Tutorial examples

Section 3: Tutorial examples of resonator calculations

Variation of cavity length for two Case I mirrors; no misalignment

First, how do the resonator losses depend on guide length for a “dual Case I” resonator (z1 = z2 = 0)? This is a simple introduction with no exciting results: the losses rise monotonically with guide length, the loss curves never intersect, there is zero loss at the mirrors, and the differences in round-trip phase shift (modulo 2π) between the resonator modes evolve smoothly. Later we will tilt the mirror and move it away from the guide.

(See the “Reviews” in the References at the end for the terms Case I, Case II and Case III in waveguide resonator studies. They were introduced as (waveguide + mirror) configurations of low loss for the fundamental waveguide mode EH11).

We use wavelength λ = 10.59 μm and guide halfwidth a = 1.025 mm, typical for CO2 devices, so that the length 2a2/λ (see below) is roughly 198 mm, and the guide loss factors are chosen to give realistic losses of a few per cent per metre.

Figure 2: Round-trip losses for the first three resonator modes of the well-aligned dual Case I resonator as the guide length varies. Most 1980s / 1990s CO2 waveguide laser structures lay between the length limits shown here (50 mm to 550 mm). Apart from small numerical and truncation errors, the only source of loss is EHmn guide attenuation. Only variations in the X dimension are considered (see text) so these “first three” modes are essentially EH11, EH21 and EH31.

The evolution in more than two dimensions can be inferred if we allow some artistic licence. For convenience we will show MATLAB plots in Cartesian axes where two axes (nominally X and Y) represent the phasor plane and the third axis represents a varying parameter of interest.

Figure 3: Complex eigenvalues for the first three resonator modes of the well-aligned dual Case I resonator as the guide length varies from 50 mm to 550 mm.

Another view angle shows more clearly that the eigenvalues remain within a unit circle as they spiral round the third (guide length) axis at different rates:

Figure 4: Complex eigenvalues for the first three resonator modes of the well-aligned dual Case I resonator as the guide length varies.

Readers familiar with plotting and animation software will imagine refinements. The behaviour of the resonator mode “phasors” or eigenvalues can be followed in magnitude and phase as we apply various changes or perturbations. The frequencies of different modes may approach one other and cross; so may the losses; how we label and keep track of the modes is up to us. We might continue to label as the “fundamental” resonator mode whichever valid solution has the lowest loss for any particular set of resonator parameters; this may mean discontinuous jumps at crossing points. Or we might label each solution initially with some mode number (in ranking order of loss, say) and then allow it to keep that label as it evolves continuously, even though its ranking may change many times.

For some practical purposes it is the relative phase that matters. So we can choose one mode as a reference, with its phasor always pointing along the X axis and assigned a phase of zero. This might be natural notation, for example, in a cavity that is constantly being micro-tuned by fractions of a wavelength so that the desired mode always lies near CO2 line centre (where the laser gain is maximum). A phase difference of π (3π, 5π etc.), per round trip, for two resonator transverse modes, implies that they lie as far apart as possible (half the free spectral range c/2L) – if the difference modulo 2π increases further, becoming slightly more than π, this means that one transverse mode (with axial mode j, say), is now approaching another that belongs with a different axial mode (j’, say).

Variation with cavity length for one case I mirror and one distant mirror; no misalignment

Now we introduce some diffraction and recoupling physics. Again the variable parameter is the guide length, and the modes are labelled by colours throughout their evolution in 2-D and pseudo 3-D plots, but we place one plane mirror (still perfectly aligned) at a distance z2 from the guide entrance. In the next three Figures that distance is 5 mm.

Figure 5: Round-trip losses for the first three resonator modes of the well-aligned resonator with one Case I mirror and one plane mirror moved 5 mm away from the guide.

Characteristic “kinks” of the mode losses are now seen. Here the interference, constructive or destructive, involves only modes of the same parity (for example EH11 and EH31) because the resonator is still well aligned. For example, as the guide length increases the first strong kink for the third resonator mode (mainly EH31, but in these modified conditions not exclusively EH31) occurs near 95 mm. But it does not coincide with a kink in the first resonator mode (mainly EH11) – for that we must wait until L ~ 2a2/λ.

Figure 6: Complex eigenvalues for the first three resonator modes of the well-aligned resonator with one Case I mirror and one plane mirror moved 5 mm away from the guide.

Figure 7: Magnitude of low-order guide mode content (x-direction sinusoidal modes of order 1, 2 and 3) for the first two even-parity resonator modes of the well-aligned resonator with one Case I mirror and one plane mirror moved 5 mm away from the guide.

Small distortions of the phasor curves are just visible in Figure 6. Figure 7, for the magnitudes of the first few guide modes in the two lowest-loss even-parity resonator modes, shows more clearly how the coupling has noticeable effects around L = 2a2/λ and L = 4a2/λ. That is, near to these L values an appreciable amount of EH31 is present in the fundamental resonator mode (which is still dominated by EH11), and an appreciable amount of EH11 is present in the next even-parity resonator mode (which is still dominated by EH31).

(The EHmn mode fields for odd integers are defined with cosine functions

cos(mπx/(2a))

whose “parity” is even in the sense that the field is symmetric about a maximum at the guide middle x = 0. But free-space Gaussian modes use odd integers, as in TEM11 or ψ11, for odd parity. This is a longstanding convention and cannot be helped).

