Abstract

By extension of the transitional operator method developed by Wünsche, the Rayleigh–Sommerfeld and Kirchhoff solutions to the diffraction of a converging spherical (or cylindrical) wave are expressed in terms of a series of derivatives of the field estimate that follows from the Fresnel approximation. This result allows a systematic assessment of the error associated with the paraxial wave model for focused fields and offers simple corrections to this model. In particular, for simple diffracting masks, the Fresnel approximation leads to estimates of the field that have a relative error near focus that is of the order of one on the square of the f-number. The number of significant digits in the field estimate is shown to be doubled by retaining just the first of the series of corrections derived here.

© 1999 Optical Society of America

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References

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  1. A. Wunsche, “Transition from the paraxial approximation to exact solutions of the wave equation and application to Gaussian beams,” J. Opt. Soc. Am. A 9, 765–774 (1992).
    [CrossRef]
  2. G. W. Forbes, D. J. Butler, R. L. Gordon, A. A. Asatryan, “Algebraic corrections for paraxial wave fields,” J. Opt. Soc. Am. A 14, 3300–3315 (1997).
    [CrossRef]
  3. M. Lax, W. H. Louisell, W. B. McBright, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
    [CrossRef]
  4. G. P. Agrawal, M. Lax, “Free-space wave propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
    [CrossRef]
  5. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, 1st ed. (Cambridge U. Press, New York, 1995), Sec. 3.2, pp. 109–127.
  6. See Ref. 5, Sec. 3.2.4, pp. 120–124.
  7. See Ref. 5, Sec. 3.2.5, pp. 125–127.
  8. W. H. Southwell, “Validity of the Fresnel approximation in the near field,” J. Opt. Soc. Am. 71, 7–14 (1981).
    [CrossRef]
  9. Equation (4.8) is the familiar paraxial wave equation, and it is straightforward to verify that U¯F satisfies this equation by using definition (2.8).
  10. G. W. Forbes, “Scaling properties in the diffraction of focused waves and an application to scanning beams,” Am. J. Phys. 62, 434–443 (1994).
    [CrossRef]
  11. H. Osterberg, L. Smith, “Closed solutions of Rayleigh’s diffraction integral for axial points,” J. Opt. Soc. Am. 51, 1050–1054 (1961).
    [CrossRef]
  12. The required expressions can be found either by returning to Eq. (5.2c) and taking the derivatives before evaluating at focus or by using Eqs. (4.8) and (4.11). Equivalently, since Eq. (5.5) is evaluated at z=f, we can take the partial f derivatives as ∂U¯F/∂f=dU¯F/df-∂U¯F/∂z, where Eq. (4.8) is then used for taking the partial z derivative. As an example, the first correction term given in Eq. (4.2a) can be found in this way to be   CI,1=if2k ddf ikρ ∂∂ρ ρ ∂U¯F∂ρ-dU¯Fdf=RU08f3kρ2 [8f2+4ikf(2ρ2+R2)-k2ρ2(ρ2+6R2)]ρJ1kRf ρ+4iR[2 f2-k2ρ2(ρ2+R2)]J2kRf ρexpik2 f ρ2.  
  13. G. W. Forbes, A. A. Asatryan, “Reducing canonical diffraction problems to singularity-free one-dimensional integrals,” J. Opt. Soc. Am. A 15, 1320–1328 (1998).
    [CrossRef]
  14. Such contour plots have been given recently, for example, in Ref. 2 and in G. W. Forbes, “Validity of the Fresnel approximation in the diffraction of collimated beams,” J. Opt. Soc. Am. A 13, 1816–1826 (1996).
    [CrossRef]

1998 (1)

1997 (1)

1996 (1)

1994 (1)

G. W. Forbes, “Scaling properties in the diffraction of focused waves and an application to scanning beams,” Am. J. Phys. 62, 434–443 (1994).
[CrossRef]

1992 (1)

1983 (1)

G. P. Agrawal, M. Lax, “Free-space wave propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
[CrossRef]

1981 (1)

1975 (1)

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

1961 (1)

Agrawal, G. P.

G. P. Agrawal, M. Lax, “Free-space wave propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
[CrossRef]

Asatryan, A. A.

Butler, D. J.

Forbes, G. W.

Gordon, R. L.

Lax, M.

