Resonator eigenvectors and similar triangles

Section 4: Resonator eigenvectors and similar triangles: round-trip consistency as a problem in trigonometry

These plots have some striking periodic features. The perturbations involving the second and third modes near L = 312 mm, and the first and second modes near L = 525 mm, are obvious even for a “small” tilt of 50 microradians. How do they arise? This section discusses them, and introduces a trigonometric form of the resonator round-trip equation that I have found useful.

Various devices deliberately based on modulo 2π phase behaviours are mentioned below. The 1990 paper with Alan Colley says:

Figure 24: Top: Text from Hill and Colley (1990); Bottom: Sketch from Hill and Hall (1986) of one round trip in a dual Case I resonator.

That is, we find the difference in phase shift for two routes round the resonator. In this example, one route is for light coupled at the tilted mirror from EH11 to EH11, then propagating through 2L and coupled back into EH11; the other is for light coupled from EH11 to EH21, then propagating through 2L and coupled back from EH21 to EH11. The “π out of phase after a round-trip” condition is that this difference equals an odd multiple of π, and this yields a set of L values of which the smallest positive one is 525 mm. Extending this calculation to the pair EH21 and EH31 we obtain a smallest positive length of 312 mm, which again agrees with the above Figures. We do not expect to see that 2nd/3rd mode-crossing feature in the laser output, although it could be revealed by “sub-threshold” probing (Pearson et al. 1995, Shackleton et al. 1995) or “hooting” (see below). The calculation requires the phases of the coupling coefficients, which for small mirror separations and small tilts are close to 0 (modes of same parity) or π/2 (modes of different parity) - see Eqn (9) of Hill (1988). These lengths (e.g. 525 mm ~ (16/3)a2/λ for modes 1 and 2, and 312 mm ~ (16/5) a2/λ for modes 2 and 3) are the same as those for “coincidence”, or equality of resonant frequencies in the unperturbed resonator, or “mode regeneration” in a hollow waveguide Talbot-type device of length 2L.

The eigenvalue plots for a well-aligned dual Case I cavity show the near-coincidences around L = 525 mm:

Figure 25: Complex eigenvalues for the first five resonator modes of the well-aligned resonator with two Case I mirrors.

With only the two lowest-order modes (m = 1, 2) included, and a small tilt, we see:

Figure 26: Complex eigenvalues for resonator modes composed of only two x-direction guide modes (m = 1 and 2), for a resonator with one well-aligned Case I mirror and one plane mirror at z2 = 5 mm with tilt φ/2 = 0.1 mrad.

Figure 27: Round-trip losses for the two resonator modes of Figure 26.

The paths of the phases for these two modes approach but do not cross:

Figure 28: Eigenvalue phases for the two resonator modes of Figure 26.

The exact evolution of the eigenvectors is probably too complicated to follow in our heads when the resonator is realistic. If we reduce to this simplest case, with only two modes included, the two eigenvalues are roots of a quadratic equation:

where

A = exp(i2γ1L).c11(φ)

B = exp(i2γ1L).c12(φ)

C = exp(i2γ2L).c21(φ)

D = exp(i2γ2L).c22(φ)

are the elements of the 2x2 round-trip matrix. These λ, λ' and the corresponding eigenvectors can be evaluated after some algebra; reassuringly, they agree with the values from numerical matrix diagonalisation. The round-trip losses 1 - |λ|2 and 1 - |λ'|2 can be evaluated (and, for example, differentiated with respect to L) after some more algebra. The eigenvalues are equal when the mean (A+D)/2 of the diagonal elements equals the square root of the determinant (AD-BC). In the limit of small perturbations, as the self-coupling magnitudes approach unity and the cross-coupling magnitudes approach zero, this degeneracy occurs when

exp(i4k1L) + exp(i4k2L) – 2exp(i2(k1 + k2)L = 0

i.e. when these three phasors, two of length 1 and one of length 2, are collinear.

