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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