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.

Near Field and Far Field

1. In diffraction optics,

Fresnel number:

F =

R2 λz

(1)

where R is aperture's radius.

Far field:
     • F << 1.
     • It is the Fraunhofer diffraction area.
     • The diffraction pattern remains its shape; only its size gets bigger at farther distance.  

Near field:
     • F ≥ 1.
     • It is the Fresnel diffraction area.
     • The diffraction pattern changes shape as a function of distance.

 There is also an "extreme near field" region where the field observed behind the aperture is simply a geometrical projection of the aperture. Geometrical optics zone is when F >> 1.

2. For laser beam,

Rayleigh range:

z0 =

πW02 λ

=

π w02 λ M2

(2)

where W₀ is 1/e² waist radius and w₀ = W₀/M is the 1/e² waist radius of the embedded fundamental Gaussian beam. (i.e., a Gaussian beam and its embedded Gaussian beam have the same Rayleigh range).

Near field is near the laser beam waist, within its Rayleigh range.
Far field is outside of the Rayleigh range.

References:
[1] http://en.wikipedia.org/wiki/Fresnel_number
[2] 黄婉云，《傅里叶光学教程》，1985年第1版. Page 95.