This is another expression of a dual Case I “coincidence” condition. As L is varied, the phasors rotate at angular rates proportional to their phase propagation terms – in this case proportional to 12 =1, 22 = 4, and the mean of the squares (12 + 22)/2 = 2.5. At certain L values all three phasors align and the resultant is as short as (locally) possible. In our toy example with only two modes in a well-aligned dual Case I resonator, the magnitude of the difference between the two eigenvalues is illustrated by:

Figure 29: Distance between the two eigenvalues in the complex plane, for the resonator of Figure 26.

There are local minima near L = 16a2/(3λ) ~ 525 mm (already mentioned) and at multiples of this length. The minimum values increase steadily, as the guide length and thus attenuation increase and the phasors become less well matched.

The 1990 paper did not pursue the “origin” explanation in depth. My main concern was that the undoubted bad behaviour (such as mode-hopping, line-hopping despite grating tilts of several mrad, hooting, and distorted mode shapes) suggested that the associated distortions and perturbations were both common and severe, so that simplifying assumptions such as “in the limit of small tilt angles” were often wrong. Thus a multimode approach (well beyond a 2x2 matrix approach, and not amenable to a simple explicit calculation) seemed necessary. An “account of the fine detail” would involve the interactions of more modes (i.e. a larger complex matrix problem, with several phasors not of unit length, and with non-negligible cross-coupling terms) and the sensitivities of the eigenvalues to other parameters such as attenuation and guide-to-mirror separation. Raising the maximum mode order typically introduces significant extra detail at each step. And again these are oversimplified round-trip matrices within a static model. Their eigenvalues and eigenvectors can be tracked as we vary one or more parameters, but we are not modelling time evolution or laser dynamics.

Another look at 2x2 matrix results and “mode coincidence”

We saw that successive “kinks” in plots of resonator parameters vs guide length are separated by the “beat length” between two guide modes – this periodicity was familiar since the 1970s and explicit in several early papers e.g. Roullard and Bass (1975). I was not innovating in conference and journal papers when I noted such kinks and their spacings. But their positions and shapes (not just their separations / periodicities) are crucial in practice, and our 1990 comments above about the “origin” of perturbed-resonator kinks were apparently novel: “In the limit of small tilt angles, the kinks appear where the dominant mode EH11 and another important mode are π out of phase after a round-trip”.

Unlike the researchers just cited, we were treating tilted-mirror resonators, where odd and even modes interact. We considered light that travels from the point where it is about to leave EH11 and couple into (say) EH41, to the point where it has coupled back into EH11. That second coupling takes us one stage beyond the “round trip” drawn in four stages 1-2-3-4 in Figure 24. This is why the propagation phase shift difference is an even (not odd) multiple of π, combining with the two tilted-mirror-coupling phase shifts (π/2 each) to satisfy “π out of phase”. The round-trip matrix includes only one such event, and the geometrical argument in Section 4 below ends after those four stages (1-2-3-4 or 4-3-2-1; the order does not matter); so it involves only one π/2 phase shift. The two arguments, as we would hope, lead to the same conclusions about where kinks originate. In 1987, before publishing the final versions of Hill (1987) and Hill et al. (1988), I decided that, since the origin-of-kinks argument was apparently not clear to a very helpful and astute referee, it might not be clear to me and might need rewriting: the emphasis of “one complete round trip” was intentional. Readers may wish to check, using the simplified expressions for mode propagation phase shifts and coupling coefficients, that this argument – see Eqn (12) of Hill, Monk and Hall (1987) – does yield the “kink” positions.

Having seen no explanations of the form of the kinks, rather than simple notes of their existence and Talbot-type periodicity, I will try to clarify and expand what I wrote. The next plots are again for a dual Case I example with guide halfwidth a = 1.025 mm and λ = 10.59 μm. One mode coincidence occurs when the difference in round-trip propagation phase shift equals 2π for EH11 and EH21, that is, when π(22-1)/8 = 2πN where N is the “Fresnel number” a2/(λL). Hence L = (16/3)a2/λ which is about 529.1 mm for our values.

