Abstract

Asymptotic analysis of the angular spectrum solution for diffraction is used to establish the validity of a standard, formal series for nonparaxial wave propagation. The lowest term corresponds to the field in the Fresnel approximation, and this derivation clarifies some of the remarkable aspects of Fresnel validity for both small and large propagation distances. This asymptotic approach is extended to derive simple, generic algebraic corrections to the field estimates found by using the paraxial model, i.e., the Fresnel approximation. Contour maps of the field errors associated with the diffraction of collimated beams—both uniform and Gaussian—in two and three dimensions demonstrate the effectiveness of these corrections.

© 1997 Optical Society of America

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References

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  1. M. Lax, W. H. Louisell, W. B. McKnight, “From Maxwell to paraxial wave optics,” Phys. Rev. A 11, 1365–1370 (1975).
    [Crossref]
  2. T. Takenaka, M. Yokota, O. Fukumitsu, “Propagation of light beams beyond the paraxial approximation,” J. Opt. Soc. Am. A 2, 826–829 (1985).
    [Crossref]
  3. G. P. Agrawal, M. Lax, “Free-space wave propagation beyond the paraxial approximation,” Phys. Rev. A 27, 1693–1695 (1983).
    [Crossref]
  4. S. Nemoto, “Nonparaxial Gaussian beams,” Appl. Opt. 29, 1940–1946 (1990).
    [Crossref] [PubMed]
  5. A. Wünsche, “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]
  6. In essence, while a spatially nonuniform, but purely transverse, linearly polarized paraxial field is not divergence free, it can be made so by extending it in the form of a series in (λ/R), where R is a measure of the beam width at its waist. If the +z axis is taken to be in the general direction of propagation and the y axis aligned to the transverse component of the electric field, Ey is then effectively a paraxial scalar field. Since ∂Ey/∂z≈ikEy, the condition that the field be divergence free—viz., ∂Ey/∂y+∂Ez/∂z=0—is approximately satisfied if Ez is taken to be given by Ez=(i/k)∂Ey/∂y=(i/2π)(λ/R)∂Ey/∂η, where η=y/R. That is, the longitudinal component is a first-order correction; higher-order terms in (λ/R) are added to both Ey and Ez to complete the series.
  7. A. M. Steane, H. N. Rutt, “Diffraction calculations in the near field and the validity of the Fresnel approximation,” J. Opt. Soc. Am. A 6, 1809–1814 (1989).
    [Crossref]
  8. C. J. R. Sheppard, M. Hrynevych, “Diffraction by a circular aperture: a generalization of Fresnel diffraction theory,” J. Opt. Soc. Am. A 9, 274–281 (1992).
    [Crossref]
  9. G. W. Forbes, “Validity of the Fresnel approximation in the diffraction of collimated beams,” J. Opt. Soc. Am. A 13, 1816–1826 (1996).
    [Crossref]
  10. F. D. Feiock, “Wave propagation in optical systems with large apertures,” J. Opt. Soc. Am. 68, 485–489 (1978).
    [Crossref]
  11. W. H. Southwell, “Validity of the Fresnel approximation in the near field,” J. Opt. Soc. Am. 71, 7–14 (1981).
    [Crossref]
  12. W. G. Rees, “The validity of the Fresnel approximation,” Eur. J. Phys. 8, 44–48 (1987).
    [Crossref]
  13. Also see Ref. 7.
  14. It also turns out that the numerical computation of both the exact and the Fresnel diffracted field is made easier by using a Green’s function approach for larger propagation distances but an angular spectrum method for points close to the aperture. Both methods are used for the numerical examples given in Sections 4 and 5.
  15. See, for example, L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995), Sec. 3.2.
  16. This kernel takes the form (-ik/2π)(z/S)(1+i/kS)×exp(ikS)/S, where S=[(x′-x)2+(y′-y)2+z2]1/2. [See Eq. (3.2–78) of Ref. 15.]
  17. This observation explains some of the simplifications that seemed so fortuitous in the analysis of Ref. 9.
  18. See, for example, J. J. Stamnes, Waves in Focal Regions (Adam Hilger, Bristol, UK, 1986), Sec. 5.2.
  19. This follows from the fact that the angular spectrum shrinks toward the origin as R/λ is increased.
  20. NR is just the inverse of the parameter used in Ref. 1 as the basis for their series. When the order of integration in Eq. (2.6) is reversed, the integral over u and v can be done in closed form, giving the Rayleigh–Sommerfeld solution: U(ξ, η, ζ)=-iζ∬U(ξ′, η′, 0)1T21+i2πNR2ζT×expi2π(1+T)ζ[(ξ-ξ′)2+(η-η′)2]dξ′dη′, where T={1+[(ξ-ξ′)2+(η-η′)2]/(ζNR)2}1/2. This is a convenient form for computation and error analysis along the lines presented in Ref. 9 and is more appropriate here than the form given in Eq. (3.1) of that work. Notice that in the limit of large NR,T goes to unity, and this expression for U(ξ, η, ζ) approaches the Fresnel form given in Eq. (2.7).
