Fresnel Diffraction
For small Fresnel numbers, the appropriate theory is Fresnel diffraction. For a complete discussion and derivation, see the Goodman reference given in the introduction. The key assumptions in Fresnel theory require that the field being computed is not too close to the initial field, namely, if z2 – z1 = Δz , then Δz is large compared to the region over which the field at z2 is to be determined. Another way of saying this is that the beam cannot diverge too quickly; very fast F/# beams cannot be accurately modeled with Fresnel diffraction theory (for more information on these approximations search the help files for "Algorithm assumptions").
In the Fresnel region the electric field distribution is given by
Where
Each of the terms in the above expression has a useful physical interpretation. The leading term indicates that as the beam propagates, the phase changes along the z axis, just like the plane wave described earlier. The amplitude also decreases linearly with distance, or the intensity (E*E) falls quadratic ally.
The expression for q( r, Δz ) , called the quadratic phase factor, indicates that the phase is referenced to a sphere of radius Δz (strictly speaking it is a parabola, but we have already assumed in the Fresnel development that r2 « Δz ).
This is a very useful property; all that is required in the representation of the electric field is the phase difference relative to the reference sphere. This significantly reduces the number of sample points needed to accurately define the phase of the beam. When using the Fresnel propagator, the phase of the electric field is measured relative to a reference sphere with a radius equal to that of the distance from the beam waist. This is not the same radius as the phase radius of curvature of the Gaussian beam. Positive phase indicates the wavefront is advanced along the local +z axis relative to the reference sphere, regardless of the direction of propagation.
Another important property of q( r, Δz ) is that as Δz gets larger, q( r, Δz ), varies more slowly in phase.
This is the opposite of the T(Δz) operator, which varies rapidly in phase as Δz gets larger.
Accordingly, Fresnel diffraction is useful when the Fresnel number is small.
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