One mirror is well aligned and the other is given a small tilt φ/2 = 0.0125 mrad. The propagation matrix is 2x2 (only the EH11 and EH21 modes are included) in the interests of simplicity. The two resonator modes are almost pure EH11 and EH21, with guide losses rising monotonically as the guide length L increases – except for the small kinks near the mode coincidence length L = (16/3)a2/λ. We gradually restrict the range of L so that these kinks are shown more clearly:

Figure 30: “2x2” matrix model predictions for round-trip losses: only EH11 and EH21 are included. The two mirrors are Case I, one well aligned and the other tilted by φ/2 = 0.0125 mrad.

Figure 31: As Figure 30 with closer inspection of the kink near L = (16/3)a2/λ.

Figure 32: As Figures 30 and 31 with yet closer inspection of the kink near L = (16/3)a2/λ.

Figure 33: The phase of the EH11 main component of the fundamental resonator mode is conveniently set to zero. As L passes through the mode coincidence length near 529 mm, the relative phase of the EH21 mode (called ϕ2 – ϕ1 below) changes smoothly but rapidly.

Figure 34: This quotient of absolute values is the reciprocal of what we call |a1|/|a2|below. It varies smoothly but rapidly as L passes through the mode coincidence length near 529 mm.

The next two plots are for a 6x6 MATLAB model, and they confirm that the perturbation φ/2 = 0.0125 mrad is very weak in the sense that the 2x2 and 6x6 results for the lowest two modes are very similar. Thus a simple model – 2x2 is not what I called “multimode” – can be reasonably accurate for very small perturbations, and may help our understanding.

Figure 35: “6x6” matrix model predictions for round-trip losses: modes from EH11 to EH61 are included. The two mirrors are Case I, one well aligned and the other tilted by φ/2 = 0.0125 mrad.

Figure 36: As Figure 35: magnitudes and relative phases of EH11 and EH21 in the fundamental resonator mode.

Let the two-mode eigenvector be a1E1 + a2E2 where a1 and a2 are complex numbers (“amplitudes”) and E1 and E2 are the electric fields of two EHmn waveguide modes. We may show a1 and a2, for example, as phasors on a 2-D Argand diagram (with sum a1 + a2), or as elements in a complex eigenvector matrix [a1 a2]T. Initially we are taking EH11 as the first mode, and EH21 as the second mode to which it is weakly coupled; EH31 is another possibility to be considered later.

The algebraic solution for the two “eigentriples” (λ, a1, a2) was discussed above. But, for a simple graphical view, consider what is happening at and near this “mode coincidence” (L ~ 529 mm). The resonator seeks a combination of EH11 amplitude and EH21 amplitude that produces a self-repeating field, and we know:

- the two round-trip propagation phase shifts are nearly equal (modulo 2π) because the length L is near a mode-coincidence length

- one mirror is slightly tilted so there is a weak cross-coupling whose phase is near π/2.

After a round-trip propagation through the guide length 2L, both mode fields are attenuated, EH21 more than EH11. Then recoupling from the tilted mirror provides the four terms already introduced:

[1] EH11 self-coupling exp(i2γ1L).c11(φ).a1 = A.a1

[2] EH21 self-coupling exp(i2γ2L).c22(φ).a2 = D.a2

[3] EH11 to EH21 cross-coupling exp(i2γ1L).c21(φ).a1 = C.a1

[4] EH21 to EH11 cross-coupling exp(i2γ2L).c12(φ).a2 = B.a2

The geometry problem, as stated here and shown below, is not hard, and some readers may prefer to skip several pages; the problem is more to convince oneself that it is correctly stated.