G. P. Agrawal, M. Lax, “Free-space wave propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
[CrossRef]

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

Louisell, W. H.

M. Lax, W. H. Louisell, W. B. McBright, “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, 1st ed. (Cambridge U. Press, New York, 1995), Sec. 3.2, pp. 109–127.

McBright, W. B.

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

Osterberg, H.

Smith, L.

Southwell, W. H.

Wolf, E.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, 1st ed. (Cambridge U. Press, New York, 1995), Sec. 3.2, pp. 109–127.

Wunsche, A.

Am. J. Phys. (1)

G. W. Forbes, “Scaling properties in the diffraction of focused waves and an application to scanning beams,” Am. J. Phys. 62, 434–443 (1994).
[CrossRef]

J. Opt. Soc. Am. (2)

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

Phys. Rev. A (2)

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

G. P. Agrawal, M. Lax, “Free-space wave propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
[CrossRef]

Other (5)

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics, 1st ed. (Cambridge U. Press, New York, 1995), Sec. 3.2, pp. 109–127.

See Ref. 5, Sec. 3.2.4, pp. 120–124.

See Ref. 5, Sec. 3.2.5, pp. 125–127.

Equation (4.8) is the familiar paraxial wave equation, and it is straightforward to verify that U¯F satisfies this equation by using definition (2.8).

The required expressions can be found either by returning to Eq. (5.2c) and taking the derivatives before evaluating at focus or by using Eqs. (4.8) and (4.11). Equivalently, since Eq. (5.5) is evaluated at z=f, we can take the partial f derivatives as ∂U¯F/∂f=dU¯F/df-∂U¯F/∂z, where Eq. (4.8) is then used for taking the partial z derivative. As an example, the first correction term given in Eq. (4.2a) can be found in this way to be   CI,1=if2k ddf ikρ ∂∂ρ ρ ∂U¯F∂ρ-dU¯Fdf=RU08f3kρ2 [8f2+4ikf(2ρ2+R2)-k2ρ2(ρ2+6R2)]ρJ1kRf ρ+4iR[2 f2-k2ρ2(ρ2+R2)]J2kRf ρexpik2 f ρ2.  

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Figures (5)

Fig. 1
Fig. 1

On-axis relative errors of the Fresnel and the corrected estimates, as well as the level of uncertainty associated with the Kirchhoff boundary conditions. These plots correspond to the case of a spherical wave truncated by a circular aperture, for f/#=2, 10, 50 and NR=102, 103, 104.

Fig. 2
Fig. 2

Radial relative errors at the focal plane of the Fresnel and the corrected estimates, for a spherical wave truncated by a circular aperture, for f/#=2, 10, 50 and NR=102, 103, 104.

Fig. 3
Fig. 3

On-axis relative errors of the Fresnel and the corrected estimates, together with the level of uncertainty associated with the Kirchhoff boundary conditions for a converging spherical wave passing through the Gaussian mask defined in Eq. (5.6), where f/#=2, 10, 50 and NR=102, 103, 104.

Fig. 4
Fig. 4

Radial relative errors at the focal plane of the Fresnel and the corrected estimates for a Gaussian beam, where f/#=2, 10, 50 and NR=102, 103, 104.

Fig. 5
Fig. 5

On-axis relative errors of the Fresnel estimates for a collimated field [see Eq. (5.9)] and a focused field with f/#=5000, respectively, after passing through the Gaussian mask defined in Eq. (5.6), for NR=102. The f that appears in the definition of the abscissa corresponds to the focal length of the focused field.

Equations (104)