  21. When ∊≪1/l2, the series in Eq. (3.3) converges rapidly for all |x|<l provided that b∊l4≪1. This means that the condition given in the text, namely, b≪1/l2, is actually stronger than necessary.
  22. See Ref. 18, Chap. 8.
  23. When the contour of Eq. (3.1) is deformed onto the tangent to the curve of steepest descent, the integrals that appear in the new analog of Eq. (3.6) involve a Gaussian factor in place of the quadratic phase term, and this is the justification for the replacement given in Eq. (3.7). The required integral follows from the results given in Appendix A by taking p=0 and noting that Γ[(2m+1)/2]=(2m-1)!!π/2m.
  24. See Ref. 18, Chap. 9.
  25. For example, if f′′(x)=[f(x+Δ)-2 f(x)+f(x-Δ)]/Δ2 is used to estimate the first correction, an interesting option exists that leads to a two-term approximation: I(p, b, ∊)=12[F(p, b-ib∊)+F(p, b+ib∊)]+O(∊2). Here we have simply chosen Δ=ib∊. Although this saves one evaluation of F, complex values of the second argument are not always as manageable. For Eq. (3.9), this becomes U¯(x, y, z)≈12[U¯F(x, y, z-iz/k)+U¯F(x, y, z+iz/k)].
  26. There is another stationary point on the other sheet of the integrand’s double-valued phase, but this does not contribute.
  27. The differences between the expressions given in Eqs. (3.12) and (3.13) parallel the distinction (in the far field) between the r-type and z-type approximations discussed in Ref. 9. To translate the expressions given here, recall that we have ∊=1/(2πNR)2,b=ζ/2π=1/(2πNz), and p=ξ. The difference between the first phase factors of the two expressions is readily seen to be the error labeled ∊Φ in Ref. 9. Given that L(u) is just sinc(u)/π for a uniform aperture, the difference between the arguments of L can be seen to correspond to ∊θ. Finally, the remaining amplitude factor is readily seen to correspond to ∊C.
  28. It is easier to appreciate the origins of the phase factor in Eq. (3.16) when its argument is written as bg2(p, b)/[1+r]-12bg2(p, b)/r2, but this form can cause problematic cancellation. The result follows by considering F(p/r, b) in Eq. (3.13) and then adding factors to match the resulting expression to the one given in Eq. (3.12). Another option follows similarly: G3(p, b, ∊) := exp{i12b∊g4(p, b)/[r(1+r)2]}r-1F[p, br]. Unlike G2, when translated to the diffraction context this form involves evaluation of the Fresnel field over a curved surface to estimate the diffracted field on a plane of fixed z. It therefore appears to be less convenient for propagation with fast-Fourier-transform-based methods. Alternatively, the arguments of F can be chosen to match all but the amplitude factor, and this leads to G4(p, b, ∊) := 2r(1+r) F[p2r/(1+r), b2r2/(1+r)]. Notice that, in view of the comments in Note 27, the amplitude correcting factors in G2,G3, and G4 are significant only in the absence of the uncertainty associated with the Kirchhoff boundary conditions. These three all give similar results.
  29. See Chap. 9 of Ref. 18 for references and a formal treatment of the method required here.
  30. The analogs of the alternatives given in Note 28 take the form G3(p, q, b, ∊) := exp{i12b∊t2/[s(1+s)2]}s-1F[p, q, bs] [note that, unlike Eq. (3.22), the amplitude factor is now the same as in the corresponding two-dimensional case], and G4(p, q, b, ∊) :=21+sF[p2s/(1+s), q2s/(1+s), b2s2/(1+s)].
  31. Consider approaching the aperture along the axis. Because of the cancellation between adjacent Fresnel zones, the field value in the Fresnel approximation ends up depending solely on the value of the field in an infinitessimal rim around the aperture’s edge, and this is precisely where the validity of the Kirchhoff boundary conditions is at its worst.
  32. This result is found by expanding Eq. (3.14) to first order in ∊ and finding the difference between the result and the correction given in Eq. (4.4). Equation (4.5) is just the highest term in an expansion of this difference in 1/ζ, upon taking ξ=v(1+ζ2)1/2. An approximation that is better where the expression in Eq. (4.5) goes to zero follows upon retaining an additional term: ∊≈|(3-4πv2)v2+i3/(4πζ)|/(4NR2).
  33. The exact field takes the form U(ξ, ζ)=1-exp{i2π/[(R+1)ζ]}/R, where R=[1+(ζNR)-2]1/2.
  34. This is crucial in, for example, R. L. Gordon, G. W. Forbes, “Apodization for systems of high f-number,” J. Opt. Soc. Am. A 14, 1243–1254 (1997).
    [Crossref]
  35. I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, New York, 1980). See 3.462.3 and 9.253.
  36. For even values of n, this follows from 6.631.10 of Ref. 35 after converting the variables of integration to polar form in Eq. (A5).

