Contradiction within wave optics and its solution within a particle picture: comment

Xavier Délen; Marc Hanna; François Balembois; Patrick Georges; Fabien Bretenaker

doi:10.1364/OE.24.002106

A recent article by Altmann claims to point out a contradiction in wave optics, by assessing the resonance frequencies of different laser resonators. In [1], the author first considers a plano-concave cavity of length L that fixes a stable family (fundamental and higher-order) of Gaussian beam modes, as depicted in Fig. 1.

Fig. 1 Schematic layout of the full cavity of length L and the subcavity of length (L-L_s).

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The resonance frequencies of this first cavity, closed by the mirrors located at z = 0 and z = L, are given by [2] (also Eq. (1) in [1]):

ω_{q n m} = \frac{2 π c q}{2 L} + (1 + n + m) \frac{c}{L} (ψ (L) - ψ (0)),

where c is the speed of light, qnm are the axial and transverse mode numbers, and ψ is the axial Gouy phase function associated to the family of Gaussian beams that are stable in the cavity. Consider a particular mode corresponding to mode numbers qnm. The central argument in ref [1]. is that the subcavity formed between mirrors (with radii of curvature such that the stable family of Gaussian beam is unchanged compared to the full cavity) located at L_s and L, with

L - L_{S} = λ_{q} = 2 π c / ω_{q n m}

, should exhibit a resonance frequency at ω_qnm (see the sentence between Eqs. (9) and 10 in [1]). The author then proceeds and shows that it is not the case. From that, a rather exotic theory is developed.

Our opinion is that there is a flaw is the reasoning presented above: if one wants the subcavity located between the two spherical mirrors to have a resonance frequency at ω_qnm, then the length (L - L_s) of this subcavity should be calculated taking into account the Gouy phase shift corresponding to the propagation between these two mirrors. To determine this length, we first write the resonance angular frequencies of the subcavity ω’ using Eq. (1):

ω'_{\tilde{q} n m} = \frac{2 π c \tilde{q}}{2 (L - L_{S})} + (1 + n + m) \frac{c}{L - L_{S}} (ψ (L) - ψ (L_{S})),

where

\tilde{q}

denotes the longitudinal mode number of the subcavity resonance considered. In [1], the author considers

\tilde{q} = 2

, so that the cavity length is approximately equal to the wavelength. To match the resonance frequencies of the full cavity and the subcavity, we write

ω'_{2 n m} = ω_{q n m}

, which translates into the following condition for the length L_S:

\frac{ω_{q n m}}{c} (L - L_{S}) - (1 + n + m) (ψ (L) - ψ (L_{S})) = 2 π .

Again, the subcavity mirrors should exhibit radii of curvature such that the stable family of Gaussian beams supported by the subcavity are the same as for the full cavity. This definition of the subcavity length automatically matches the frequency resonances of both cavities, and differs slightly from the wavelength

λ_{q} = 2 π c / ω_{q n m}

because of the Gouy phase shift. The condition

ω'_{2 n m} = ω_{q n m}

is impossible to satisfy with the length

L - L_{S} = λ_{q}

taken by Altmann. Our conclusion is that there is no such contradiction within wave optics theroretical framework, and no need for a new theory.

References and links

1. K. Altmann, “Contradiction within wave optics and its solution within a particle picture,” Opt. Express 23(3), 3731–3750 (2015). [CrossRef] [PubMed]

2. A. E. Siegman, Lasers (University Science Books, 1986) pp. 761–763.

Contradiction within wave optics and its solution within a particle picture: comment

Abstract

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