lagasse, “Analysis of geodesic lenses by beam propagation method,” Electron. Lett. 16, 292–294 (1980).
[CrossRef]

Lawrence, G. N.

McMullin, J. N.

Mickelson, A. R.

A. R. Mickelson, Physical Optics (Van Nostrand Reinhold, New York, 1992), Chap. 7.
[CrossRef]

Pearson, C. E.

C. E. Pearson, Numerical Methods in Engineering and Science (Van Nostrand Reinhold, New York, 1986), Chap. 6.

Pozzi, G.

Siegman, A.

A. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986), Chap. 20.

Tanaka, K.

Torre, A.

G. Dattoli, J. C. Gallardo, A. Torre, “An algebraic view to the operatorial ordering and its application to optics,” Riv. Nuovo Cimento 11, 1–79 (1988).
[CrossRef]

Tovar, A. A.

van der Donk, J.

J. Van Roey, J. van der Donk, P. E. Lagasse, “Beam-propagation method: analysis and assessment,” J. Opt. Soc. Am. 71, 803–810 (1981).
[CrossRef]

J. van der Donk, P. E. Lagasse, “Analysis of geodesic lenses by beam propagation method,” Electron. Lett. 16, 292–294 (1980).
[CrossRef]

Van Roey, J.

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1987), Chap. 4.

Adv. Imaging Electron Phys.

G. Pozzi, “Multislice approach to lens analysis,” Adv. Imaging Electron Phys. 93, 173–217 (1995).
[CrossRef]

Appl. Opt.

Electron. Lett.

J. van der Donk, P. E. Lagasse, “Analysis of geodesic lenses by beam propagation method,” Electron. Lett. 16, 292–294 (1980).
[CrossRef]

J. Lightwave Technol.

L. W. Casperson, “Beam propagation in tapered quadratic index waveguides: numerical solutions,” J. Lightwave Technol. LT-3, 256–263 (1985).
[CrossRef]

J. Opt. Soc. Am.

Opt. Lett.

Riv. Nuovo Cimento

G. Dattoli, J. C. Gallardo, A. Torre, “An algebraic view to the operatorial ordering and its application to optics,” Riv. Nuovo Cimento 11, 1–79 (1988).
[CrossRef]

Other

A. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986), Chap. 20.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978), Chap. 9.

Ref. 14, Chap. 10.

C. E. Pearson, Numerical Methods in Engineering and Science (Van Nostrand Reinhold, New York, 1986), Chap. 6.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, UK, 1987), Chap. 4.

A. R. Mickelson, Physical Optics (Van Nostrand Reinhold, New York, 1992), Chap. 7.
[CrossRef]

mathematica V. 2.2 (Wolfram Research, Champaign, Ill., 1994).

Ref. 7, Chap. 7.

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Figures (4)

Fig. 1
Fig. 1

Logarithmic contour map of the intensity of a Gaussian beam propagating through a Maxwell fish-eye lens, as calculated according to (a) the IBPM and (b) the standard BPM.

Fig. 2
Fig. 2

Intensity I, in arbitrary units (solid curves), and phase ph, in radians (dashed line and curves), profiles of a Gaussian beam propagating through a Maxwell fish-eye lens at four different values of z. The upper and lower intensity profiles are calculated by means of the IBPM and of the ABCD matrix approach of Lawrence and Hwang,5 respectively.

Fig. 3
Fig. 3

Peak intensity along the optical axis of a Gaussian beam propagating through a Maxwell fish-eye lens, as calculated according to the IBPM (upper curve) and the ABCD matrix approach of Lawrence and Hwang5 (lower curve).

Fig. 4
Fig. 4

Intensity distribution of a Gaussian beam propagating in a classical Luneburg lens, as calculated by means of (a) the improved and (b) the standard BPM.