1997 (1)

1996 (1)

1992 (2)

1990 (1)

1989 (1)

1987 (1)

W. G. Rees, “The validity of the Fresnel approximation,” Eur. J. Phys. 8, 44–48 (1987).
[Crossref]

1985 (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)

1978 (1)

1975 (1)

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

Agrawal, G. P.

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

Feiock, F. D.

Forbes, G. W.

Fukumitsu, O.

Gordon, R. L.

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, New York, 1980). See 3.462.3 and 9.253.

Hrynevych, M.

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. 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.

See, for example, L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995), Sec. 3.2.

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]

Nemoto, S.

Rees, W. G.

W. G. Rees, “The validity of the Fresnel approximation,” Eur. J. Phys. 8, 44–48 (1987).
[Crossref]

Rutt, H. N.

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, New York, 1980). See 3.462.3 and 9.253.

Sheppard, C. J. R.

Southwell, W. H.

Stamnes, J. J.

See, for example, J. J. Stamnes, Waves in Focal Regions (Adam Hilger, Bristol, UK, 1986), Sec. 5.2.

Steane, A. M.

Takenaka, T.

Wolf, E.

See, for example, L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995), Sec. 3.2.

Wünsche, A.

Yokota, M.

Appl. Opt. (1)

Eur. J. Phys. (1)

W. G. Rees, “The validity of the Fresnel approximation,” Eur. J. Phys. 8, 44–48 (1987).
[Crossref]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (6)

Phys. Rev. A (2)

M. Lax, W. H. Louisell, W. B. McKnight, “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 (24)

In essence, while a spatially nonuniform, but purely transverse, linearly polarized paraxial field is not divergence free, it can be made so by extending it in the form of a series in (λ/R), where R is a measure of the beam width at its waist. If the +z axis is taken to be in the general direction of propagation and the y axis aligned to the transverse component of the electric field, Ey is then effectively a paraxial scalar field. Since ∂Ey/∂z≈ikEy, the condition that the field be divergence free—viz., ∂Ey/∂y+∂Ez/∂z=0—is approximately satisfied if Ez is taken to be given by Ez=(i/k)∂Ey/∂y=(i/2π)(λ/R)∂Ey/∂η, where η=y/R. That is, the longitudinal component is a first-order correction; higher-order terms in (λ/R) are added to both Ey and Ez to complete the series.