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(2+k2)U(r)=0,
U(x, y, z)=A(α, β)exp[ik(αx+βy+z1-α2-β2)]dαdβ,
A(α, β)=k24π2 U(x, y, 0)×exp[-ik(αx+βy)]dxdy.
A(α, β)=-ik4π2  Uz (x, y, 0)×exp[-ik(αx+βy)]1-α2-β2 dxdy.
U(x, y, z)=k24π2 U(x, y, 0)× exp{ik[α(x-x)+β(y-y)+z1-α2-β2]}dαdβdxdy,
U(x, y, z)=-ik4π2 Uz (x, y, 0)× exp{ik[α(x-x)+β(y-y)+z1-α2-β2]}1-α2-β2 dαdβdxdy.
G(r; r) exp(ik|r-r|)4π|r-r|=ik8π2 exp{ik[α(x-x)+β(y-y)+(z-z)1-α2-β2]}1-α2-β2 dαdβ.
URSI(x, y, z)=-2U(x, y, 0)×Gz(x, y, z; x, y, 0)dxdy,
URSII(x, y, z)=-2 Uz (x, y, 0)×G(x, y, z; x, y, 0)dxdy.
U(x, y, z)UF(x, y, z) k24π2 U(x, y, 0)× expikα(x-x)+β(y-y)+z1-α2+β22dαdβdxdy=-ik exp(ikz)2πz U(x, y, 0)×expik (x-x)2+(y-y)22zdxdy.
U(x, y, 0)=fU0f2+x2+y2 M(x, y)×exp[ik(f-f2+x2+y2)],
Uz (x, y, 0)=f2U0M(x, y)f2+x2+y2 ik+1f2+x2+y2×exp[ik(f-f2+x2+y2)].
URSI(x, y, z)
=k2 exp(ikf )4π2  fU0M(x, y)f2+x2+y2×exp{ik[z1-α2-β2-f2+x2+y2+α(x-x)+β(y-y)]}dαdβdxdy,
URSII(x, y, z)
=-ik exp(ikf )4π2× f2U0M(x, y)(f2+x2+y2)1-α2-β2×ik+1f2+x2+y2exp{ik[z1-α2-β2-f2+x2+y2+α(x-x)+β(y-y)]}dαdβdxdy.
UF(x, y, z)
=k24π2 U0M(x, y)×expikz1-α2+β22-x2+y22 f+α(x-x)+β(y-y)dαdβdxdy.
URSI(ξ, η, ζ)
= exp-iζ(u2+v2)/2π1+1-(u2+v2)/(2πNR)2+i(uξ+vη) U0(ξ, η)exp[-i(uξ+vη)]4π21+(ξ2+η2)(Nf/NR)2×exp-i2πNf(ξ2+η2)1+1+(ξ2+η2)(Nf/NR)2×dξdηdudv,
URSII(ξ, η, ζ)
= 11-(u2+v2)/(2πNR)2×exp-iζ(u2+v2)/2π1+1-(u2+v2)/(2πNR)2+i(uξ+vη) U0(ξ, η)4π2[1+(ξ2+η2)(Nf/NR)2]×1-iNf2πNR21+(ξ2+η2)(Nf/NR)2×exp-i2πNf(ξ2+η2)1+1+(ξ2+η2)(Nf/NR)2exp[-i(uξ+vη)]dξdηdudv,
UF(ξ, η, ζ)
= exp[-iζ(u2+v2)/4π+i(uξ+vη)]× U0(ξ, η)4π2 exp[-iπNf(ξ2+η2)]×exp[-i(uξ+vη)]dξdηdudv,
U(ξ, η, ζ)  exp[-i2πNR2ζ]U(ξR, ηR, ζR2/λ),
U0(ξ, η)  U0M(ξR, ηR).
UF(ξ, η, ζ)=limNR URSI(ξ, η, ζ)=limNR URSII(ξ, η, ζ).
I(a, b, ε, s)  H(p)1+εa2p2×exp-iap21+1+εa2p2-ipqdp×exp-ibq21+1-εq2+iqsdq,
II(a, b, ε, s) 11-εq2  H(p)1+εa2p2×1-iεa1+εa2p2×exp-iap21+1+εa2p2-ipqdp×exp-ibq21+1-εq2+iqsdq,