Equations (103)

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2Er+4π2λ2 n2rEr=0,
Er=Urexpi 2πλz0z n0, zdz
2Uρ, z+i 4πλ n0, zUzρ, z+4π2λ2n2ρ, z-n20, zUρ, z+i 2πλn0, zz Uρ, z=0,
22x2+2y2.
Φz=n0, z-+ |Uρ, z|2d2ρ
ψρ, z=n0, z1/2Uρ, z
2ψρ, z+i 4πλ n0, zψzρ, z+4π2λ2n2ρ, z-n20, zψρ, z=0.
ψk+1=ψk+ψkz δz+Oδz2.
ψkz=iλ4πn0, k 2ψk+i πλnk2-n0, k2n0, k ψk,
ψk+1=1+i λδz4πn0, k 2+i πλnk2-n0, k2n0, k δz×ψk+Oδz2.
ψk+1=SˆkPˆkψk+Oδz2,
Pˆkexpi λδz4πn0, k 2=1+i λδz4πn0, k 2+Oδz2,
Sˆkexpi πλnk2-n0, k2n0, k δz=1+i πλnk2-n0, k2n0, k×δz+Oδz2.
ψk=ψρ, zk=-+ Ψf, zkexpi2πf·ρd2f,
Pˆkψk=-+ Ψf, zkexp-iπ λδzn0, k f2expi2πf·ρd2f.
Pˆkψk=n0, kiλδz-+ ψkρexpiπ n0, kλδzρ-ρ2d2ρ.
ψρ, zN=SˆN-1PˆN-1SˆN-2PˆN-2  Sˆ1Pˆ1Sˆ0Pˆ0ψρ, z0+Oδz.
ψk+1=ψk+ψkz δz+122ψkz2 δz2+Oδz3.
2ψkz2=iλ4πn0, k 2ψkz-iλ4πn0, k2dn0, kdz 2ψk+i πλnk2-n0, k2n0, kψkz+i 2πλnkn0, knkz-dn0, kdz×ψk-i πλnk2-n0, k2n0, k2dn0, kdz ψk.
2ψkz2=-λ216π2n0, k2 4ψk-14n0, k 2nk2-n0, k2n0, k ψk-iλ4πn0, k2dn0, kdz 2ψk-nk2-n0, k24n0, k2 2ψk-π2λ2nk2-n0, k22n0, k2 ψk+i 2πλnkn0, knkz ψk-i πλdn0, kdz1+nk2n0, k2ψk.
ψk+1=Pˆk+11/2Sˆk+1/2Pˆk1/2ψk+Oδz3,
Pˆk1/2expi λδz8πn0, k 2=1+i λδz8πn0, k 2-λ2δz2128π2n0, k2 4+Oδz3,
Pˆk+11/2expi λδz8πn0, k+1 2=1+i λδz8πn0, k 2-i λδz28πn0, k2dn0, kdz 2-λ2δz2128π2n0, k2 4+Oδz3,
Sˆk+1/2expi πλnk+1/22-n0, k+1/22n0, k+1/2 δz=1+i πλnk2-n0, k2n0, k δz+i πλnkn0, knkz δz2-i π2λdn0, kdz1+nk2n0, k2δz2-π22λ2nk2-n0, k22n0, k2×δz2+Oδz3.
ψρ, zN=PˆN1/2SˆN-1/2PˆN-11/2PˆN-11/2SˆN-3/2PˆN-21/2 Pˆ21/2Sˆ3/2Pˆ11/2Pˆ11/2Sˆ1/2Pˆ01/2ψρ, z0+Oδz2.
ψρ, zN=PˆN1/2SˆN-1/2PˆN-1SˆN-3/2 Pˆ2Sˆ3/2Pˆ1Sˆ1/2Pˆ01/2ψρ, z0+Oδz2.
Eρ, zk+1=n0, zkn0, zk+11/2×expi πλn2ρ, zk+n20, zkn0, zk δz×n0, zkiλδz-+ Eρ, zkexpi πn0, zkλδz×ρ-ρ2d2ρ+Oδz2.
nρ, z=nref+δnρ, z,
Er=Urexpi 2πλ nrefz
Eρ, zk+1=expi 2πλ δnρ, zδzexpi 2πλ nrefδz×nrefiλδz-+ Eρ, zkexpi πnrefλδzρ-ρ2×d2ρ+Oδz2.
nρ, z=n0z-12n2zρ2.
Eρ, z=ψρ, zn0, z1/2expi 2πλz0z n0, zdz.
ϕρ, z=azexpi 2πλγz+αz·ρ+βzρ2,
 -+expipx2+y2+iqx+rydxdy=iπpexp-i q2+r24p.
az+δz=azn0zn0z+2δzβz+Oδz2,
γz+δz=γz-12δzα2zn0z+2δzβz+Oδz2,