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, New York, 1980). See 3.462.3 and 9.253.

For even values of n, this follows from 6.631.10 of Ref. 35 after converting the variables of integration to polar form in Eq. (A5).

Also see Ref. 7.

It also turns out that the numerical computation of both the exact and the Fresnel diffracted field is made easier by using a Green’s function approach for larger propagation distances but an angular spectrum method for points close to the aperture. Both methods are used for the numerical examples given in Sections 4 and 5.

See, for example, L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, 1995), Sec. 3.2.

This kernel takes the form (-ik/2π)(z/S)(1+i/kS)×exp(ikS)/S, where S=[(x′-x)2+(y′-y)2+z2]1/2. [See Eq. (3.2–78) of Ref. 15.]

This observation explains some of the simplifications that seemed so fortuitous in the analysis of Ref. 9.

See, for example, J. J. Stamnes, Waves in Focal Regions (Adam Hilger, Bristol, UK, 1986), Sec. 5.2.

This follows from the fact that the angular spectrum shrinks toward the origin as R/λ is increased.

NR is just the inverse of the parameter used in Ref. 1 as the basis for their series. When the order of integration in Eq. (2.6) is reversed, the integral over u and v can be done in closed form, giving the Rayleigh–Sommerfeld solution: U(ξ, η, ζ)=-iζ∬U(ξ′, η′, 0)1T21+i2πNR2ζT×expi2π(1+T)ζ[(ξ-ξ′)2+(η-η′)2]dξ′dη′, where T={1+[(ξ-ξ′)2+(η-η′)2]/(ζNR)2}1/2. This is a convenient form for computation and error analysis along the lines presented in Ref. 9 and is more appropriate here than the form given in Eq. (3.1) of that work. Notice that in the limit of large NR,T goes to unity, and this expression for U(ξ, η, ζ) approaches the Fresnel form given in Eq. (2.7).

When ∊≪1/l2, the series in Eq. (3.3) converges rapidly for all |x|<l provided that b∊l4≪1. This means that the condition given in the text, namely, b≪1/l2, is actually stronger than necessary.

See Ref. 18, Chap. 8.

When the contour of Eq. (3.1) is deformed onto the tangent to the curve of steepest descent, the integrals that appear in the new analog of Eq. (3.6) involve a Gaussian factor in place of the quadratic phase term, and this is the justification for the replacement given in Eq. (3.7). The required integral follows from the results given in Appendix A by taking p=0 and noting that Γ[(2m+1)/2]=(2m-1)!!π/2m.

See Ref. 18, Chap. 9.

For example, if f′′(x)=[f(x+Δ)-2 f(x)+f(x-Δ)]/Δ2 is used to estimate the first correction, an interesting option exists that leads to a two-term approximation: I(p, b, ∊)=12[F(p, b-ib∊)+F(p, b+ib∊)]+O(∊2). Here we have simply chosen Δ=ib∊. Although this saves one evaluation of F, complex values of the second argument are not always as manageable. For Eq. (3.9), this becomes U¯(x, y, z)≈12[U¯F(x, y, z-iz/k)+U¯F(x, y, z+iz/k)].

There is another stationary point on the other sheet of the integrand’s double-valued phase, but this does not contribute.

The differences between the expressions given in Eqs. (3.12) and (3.13) parallel the distinction (in the far field) between the r-type and z-type approximations discussed in Ref. 9. To translate the expressions given here, recall that we have ∊=1/(2πNR)2,b=ζ/2π=1/(2πNz), and p=ξ. The difference between the first phase factors of the two expressions is readily seen to be the error labeled ∊Φ in Ref. 9. Given that L(u) is just sinc(u)/π for a uniform aperture, the difference between the arguments of L can be seen to correspond to ∊θ. Finally, the remaining amplitude factor is readily seen to correspond to ∊C.

