Diffraction theory summary

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:

In figure above, Σ is the diffration aperture, P₁ is a point in it and U₀(P₁) is the complex amplitude waiting to be diffracted, P is the point we want to solve. The complex amplitude at P is:

U(P) =

1 jλ

∫∫ U0(P1)

ejkr r

cos(n, r) ds

(1)

The approximation needed here is r >> λ . This is the Rayleigh-Summerfeld solution.

In frequency domain,  let G_z(f_x, f_y) and G_z=0(f_x, f_y) 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):
     G_z(f_x, f_y) = G₀(f_x, f_y) exp[jkz(1-α²+β²)^1⁄₂]                     (2)
where
    f_x = α/λ,  f_y = 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)

where

h(x, y) = exp(jk

x2+y2 2z

)

(5b)

Eq.(5)-(5b) have the following physical meaning:
• 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)

Eq.(6) means:
• 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)

then

U(x, y) =

ejkz jλz

exp(jk

x2+y2 2z

) F[U0(x1, y1)]

(8)

So Fraunhofer diffraction is simple: the diffracted field is simply a Fourier transform of the initial field.

Eq.(7) is more often expressed by "Fresnel number":

F =

R2 λz

(7a)

where R is aperture's radius. Fraunhofer diffraction is when F << 1. 

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.