αz+δz=αzn0zn0z+2δzβz+Oδz2,
βz+δz=n0zβzn0z+2δzβz-δz2 n2z+Oδz2.
az=-2azβzn0z,
γz=-α2z2n0z,
αz=-2αzβzn0z,
βz=-2 β2zn0z-12 n2z,
βz=n0zμz2μz
ddzn0zμz+n2zμz=0.
n0zσzτz-σzτz=K,
βz=n0zσz2σz,
az=k1σz,
αz=K2σz,
ddzτzσz=-Kn0zσ2z,
γz=-K222n0zσ2z,
γz=K222Kτzσz.
ϕρ, z=k1σzexpi πλσzK22τzK+2K2·ρ+n0zσzρ2.
ψρ, z=-+AK2σzexpi πλσzK22τzK+2K2·ρ+n0zσzρ2d2K2,
gz0=1,hz0=0,gz0=0,hz0=1.
σz=gz,  τz=hz,
ϕρ0, z0=expi2πλK2·ρ0.
K2=λf,  AK2=Ψf, z0/λ2.
ψρ, z=1gzexpiπλgzn0zgz ρ2-+ Ψf, z0×exp-iπλ hzgzn0z0 f2+i2π f·ρgzd2f
ψρ, z=n0z0iλhzexpiπ n0zhzλhz ρ2-+ ψρ0, z0×expiπ n0z0λhzgzρ02-2ρ0·ρd2ρ0.
σz=hz,  τz=gz.
gzn0zgz=ACDgz0n0z0gz0,hzn0zhz=ACDhz0n0z0hz0,
gz=A,gz=C/n0z,hz=n0z0,hz=n0z0D/n0z.
Eρ, z=n0z0n0z1/21iλexpi 2πλz0z n0zdz×-+ Eρ0, z0expi πλAρ02-2ρ0·ρ+Dρ2d2ρ0.
-+ |Eρ, z|2d2ρ=-+ |Eρ0, z0|2d2ρ0
n0z-+ |Eρ, z|2d2ρ=n0z0-+ |Eρ0, z0|2d2ρ0,
EGρ, z0=Az0expi 2πλ βz0ρ2,
EGρ, z=Az0n0z03/2n0z0gz+2βz0hz1n0z1/2×expi 2πλz0z n0zdz×expiπn0zλ×n0z0gz+2βz0hzn0z0gz+2βz0hzρ2,
n0znref.
nρ, z=ñ1+ρ2+z2a2.
nρ, z=ña2a2+z2-ña2a2+z22 ρ2,
n0z=ña2a2+z2,
n2z=2ña2a2+z22.
a2+z2d2μdz2-2z dμdz+2μ=0,
gz=-z2+2z0z+a2a2+z02,
hz=z0z2+a2-z02z-a2z0a2+z02.
a2+z22d2μdz2+2 ña2nref μ=0,
2πλz0z n0zdz
nr=n02-rr021/2n0if rr0if r>r0,
nρ, z=n02-zr021/2-ρ22r022-zr021/2for |z|r0,
gz=12-zr0+2-zr021/21-zr0for |z|r0for z>r0,
hz=12z+r02-zr021/2r0for |z|r0for z>r0.
Eρ, z=Uρ, zexpi 2πλ nrefz
2U+i 4πλ nrefUz+4π2λ2n2-nref2U=0.
Eρ, z=Aρ, zexpi 2πλ Sρ, z,
Uρ, z=Aρ, zexpi 2πλ Sρ, zexp-i 2πλ nrefz.
2A-4π2λ2 A|S|2-8π2λ2 nrefA Sz+4π2λ2×nref2+n2A=0,
|S|2+2nrefSz=n2+nref2.
|S|2=n2+nref-Sz2.
S=n2+nref-Sz21/2drds,
Sz=n2+nref-Sz21/2,
Sz=nref2+n22nref.
|S|2=nref2+n22nref2.
neqρ, z=nref2+n2ρ, z2nref.
neqρ, z=nref2+n02z2nref-12n0zn2znref ρ2.
neqρ, z=nref+δn0z-12n2zρ2,
ddznref+δn0zdμdzz+n2zμz=0.
d2μdz2z+n2znref μz=0.
Uρ, z=Aρ, z×expi 2πλ Sρ, zexp-i 2πλz0z n0, zdz
2A-4π2λ2 A|S|2-8π2λ2 n0A Sz+4π2λ2×n02+n2A=0,
|S|2+2n0Sz=n2+n02.
|S|2=n02+n22n02.
neqρ, z=n02z+n2ρ, z2n0z.
neqρ, z=n0z-12n2zρ2,

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