This is a summary of the scalar diffraction theory after reading Ref. 1 and 2.
0. Scalar diffraction theory neglects the vectorial nature of the e&m wave. Two conditions need be met for scalar theory to produce accurate result: (1) the diffracting aperture larger than wavelength; and (2) the observing field not too close to the aperture. (Ref.1, p.35)
1. Scalar diffraction formula with least approximation:
| U(P) = | 1 jλ |
∫∫ U0(P1) | ejkr r |
cos(n, r) ds | (1) |
In frequency domain, let Gz(fx, fy) and Gz=0(fx, fy) be the Fourier transform of U(P) and U0(P1) respectively; they are the amplitudes of each fundamental plane-waves (decomposed by Fourier transform). By solving Helmholtz wave equation, Eq.(1) in frequency domain becomes (Ref.2, p.83):
Gz(fx, fy) = G0(fx, fy) exp[jkz(1-α2+β2)1⁄2] (2)
where
fx = α/λ, fy = β/λ (2a)
are the spatial frequencies on 2-D planes normal to the z-axis; they depend on these plane-wave's propagation direction and α, β are the directional cosines relative to x and y axes respectively. Eq.(1) analyzed in frequency domain is called "Angular Spectrum" approach (Ref.1, p.55).
2. Fresnel diffraction:
If further approximation is made:
cos(n,r) ≈ 1 (3)
then
| U(x, y) = | ejkz jλz |
∫∫ U0(x1, y1) exp[jk | (x-x1)2+(y-y1)2 2z |
] dx1dy1 | (4) |
2.1
Eq.(4) can be written as:
| U(x, y) = | ejkz jλz |
∫∫ U0(x1, y1) h(x-x1, y-y1) dx1dy1 | (5) |
| = | ejkz jλz |
U0(x, y) ∗ h(x, y) | (5a) |
| h(x, y) = exp(jk | x2+y2 2z | ) | (5b) |
• Eq.(5b): the impulse response h is the complex amplitude of the spherical wave from a unit point source at (x, y) = (0, 0).
• Eq.(5a): the diffracted field is simply a convolution between the initial field U0 and the impulse response h.
• Fresnel diffraction is a space-invariant linear system.
2.2
Eq.(4) can also be written as:
| U(x, y) = | ejkz jλz |
exp(jk | x2+y2 2z | ) F[U0(x1, y1) exp(jk | x12+y12 2z | )] | (6) |
• The diffracted field is a Fourier transform of U0 multiplied by a diverging spherical wave.
3. Fraunhofer diffraction:
If an even more stringent approximation is made:
| z >> | (x12+y12)max 2λ | (7) |
| U(x, y) = | ejkz jλz | exp(jk | x2+y2 2z | ) F[U0(x1, y1)] | (8) |
Eq.(7) is more often expressed by "Fresnel number":
| F = | R2 λz | (7a) |
In summary,
• When diffraction distance >> λ , scalar theory can be used. In frequency domain, the angular spectrum approach is very physically intuitive.
• When diffraction angle is small (Eq.(3)), Fresnel diffraction is used.
• When Eq. (3) and (7) are both satisfied, Fraunhofer diffraction can be used.
References:
[1] Goodman, "Introduction to Fourier Optics", 2nd Ed.
[2] 黄婉云，《傅里叶光学教程》，1985年第1版.