It is easier to appreciate the origins of the phase factor in Eq. (3.16) when its argument is written as bg2(p, b)/[1+r]-12bg2(p, b)/r2, but this form can cause problematic cancellation. The result follows by considering F(p/r, b) in Eq. (3.13) and then adding factors to match the resulting expression to the one given in Eq. (3.12). Another option follows similarly: G3(p, b, ∊) := exp{i12b∊g4(p, b)/[r(1+r)2]}r-1F[p, br]. Unlike G2, when translated to the diffraction context this form involves evaluation of the Fresnel field over a curved surface to estimate the diffracted field on a plane of fixed z. It therefore appears to be less convenient for propagation with fast-Fourier-transform-based methods. Alternatively, the arguments of F can be chosen to match all but the amplitude factor, and this leads to G4(p, b, ∊) := 2r(1+r) F[p2r/(1+r), b2r2/(1+r)]. Notice that, in view of the comments in Note 27, the amplitude correcting factors in G2,G3, and G4 are significant only in the absence of the uncertainty associated with the Kirchhoff boundary conditions. These three all give similar results.

See Chap. 9 of Ref. 18 for references and a formal treatment of the method required here.

The analogs of the alternatives given in Note 28 take the form G3(p, q, b, ∊) := exp{i12b∊t2/[s(1+s)2]}s-1F[p, q, bs] [note that, unlike Eq. (3.22), the amplitude factor is now the same as in the corresponding two-dimensional case], and G4(p, q, b, ∊) :=21+sF[p2s/(1+s), q2s/(1+s), b2s2/(1+s)].

Consider approaching the aperture along the axis. Because of the cancellation between adjacent Fresnel zones, the field value in the Fresnel approximation ends up depending solely on the value of the field in an infinitessimal rim around the aperture’s edge, and this is precisely where the validity of the Kirchhoff boundary conditions is at its worst.

This result is found by expanding Eq. (3.14) to first order in ∊ and finding the difference between the result and the correction given in Eq. (4.4). Equation (4.5) is just the highest term in an expansion of this difference in 1/ζ, upon taking ξ=v(1+ζ2)1/2. An approximation that is better where the expression in Eq. (4.5) goes to zero follows upon retaining an additional term: ∊≈|(3-4πv2)v2+i3/(4πζ)|/(4NR2).

The exact field takes the form U(ξ, ζ)=1-exp{i2π/[(R+1)ζ]}/R, where R=[1+(ζNR)-2]1/2.

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Figures (9)

Fig. 1
Fig. 1

Contour plot of the relative error in a cylindrical paraxial Gaussian beam where the waist size is given by NR=πw/λ =100. The coordinates used in all the figures here are those introduced in Ref. 9: v=ξ/(1+ζ2)1/2 and u=log2(ζ). In the near field (i.e., u large and negative) v is roughly ξ, but it approaches ξ/ζ in the far field. The field modulus in each plane of fixed u is proportional to exp(-πv2) [see Eq. (4.2)], so the beam width is just a horizontal band in the coordinate plane shown here. Since the field modulus is down from its central value by a factor of exp(-4π)10-5.5 at the top and bottom edges of the plot, there is no need to go outside this domain for a Gaussian. This error map is found by comparing the paraxial field with an exact solution—numerically determined to more than ten significant figures—over a grid of 300×320 points. Since the error is proportional to NR-2, this figure can be used to find the relative error for other waist sizes.

Fig. 2
Fig. 2

Contour plots of the relative error of the beam considered in Fig. 1, but now the first term in the Lax–Wünsche series [i.e., Eq. (4.4)] is retained. The gray levels in Fig. 2(a) are the same as those in Fig. 1 but, in this case, the 10-8, 10-9, 10-10, and 10-11 contours are overlaid as white lines. The error reduction in the near field is impressive—in fact, the numerical noise in the exact data becomes obvious now for ζ<1 and ξ>1. For comparison, the contours predicted by the error estimate that follows from the term of order NR-4 in Eq. (4.4) are shown in (b). As in Fig. 1, the error is evidently arbitrarily large over essentially the whole field for sufficiently large ζ. (Note that the error is now proportional to NR-4, so when NR=10, for example, the white lines then correspond to relative errors of 10-4, 10-5, 10-6, and 10-7.)

