Paraxiality

The previous post on the Gaussian beam noted that this beam is a solution to the paraxial wave equation and thus only accurate for small divergence angles. The limit given in [1] was $w_0 \gt 2\lambda/\pi$ or $\theta \lt 0.5\mathrm{rad}$ or $\approx 29^\circ$. The Gaussian beam model is of course not exact up to that limit angle and then suddenly becomes inaccurate. There actually is always a small error which grows as the divergence angle increases.

To get a notion of how accurate the approximation is for various beam waists, the Fourier diffraction propagation method as explained in this post can help, unlike the Fresnel or Fraunhofer propagation methods which were shown in this post and which also make the paraxial approximation are thus not of great use here.

Using Fourier propagation on a Gaussian field distribution with $w_0 = \lambda$ in the source plane, we can compare the exact field and the Gaussian beam model at any propagation distance. This is the result at $10\lambda$ from the waist position:



Plot of the normalized transverse amplitude at longitudinal position z = 10λ from the beam waist w0 = λ for the Gaussian beam (blue) and the exact field (red).



Plot of the transverse phase evolution at longitudinal position $z$ = 10μm from beam waist w0 = λ for the Gaussian beam (blue) and the exact field (red).

The Gaussian and exact fields differ significantly away from the optical axis. The difference between the exact and Gaussian fields is plotted below for various beam waist to wavelength ratios. Refer to this post to calculate any other beam parameters for these cases.

The amplitude error is normalized to the exact on-axis field. This way the errors at different distances remain comparable without giving undue significance to the weak fields far away from the axis. The phase difference is not normalized. All plots show a transverse cross section of the beam up to a radius of $2 w(z)$ as computed from the Gaussian beam model.

Click on the buttons to see the results for the respective beam:
w0 = λ/2w0 = 2λ/πw0 = λw0 = 2λ



Amplitude difference between (paraxial) Gaussian beam and exact field, normalized to the on-axis field, vs. transverse position for the near field at $z$ = λ (red) and the far field at $z = 10 z_R$ (blue).



Phase difference between (paraxial) Gaussian beam and exact field vs. transverse position for the near field at $z$ = λ (red) and the far field at $z = 10 z_R$ (blue).

Clearly, the validity limit $w_0 = 2\lambda/\pi$ for the Gaussian beam model already has a significant phase error in the far field, which could be relevant for mode coupling calculations. This should be kept in mind before blindly trusting such calculations.

References

[1] Gaussian beam [Wikipedia]


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