Consider the sum of the first three terms, and temporarily neglect the fourth. This is the simplest approach that still contains some mode-coupling physics. The fourth term involves the product of c12 (weak coupling) and a2 (small amplitude), so the additional resulting EH11 amplitude is smaller still in any realistic case of very small perturbations (see for examples the Figures here). Despite this neglect – which for general perturbations would be unwise – the remaining three-term vector sum must still be a reasonably close, shape-preserving, scaled copy of the two-term sum (a1 + a2) from one round trip earlier. The first term is a new EH11 amplitude a1new, and the second and third terms sum to give a new EH21 amplitude a2new. We can evaluate the phase arg(a1 + a2) and the “magnitude quotient” or relative mode strength|a1|/|a2|for this a1 + a2 or any such propagating two-mode sum. “Shape-preserving” means that we have similar triangles: there may be an overall scaling factor and an overall rotation (represented by the complex eigenvalue of the round-trip matrix), but the relative magnitudes of the vectors a1 and a2, and the angle between them, must be the same as for a1new and a2new if the two-mode field within the guide is to be self-repeating.

Since EH11 and EH21 have different attenuation constants, and since the self-coupling magnitude is very nearly 1 for very small tilt, the magnitude quotient increases if we consider only propagation through 2L and self-coupling, so the term [3] - the cross-coupling from EH11 to EH21 – must reduce it, i.e. boost EH21 to the required strength. Since EH11 and EH21 also have different phase shifts for the extra 2L (except in the special case of exact mode coincidence), this cross-coupling term must also restore the previous phase angle. Recall again that this third term is a small EH21 vector lying at 90 degrees to the EH11 vector, because c21(φ) is near π/2 for very small tilt angle φ.

Thus we take the eigenvector [a1 a2]T with its two amplitudes (one for EH11 and one for EH21), propagate it explicitly through another round trip to obtain a new vector, and impose the condition that the new vector must be a scaled copy of the old one. We know two of the three components of the new vector (the arbitrary EH11 amplitude lying along the x-axis, and the small extra EH21 amplitude to which it gives rise), so the problem is to find the main EH21 amplitude that satisfies the round-trip condition; if there are multiple solutions, the problem is to find the one giving the lowest round-trip loss.

At exact mode coincidence, after propagation through 2L, the EH11 and EH21 amplitudes are attenuated but (by the definition of coincidence) the angle modulo 2π between them is unchanged. Unless the angle between them was initially 90 degrees (or 270 degrees), the EH21 component created by cross-coupling from EH11 to EH21 is misaligned with the main EH21 component, so their sum has a different angle and the triangles cannot be similar. So we know without calculating them that the EH11 and EH21 amplitudes are 90 degrees out of phase at exact mode coincidence. 90 degrees is a special or limiting case as we will see; in a more general case of small perturbations the same argument applies, so the two guide mode amplitudes are ϕ out of phase at exact mode coincidence, where ϕ is the phase shift of the coupling between them. (I recommend a pause if this is not clear).

Away from mode coincidence, after propagation through 2L, the EH11 and EH21 amplitudes are attenuated and the angle between them is changed. For example, by varying L slightly above or below 529 mm in our example below, we introduce a small relative rotation between EH11 and EH21 phasors; if they were initially collinear (or orthogonal etc.), they are not quite so after propagation through 2L. The resonator must select an EH21 amplitude that combines with the cross-coupling amplitude to restore both the magnitude ratio and the relative phase of the final sum, but it may now have extra freedom due to the fact that the EH11 and EH21 amplitudes need no longer be orthogonal.

Readers may wish to sketch some vector triangles, exploring what is and is not possible with these|a1|/|a2|and arg(a1 + a2) conditions.

Without loss of generality we can, as described earlier, rotate the phasors or “micro-tune” the cavity so that phase angles are presented with the x-axis (zero phase) aligned with both EH11 amplitude components (i.e. before and after the extra round trip). The Figures below show plots of these “angle” and “magnitude quotient” parameters, for a small range of L near a mode coincidence. Note that the appropriate EH21 amplitude due to cross-coupling will automatically restore both parameters – they are both implied in the idea of similar triangles.

Figure 37: Some three-term vector (phasor) sums of EH11 amplitude (in red) and EH21 amplitude (in blue) for guide lengths L near mode coincidence.