Fig. 3
Fig. 3

Contour plot of the relative error in G1[ξ,ζ/2π,1/(2πNR)2] as defined in Eq. (3.14), where F(ξ, ζ/2π) is just UF(ξ, ζ) of Eq. (4.2). This is, again, for the case considered in Fig. 1. It is evident that this phase correction successfully addresses the dominant error for ζ2. As for Fig. 1, the relative error scales with NR-2.

Fig. 4
Fig. 4

Contour plot like that of Fig. 3 but now for G2[ξ, ζ/2π, 1/(2πNR)2] as given in Eq. (3.16). The remarkable effectiveness of this correction is evident in that the error in the far field is now reduced to a level that fully mirrors the performance in the near field.

Fig. 5
Fig. 5

Relative error in a circular paraxial Gaussian beam where the waist size is given by NR=πw/λ=100. The similarity between this plot and that given in Fig. 1 for the cylindrical case carries over for the analogs of Figs. 24.

Fig. 6
Fig. 6

Relative error in the Fresnel diffracted field when a collimated beam is normally incident on a slit. The slit width is given by NR=R/λ=1,000. The field modulus falls more slowly away from the axis in this case, so a larger range of v is shown. [Far from the axis, i.e., v1, the Fresnel field modulus is down from its axial value by a factor of ζ/(πv) in the near field and 1/(2πv) in the far field.] The overlaid contours are predicted by Eq. (5.8), which evidently gives an impressively simple and globally accurate estimate of the Fresnel error in this case (except near the curves corresponding to minima in the field modulus where the relative error is anomalously high but only by about an order of magnitude).

Fig. 7
Fig. 7

Relative error of the beam considered in Fig. 6, but now the first term in the Lax–Wünsche series [i.e., Eq. (3.8)] is retained. The Fresnel field for this case is given in Eq. (5.2). Keep in mind that the error of this corrected field scales with NR-4. The overlaid contours are predicted by Eq. (5.9). The error reduction is again impressive in the near field, but the error in the far field remains as the dominant problem.

Fig. 8
Fig. 8

Contour plot like that of Fig. 6 but now for G2[ξ, ζ/2π, 1/(2πNR)2] as defined in Eq. (3.16), where F(ξ, ζ/2π) is just UF(ξ, ζ) of Eq. (5.2). Together with Fig. 4, this plot demonstrates the versatility of this correction since the error in the far field is now below the underlying uncertainty associated with the Kirchhoff boundary conditions.

Fig. 9
Fig. 9

Contour plot like that of Fig. 6, but the aperture is now circular, with its radius given by NR=R/λ=1,000. The principal difference is that the error has localized peaks on axis where the Fresnel field vanishes, i.e., when the Fresnel number is an even integer. (The Fresnel number, just 1/ζ, ranges from 64 to 1/64 as u goes from -6 to 6.) Contours of the error estimate given in Ref. 9 are overlaid for comparison. The agreement in the near field is not as close as that seen in Figs. 6 and 7.

Equations (66)

