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.