Abstract

Simulations of the Huygens, Fresnel, and Rayleigh–Sommerfeld I approximations in the case of free-space propagation of a Gaussian beam are compared with analytical solutions. The most accurate results were obtained by the Rayleigh–Sommerfeld I approximation. The study reveals that the approximations are not uniform throughout the propagation region. While the accuracies of the Huygens and Fresnel methods generally increase as the propagation distance increases, the accuracy of the Rayleigh–Sommerfeld I approximation at first starts to diminish and later recovers as the propagation distance is further increased.

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References

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  1. M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1975).
  2. A. E. Siegman, Lasers (University Science, 1986).
  3. C. Huygens, Treatise on Light (Dover, 1962).
  4. A. Sommerfeld, Optics, Vol. 3 of Lectures on Theoretical Physics (Academic, 1964).
  5. M. Totzeck, “Validity of the scalar Kirchhoff and Rayleigh-Sommerfeld diffraction theories in the near field of small phase objects,” J. Opt. Soc. Am. A 8, 27–32 (1991).
    [CrossRef]
  6. C. J. Bouwkamp, “Diffraction theory,” Rep. Prog. Phys. 17, 35–100 (1954).
    [CrossRef]
  7. L. Eyges, The Classical Electromagnetic Field (Addison-Wesley, 1972), p. 263.
  8. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 7th ed. (National Bureau of Standards, 1968).

1991 (1)

1954 (1)

C. J. Bouwkamp, “Diffraction theory,” Rep. Prog. Phys. 17, 35–100 (1954).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 7th ed. (National Bureau of Standards, 1968).

Born, M.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1975).

Bouwkamp, C. J.

C. J. Bouwkamp, “Diffraction theory,” Rep. Prog. Phys. 17, 35–100 (1954).
[CrossRef]

Eyges, L.

L. Eyges, The Classical Electromagnetic Field (Addison-Wesley, 1972), p. 263.

Huygens, C.

C. Huygens, Treatise on Light (Dover, 1962).

Siegman, A. E.

A. E. Siegman, Lasers (University Science, 1986).

Sommerfeld, A.

A. Sommerfeld, Optics, Vol. 3 of Lectures on Theoretical Physics (Academic, 1964).

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 7th ed. (National Bureau of Standards, 1968).

Totzeck, M.

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1975).

J. Opt. Soc. Am. A (1)

Rep. Prog. Phys. (1)

C. J. Bouwkamp, “Diffraction theory,” Rep. Prog. Phys. 17, 35–100 (1954).
[CrossRef]

Other (6)

L. Eyges, The Classical Electromagnetic Field (Addison-Wesley, 1972), p. 263.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 7th ed. (National Bureau of Standards, 1968).

M. Born and E. Wolf, Principles of Optics, 5th ed. (Pergamon, 1975).

A. E. Siegman, Lasers (University Science, 1986).

C. Huygens, Treatise on Light (Dover, 1962).

A. Sommerfeld, Optics, Vol. 3 of Lectures on Theoretical Physics (Academic, 1964).

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Figures (4)

Fig. 1.
Fig. 1.

Geometry for evaluation of the Huygens’ integral [after 2].

Fig. 2.
Fig. 2.

Diffraction of an incident cylindrical wave by a slit aperture [after 5].

Fig. 3.
Fig. 3.

Gaussian beam with radius of the beam (ω0) at e2 or 13.5% of its maximum intensity.

Fig. 4.
Fig. 4.

Logarithmic graphs of percent relative error versus propagation distance for maximum Gaussian beam intensity obtained from various approximations as compared to the values obtained from the exact integration. (a) Fresnel 1 and 2 approximations. (b) Rayleigh–Sommerfeld I approximation. (c) Huygens and asymptotic approximations.

Equations (30)

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E˜(r;r0)=exp[jkρ(r,r0)]ρ(r,r0),
ρ(r,r0)(xx0)2+(yy0)2+(zz0)2.
E˜(s,z)=jλS0E˜0(s0,z0)exp[jkρ(r,r0)]ρ(r,r0)cosθ(r,r0)ds0,
L(xx0)2+z2.
E˜(x,z)=jLλE˜0(x0,z0)exp(jkL)cosθdx0.
ρ(r,r0)=zz0+(xx0)2+(yy0)22(zz0)+.
E˜(x,y,z)1zz0exp[jk(zz0)jk(xx0)2+(yy0)22(zz0)].
E˜(x,y,z)j(zz0)λE˜0(x0,y0,z0)exp[jk(zz0)jk(xx0)2+(yy0)22(zz0)]cosθdx0dy0.
E˜(x,z)jzλE˜0(x0,z0)exp[jkzjk(xx0)22z]cosθdx0.
UI(r)=12πbbU0(r)n[exp(ik|rr|)|rr|]dxdy
exp(ik|rr|)|rr|dy=iπH0(k|ρρ|).
UI(ρ)=i2bbU0(ρ)nH0(k|ρρ|)dx.
nH0(k|ρρ|)=kH0(k|ρρ|)k|ρρ|(ρρ)|ρρ|·n=kH1(k|ρρ|)cos(ϑ),
UI(ρ)=ik2bbH1(k|ρρ|)cos(ϑ)U0(ρ)dx.
uz(x,y)u(x,y,z).
u0(x,y)=cexp((x2+y2)ω02),
I(x,y)=|u0(x,y)|2=c2exp(2(x2+y2)ω02).
uz(x,)=jλexp(jkz)zcexp((x2+y2)ω02)exp(jk2z[(xx)2+(yy)2])dxdy.
exp(jk2zx2)exp((1ω02+jk2z)x2)exp(jkzxx)dx.
exp(αξ2)exp(±jβξ)dξ=παexp(β24α).
α=(1ω02+jk2z)=2z+jkω022ω02z;β=kxz.
2πω02z2z+jkω02exp(j2kz4z2+k2ω04x2)exp(k2ω024z2+k2ω04x2).
uz(x,y)=cjλexp(jkz)z2πω02z2z+jkω02exp(j2kz4z2+k2ω04(x2+y2))exp(k2ω024z2+k2ω04(x2+y2)),uz(x,y)=cjkω022z+jkω02exp(jkz)exp(j2kz4z2+k2ω04(x2+y2))exp(k2ω024z2+k2ω04(x2+y2)).
exp(k2ω024z2+k2ω04(x2+y2))exp((x2+y2)ω2(z)),
ω(z)4z2+k2ω04k2ω02=ω01+(2zkω02)2.
z0kω022=πω02λ.
ω(z)=ω01+(zz0)2,
R(z)=z[1+(z0z)2].
uz(x)=cjzλ2πω02z2z+jkω02exp(jkz)exp(j2kz4z2+k2ω04x2)exp(k2ω024z2+k2ω04x2).
H11(x)2/(πx)exp(j(x0.75π)).

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