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U(x, y, z)=A(α, β)exp[ik(αx+βy+1-α2-β2z)]dαdβ,
A(α, β)=k2π2U(x, y, 0)×exp[-ik(αx+βy)]dxdy.
U(x, y, z)=U(x, y, 0)×k2π2exp{ik[1-α2-β2z-α(x-x)-β(y-y)]}dαdβdxdy.
UF(x, y, z)=exp(ikz)A(α, β)exp{ik[αx+βy-(α2+β2)z/2]}dαdβ,=-ik exp(ikz)2πzU(x, y, 0)×expik2z[(x-x)2+(y-y)2]dxdy.
U(ξ, η, ζ) := exp[-i2π(R/λ)2ζ]U(ξR, ηR, ζR2/λ),
U(ξ, η, ζ)=exp-iζ(u2+v2)/(2π)1+1-(u2+v2)/(2πNR)2×12π2U(ξ, η, 0)×exp[-i(uξ+vη)]dξdη×exp[i(uξ+vη)]dudv,
UF(ξ, η, ζ)=1(kR)2AukR,vkRexpiuξ+vη-ζ4π(u2+v2)dudv,=-iζU(ξ, η, 0)expiπζ[(ξ-ξ)2+(η-η)2]dξdη.
I(p, b, ) := -L(x)exp[-ibx2/(1+1-x2)]×exp(ipx)dx,
F(p, b) := I(p, b, 0)=-L(x)exp(-ibx2/2)×exp(ipx)dx.
exp[-ibx2/(1+1-x2)]=exp(-ibx2/2)[1-18ibx4-1128bx6(8i+bx2)2+O(3)],
I(p, b, )=F(p, b)-18ibL(x)x4 exp(-ibx2/2)×exp(ipx)dx-1128b2×L(x)x6(8i+bx2)×exp(-ibx2/2)exp(ipx)dx+O(3).
I(p, b, )=F(p, b)+12ibb2F(p, b)-212bb3F(p, b)+18b2b4F(p, b)+O(3).
I(p, b, )=Lubexpipub1-18iu4b-1128u6(8i+u2)b2+Ob3exp(-i12u2)du/b.
un exp(-i12u2)du(1-i)n+1Γn+12,neven0,nodd,
U=UF+1NR2iζ4πζ2UF-1NR4ζ8π2ζ3UF+ζ232π2ζ4UF+O1NR6,
U¯U¯F+iz2kz2U¯F-z2k2z3U¯F+14zz4U¯F.
I(γb, b, )=exp[ibγ2/(1+1+γ2)]×Lγ1+γ2+s{1+O(bs3)}×exp[-i12b(1+γ2)3/2s2]ds,
I(p, b, )=[L(x)exp(ipx)]{1+O(bx4)}×exp(-i12bx2)dx.
I(γb, b, )=exp[ibγ2/(1+1+γ2)]×-i2πb(1+γ2)3/2×Lγ1+γ2[1+O(b-1/2)].
F(γb, b)=exp(i12bγ2)-i2πb L(γ)[1+O(b-1)].
I(p, b, )G1(p, b, ) := exp{-i12bg4(p, b)/[1+1+g2(p, b)]2}F(p, b),
g(p, b) := S(b)pb,
G2(p, b, ) := exp{i12bg4(p, b)(1+2r)/[r(1+r)]2}r-3/2F(p/r, b),
I(p, q, b, ) := L(x, y)exp{-ib(x2+y2)/[1+1-(x2+y2)]}×exp[i(px+qy)]dxdy,
F(p, q, b) := I(p, q, b, 0)=L(x, y)×exp[-ib(x2+y2)/2]×exp[i(px+qy)]dxdy.
I(γ1b, γ2b, b, )exp{ib(γ12+γ22)/[1+1+(γ12+γ22)]}×-i2πb[1+(γ12+γ22)]×Lγ11+(γ12+γ22),γ21+(γ12+γ22),
F(γ1b, γ2b, b)exp[ib(γ12+γ22)/2]-i2πbL(γ1, γ2).
G1(p, q, b, ) := exp[-i12bt2(1+s)2]F(p, q, b),
G2(p, q, b, ) := exp{i12bt2(1+2s)/[s(1+s)]2}s-2F[p/s, q/s, b],