F(a, b, s) H(p)×exp-i ap22-ipqdp×exp-i bq22+iqsdq.
I(a, b, ε, s)
=H(p)1+-a22 p2+i a38 p4-i b8 q4ε++3a48 p4-i a58 p6+i a2b16 q4p2-i b16 q6-a6128 p8+a3b64 q4p4-b2128 q8ε2+O(ε3)×exp-i ap22-ipq×exp-ibq22+iqsdpdq.
(2i)m+n m+nFambn=p2mq2nH(p)×exp-i ap22-ipq×exp-i bq22+ipqdpdq.
I(a, b, ε, s)=F+iε-a2 Fa-a32 2Fa2+b2 2Fb2+ε2-3a42 2Fa2-a5 3Fa3+a2b2 3Fab2-b2 3Fb3-a68 4Fa4+a3b4 4Fa2b2-b28 4Fb4+O(ε3).
URSI=UF+i2πNR2 -Nf2 UFNf+ζ2 2UFζ2-Nf32 2UFNf2-14π2NR4 3Nf42 2UFNf2+Nf5 3UFNf3-Nf2ζ2 3UFNfζ2+ζ2 3UFζ3+Nf68 4UFNf4-Nf3ζ4 4UFNf2ζ2+ζ28 4UFζ4+O(NR-6).
U¯RSI(x, y, z)U¯F+ik -1f2 U¯F(f-1)+z2 2U¯Fz2-12 f3 2U¯F(f-1)2-1k2 32 f4 2U¯F(f-1)2+z2 3U¯Fz3-z2 f2 3U¯Fz2(f-1)+1f5 3U¯F(f-1)3+z28 4U¯Fz4-z4f3 4U¯Fz2(f-1)2+18f6 4U¯F(f-1)4=U¯F+ik z2 2U¯Fz2-f2 2U¯Ff2-1k2 z2 3U¯Fz3+f2 3U¯Ff3+z28 4U¯Fz4-fz4 4U¯Fz2f2+f28 4U¯Ff4,
U¯RSII(x, y, z)U¯F+ik -U¯Ff+U¯Fz+U¯Ff+z2 2U¯Fz2-f2 2U¯Ff2-1k2 -1f U¯Fz+3f-z2 f 2U¯Fz2+2U¯Fzf+z 3U¯Fz3+z2 3U¯Fz2f-f2 3U¯Fzf2+z28 4U¯Fz4-fz4 4U¯Fz2f2+f28 4U¯Ff4.
U¯RSI=T^1(z)T^1*(f )U¯F,
U¯RSII=T^2(z)T^0*(f )U¯F,
T^n(v)m=0i2km 1m! v2-n 2mv2m vm+n-2.
U¯RSα(x, y, z)U¯α,1(x, y, z)  U¯F(x, y, z)+Cα,1(x, y, z),
CI,1(x, y, z)  ik z2 2U¯Fz2-f2 2U¯Ff2,
CII,1(x, y, z)  ik -U¯Ff+U¯Fz+U¯Ff+z2 2U¯Fz2-f2 2U¯Ff2.
εα,F(x, y, z)  U¯F-U¯RSαU¯RSα,
εα,1(x, y, z)  U¯α,1-U¯RSαU¯RSα.
εα,F(x, y, z)ε˜α,F(x, y, z)  Cα,1U¯α,1,
εα,1(x, y, z)ε˜α,1(x, y, z)  Cα,2U¯α,2,
CI,2(x, y, z)  -1k2 z2 3U¯Fz3+f2 3U¯Ff3+z28 4U¯Fz4-fz4 4U¯Fz2f2+f28 4U¯Ff4,
CII,2(x, y, z)  -1k2 -1f U¯Fz+3f-z2 f 2U¯Fz2+2U¯Fzf+z 3U¯Fz3+z2 3U¯Fz2f-f2 3U¯Fzf2+z28 4U¯Fz4-fz4 4U¯Fz2f2+f28 4U¯Ff4.
εKBC(x, y, z)  U¯RSII-U¯RSIU¯K=2U¯RSII-U¯RSIU¯RSI+U¯RSII,
εKBC(x, y, z)ε˜KBC(x, y, z)  2CII,1-CI,1U¯I,1+U¯II,1.
U¯Fz=i2k 2U¯F.
U¯F(r, z; f2)γ(D-1)/2×expik2z (1-γ)|r|2U¯F(γr, γz; f1),
γ=1-f1-f2f1f2z-1.
U¯Ff=1f2 ik2 |r|2U¯F-zD-12U¯F+rU¯F+z U¯Fz.
U¯F(0, 0, z)=-iU0λz M(x, y)×expik1z-1f x2+y22dxdy,
U¯F(0, 0, f )=-iU0λf M(x, y)dxdy=-iU0Nf m0,