Examination shows how the relative phase of EH11 and EH31, over narrow ranges of L, “flips” in a way reminiscent of coupled-oscillator physics, although the presence here of “oscillators” (resonating modes) sharing an optical cavity may be less familiar. In order to present the interaction of EH11 and EH31 more clearly, the next Figures 8 to 11 refer to a resonator where only the lowest-order waveguide modes (m = 1, 2, 3) are supported. This is readily modelled, for example by truncating the mode expansions or by greatly increasing the guide attenuation for all higher-order modes. The guide-to-mirror distance is again 5 mm and we focus on a narrow range of guide length L near 2a2/λ.

With well-aligned mirrors we see:

Figure 8: Round-trip losses for the first three resonator modes of the well-aligned resonator with one Case I mirror and one plane mirror moved 5 mm away from the guide. Only the three lowest-order waveguide modes (m = 1, 2, 3) are included in the model, and this is clearly a poor approximation.

Figure 9: Phases of the (dominant) EH11 mode and the EH31 mode, which combine to form the fundamental resonator mode of Figure 8.

The behaviour of the eigenvalues and losses is further complicated by a mirror tilt; this couples modes of both odd and even parity. Note the factor of 2 in Figure 1: when a mirror is physically misaligned by φ/2, the reflected light travels along an axis at an “optical tilt” angle φ to the original guide axis. In all the Figures and examples here, “mirror tilt” means φ/2. For φ = 0.15 mrad, and z1 = z2 = 0, we see:

Figure 10: As Figure 8, with mirror 2 tilt φ = 0.3 mrad.

Figure 11: As Figure 9, with mirror 2 tilt φ =0.3 mrad.

If we restore the suppressed waveguide modes (up to 7th order) the behaviour is considerably different in detail. With well-aligned mirrors we see:

Figure 12: Round-trip losses for the first three resonator modes of the well-aligned resonator with one Case I mirror (z1 = 0) and one plane mirror moved away from the guide (z2 = 5 mm). The seven lowest-order waveguide modes (m = 1, 2, 3, 4, 5, 6, 7) are included in the model, giving a better but still imperfect approximation. The loss curves have changed noticeably from Figure 8.

Figure 13: Phases of the (dominant) EH11 mode and the EH31 mode, which combine to form the fundamental resonator mode of Figure 12.

With a mirror tilt of φ/2 = 0.3 mrad we see:

Figure 14: As Figure 12, with mirror 2 tilt φ/2 = 0.3 mrad.

Figure 15: As Figure 13, with mirror 2 tilt φ/2 = 0.3 mrad.

In the next two Figures the resonator is unchanged (guide modes up to EH71 are considered) except that the guide-to-mirror distance is increased to 15 mm.

Figure 16: Round-trip losses for the first three resonator modes of the well-aligned resonator with one Case I mirror and one plane mirror moved 15 mm away from the guide. The seven lowest-order waveguide modes (m = 1, 2, 3, 4, 5, 6, 7) are included in the model.

Figure 17: Complex eigenvalues for the first three resonator modes of the well-aligned resonator with one Case I mirror and one plane mirror moved 15 mm away from the guide.

Note some consequences of our labelling and plotting choices. The curves for the lowest-loss mode (in blue) are uninterrupted because this lowest-loss status is retained without any mode-crossing, although the second-lowest loss mode (in red) comes very close. But there are several regions where the curves for the second-lowest and third-lowest losses do cross, so that (according to our labelling choice) their colours are switched, and the 3D curves show gaps. The 2D curves for round-trip loss intersect at precise values of L, whereas the corresponding points in 3D (the tips of the plotted complex phasors) may be far apart. Again, readers familiar with the huge literature of matrices and eigenvalues will expect interesting degenerate or near-degenerate examples.

(Links can be made, but are not discussed here, with the broad topic of “eigenvalue repulsion” or “level avoidance”).

Studies of real-life lasing behaviour of cavities with near-equal-loss modes, and then of the subset of cavities with near-equal-loss modes which are also very close in frequency, are helped by such modelling, but of course our examples here are very simple. They omit all details of gain media and mode-competition dynamics; they are “static” solutions for the eigenvectors of these simplified matrices, and do not describe growth or decay of parameters as the “oscillators” evolve in time through phase space.

Next is a series of Figures showing more examples of switching in the order of loss of the resonator modes as the guide length is varied, for increasing values of near-Case I mirror tilt, and with z2 fixed at 5 mm. The order of the losses changes, and the colour labelling changes accordingly. As the tilt increases, the situation becomes more complicated.

Figure 18: Round-trip losses for the first three resonator modes of the well-aligned resonator with one Case I mirror and one plane mirror moved 5 mm away from the guide. The seven lowest-order waveguide modes (m = 1, 2, 3, 4, 5, 6, 7) are included in the model.

Figure 19: As Figure 18, with mirror 2 tilt φ/2 = 0.02 mrad.

Figure 20: As Figure 18, with mirror 2 tilt φ/2 = 0.05 mrad.

Figure 21: As Figure 18, with mirror 2 tilt φ/2 = 0.1 mrad.

Figure 22: As Figure 18, with mirror 2 tilt φ/2 = 0.2 mrad.

Figure 23: As Figure 18, with mirror 2 tilt φ/2 = 0.3 mrad and with four resonator modes shown.

Further plots can be produced at will. “Resonator parameter space” is extensive, even for these fairly simple linear waveguide/mirror structures.