U(ξ, ζ)=a2γπ exp[-(u/2γ)2]exp-i12πζu2/[1+1-(u/2πNR)2]exp(iuξ)du.
UF(ξ, ζ)=a1+iγ2ζ/πexp[-γ2ξ2/(1+iγ2ζ/π)].
b=ζ/(2π),=1/(2πNR)2,
p=ξ,l1.
U(ξ, ζ)=a1+iζexp[-πξ2/(1+iζ)]×1-(1/NR)2iq2πζP2-(1/NR)4iq32(πζ)2(2P3-iqP4)-(1/NR)6iq412(πζ)3(15P4-i12qP5-2q2P6)+O[(1/NR)8],
|3-4πv2|v2/4NR2,
3|1-12πv2+8π2v4|/(16πζNR2),
U(ξ, η, ζ)=a(2π)2exp-i12πζ(u2+v2)/[1+1-(u2+v2)/(2πNR)2]×exp[-(u2+v2)/4π]×exp[i(uξ+vη)]dudv,
UF(ξ, η, ζ)=a1+iζexp[-π(ξ2+η2)/(1+iζ)],
U(ξ, η, ζ)=UF(ξ, η, ζ)1-(1/NR)2iq2πζQ2-(1/NR)4iq32(πζ)2[2Q3-iqQ4]-(1/NR)6iq412(πζ)3[15Q4-i12qQ5-2q2Q6]+O[(1/NR)8],
U(ξ, ζ)=aπsinc(u)exp-i12πζu2/[1+1-(u/2πNR)2]exp(iuξ)du,
UF(ξ, ζ)=a1+i{F[(ξ+1)2/ζ]-F[(ξ-1)2/ζ]},
F(x) := 0x expiπ2t2dt.
U(ξ, ζ)=iπNR-U(ξ, 0)[H1(1)(2πζNR2T)×exp(-i2πζNR2)]1Tdξ,
H1(1)(z)=-1+iπzexp(iz)1+i38z+15128z2+O(z-3),
U(ξ, ζ)=(1-i)a2ζ-11 expi2πζ(ξ-ξ)2/(T+1)×{1+3i/(16πTζNR2)+O[1/(ζNR2)2]}T-3/2dξ.
F=[1-iπ(ξ-ξ)4/(4ζ3NR2)-3(ξ-ξ)2/(4ζ2NR2)].
=(1+3ξ2)/(4ζ5/2NR2),|ξ|<1[1+π(|ξ|-1)/ζ]|ξ|(3+ξ2)/(4ζ5/2NR2),|ξ|>1.
=π(1+21ξ2+35ξ4+7ξ6)/(32ζ11/2NR4),|ξ|<1[1+π(|ξ|-1)/ζ]π|ξ|(7+35ξ2+21ξ4+ξ6)/(32ζ11/2NR4),|ξ|>1.
In(B, p) := -xn exp(-x2/2B)exp(ipx)dx,
In(B, p)=(-iB/2)nHn(pB/2)I0,
I2m=BmPm(Bp2)I0,
Pm(x)=j=0m(-1)jmj(2m-1)!!(2j-1)!!xj.
I2m+1=ipBm+1Pm+1/2(Bp2)I0,
Pm+1/2(x)=j=0m(-1)jmj(2m+1)!!(2j+1)!!xj.
Jn(B, p, q):=--x2+y2  n×exp[-(x2+y2)/2B]×exp[i(px+qy)]dxdy,
Jn(B, p, q)=(2B)n/2Γ(n/2+1)Ln/2[B(p2+q2)/2]J0.
J2m(B, p, q)=BmQm[B(p2+q2)]J0,
Qm(x)=j=0m(-1)jmj(2m)!!(2j)!!xj,
U(ξ, η, ζ)=-iζU(ξ, η, 0)1T21+i2πNR2ζT×expi2π(1+T)ζ[(ξ-ξ)2+(η-η)2]dξdη,
I(p, b, )=12[F(p, b-ib)+F(p, b+ib)]+O(2).
U¯(x, y, z)12[U¯F(x, y, z-iz/k)+U¯F(x, y, z+iz/k)].
G3(p, b, ) := exp{i12bg4(p, b)/[r(1+r)2]}r-1F[p, br].
G4(p, b, ) := 2r(1+r) F[p2r/(1+r), b2r2/(1+r)].
G3(p, q, b, ) := exp{i12bt2/[s(1+s)2]}s-1F[p, q, bs]
G4(p, q, b, ) :=21+sF[p2s/(1+s), q2s/
(1+s), b2s2/(1+s)].

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