CI,1(0, 0, f )=U0NfNR2m02π+i3m1Nf,
CII,1(0, 0, f )=U0NfNR2-m02π+i3m1Nf,
CI,2(0, 0, f )=-U0NfNR4m1π+i 158 m2Nf,
CII,2(0, 0, f )=-U0NfNR4-m1π+i 158 m2Nf,
mn  R-2(n+1)(x2+y2)nM(x, y)dxdy.
ε˜I,F(0, 0, f )ε˜II,F(0, 0, f )3 m1m0 NfNR2=34 m1m0 (f/#)-2,
ε˜I,1(0, 0, f )ε˜II,1(0, 0, f )158 m2m0 NfNR4=15128 m2m0 (f/#)-4,
ε˜KBC(0, 0, f )NfπNR2=12πNR (f/#)-1,
M(x, y)=1,x2+y2R20,x2+y2>R2 .
U¯RSI(ρ, z)
=-U02π 0R02π zf exp[ik(f-f2+ρ2)]f2+ρ2×(ikz2+ρ2+ρ2-2ρρ cos θ-1)(z2+ρ2+ρ2-2ρρ cos θ)3/2×exp[-ik(z-z2+ρ2+ρ2-2ρρ cos θ)]×ρdθdρ,
U¯RSII(ρ, z)
=-U02π 0R02π f2(ikf2+ρ2+1)(f2+ρ2)3/2×exp[ik(f-f2+ρ2)]×exp[-ik(z-z2+ρ2+ρ2-2ρρ cos θ)]z2+ρ2+ρ2-2ρρ cos θ×ρdθdρ,
U¯F(ρ, z)
=-ikU02πz 0R02πexp-ikρ22 f-ρ2+ρ2-2ρρ cos θ2zρdθdρ.
U¯RSI(0, z)
=U0 zf exp[ik(f-f2+R2-z+z2+R2)]z2+R2(z2+R2-f2+R2)
-U0fz-f,
U¯RSII(0, z)
=U0f2 exp[ik(f-f2+R2-z+z2+R2)]f2+R2(z2+R2-f2+R2)
-U0fz-f.
U¯F(0, z)=U0fz-f expik1z-1f R22-1.
U¯F(ρ, f )=-iRU0ρ expik2 f ρ2J1kRf ρ.
M(x, y)=exp-π x2+y2R2.
U¯F(ρ, z)=U0 η(z)expik2z [1-η(z)]ρ2,
η(z)  1-zf 1-i λfR2-1.
ε˜α,FC(0, 0, z)=λ3z2π(λ2z2+R4).
U(x, y, 0)
=-M(x, y) ikfU02π 
×exp{ik[αx+βy-f(1-α2-β2-1)]}1-α2-β2
×dαdβ,
Uz (x, y, 0)
=-M(x, y) k2fU02π  exp{ik[αx+βy-f(1-α2-β2-1)]}dαdβ.
U¯σ(x, y, z)=-ik3fU0(2π)3 M(x, y)×Aσ(α2+β2, z)×Bσ*(α2+β2, f )×exp{ik[α(x-x)+β(y-y)+αx+βy]}dαdβdαdβdxdy.
ARSI(γ, v)=BRSII(γ, v)=exp[ikv(1-γ-1)],
ARSII(γ, v)=BRSI(γ, v)=11-γ×exp[ikv(1-γ-1)].
AF(γ, v)=BF(γ, v)=exp(-ikvγ/2).
T^1(v)  m=0i2km 1m! v 2mv2m vm-1,
T^2(v)  m=0i2km 1m! 2mv2m vm,
T^1(v)exp-ikvγ2=exp[ikv(1-γ-1)],
T^2(v)exp-ikvγ2=11-γ exp[ikv(1-γ-1)].
U¯RSI=fT^1(z)T^2*(f )(U¯F/f ),
U¯RSII=fT^2(z)T^1*(f )(U¯F/f ).
U¯RSI=T^1(z)T^1*(f )U¯F,
U¯RSII=T^2(z)T^0*(f )U¯F,
T^0(v)  m=0i2km 1m! v2 2mv2m vm-2.
CI,1=if2k ddf ikρ ρ ρ U¯Fρ-dU¯Fdf=RU08f3kρ2 [8f2+4ikf(2ρ2+R2)-k2ρ2(ρ2+6R2)]ρJ1kRf ρ+4iR[2 f2-k2ρ2(ρ2+R2)]J2kRf ρexpik2 f ρ2.

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