The angle between the two main vectors passes through 90 degrees as the resonator reaches exact coincidence (L ~ 529.1 mm). At lower right a close view of the vectors shows the addition of the third term (light blue, EH11 to EH21 cross-coupling, at 90 degrees to the red term).

Figure 38: The inclusion of the third term (cross-coupling from EH11 to EH21) improves the fidelity of the scaled copy of the eigenvector after one further round trip, for L values very near the mode coincidence. For values further from coincidence the fidelity is not improved and indeed a little worse. “Angle of eigenvectors” here means the angle between the phasor a1 and the summed phasor a1 + a2.

Figure 39: Magnitude quotients |a1|/|a2| and |a1new|/|a2new|for guide lengths L near mode coincidence. A close view (right) shows that fidelity improves when the cross-coupling term is included.

Within the approximations of this two-mode three-term model we can sketch similar triangles and derive, for example, the relative mode strengths given the guide losses and tilt-coupling coefficients. For clarity the guide losses, and the relative strength of EH21, are exaggerated (compare with Figure 37):

Figure 40: Sketch of phasors in similar-triangles configuration for successive round trips in a dual Case I resonator with a small tilt of one mirror, when there is a mode coincidence of EH11 and EH21. DE represents the “third term” [3] above, the coupling of EH11 to EH21, and is exaggerated for clarity, as is the quotient OA/OC, which might typically be 1.01-1.05. The one unknown in the problem is the magnitude of EH21 relative to EH11 – we already know that their relative phase is π/2. The points ABCD etc. in this diagram are not to be confused with the 2x2 matrix elements above.

OA = |a1|

OC = |a1new| = |exp(i2γ1L).c11(φ).a1|

(shown lying on x-axis)

AB = |a2|, which is to be found

(a1 and a2 are orthogonal, i.e. angle A is π/2, at mode coincidence)

DE = |a2extra| = |exp(i2γ1L).c21(φ).a1|

(EH11 to EH21 cross-coupling; this third term must be parallel with CD, otherwise their resultant will not be orthogonal to OC)

CD = |a2new| = |exp(i2γ2L).c22(φ).a2|

(rotated with a1new if necessary; a1new and a2new are orthogonal after propagation through 2L; c11 and c22 are very near 1 for small φ)

For compactness we have written these triangle side lengths as OC = xOA, CD = yAB, and DE = xgOA, understanding that each represents the length (magnitude) of a phasor that also has a direction. Here x is a real number slightly less than 1, taking account of the propagation loss for EH11; y is a smaller number, taking account of the larger propagation loss for mode EH21; g is a small number << 1, so that xg takes account of the EH11 propagation loss and the tilt cross-coupling. The self-coupling coefficients are taken as unity, which is nearly true for very small tilt. We could write more detailed functions of the guide dielectric constants and the tilt angle, but need not do so at this point.

From the similar-triangles relation

(CD+DE)/OC = AB/OA

we obtain

AB = xgOA/(x-y)

The numerical values obtained for a MATLAB run with this guide geometry (a = 1.025 mm, λ = 10.59 μm, L = (16/3) a2/λ = 529. 1 mm) and typical guide loss assumptions are:

OA = 0.9354, OC = 0.9303, AB = 0.3535, CD = 0.3458

x = 0.9945, y = 0.9782, g = 0.0055

If these are inserted it is seen that the three-mode approach agrees reasonably well, though not exactly. For example, the phase angle of the OAB sum (a1 + a2) is

tan-1(0.3535/0.9354) ~ 0.361 rad

and the collinear third term that would extend the EH21 amplitude and make OCE a similar triangle is

AB (OC / OA) – CD ~ 0.0058

which is reasonably near our numerical result g = 0.0055. The relative strength |a1|/|a2|of EH11 and EH12 is

AB/OA = xg/(x-y)

which equals 0.336, to be compared with our numerical result 0.378.

Note again the danger of neglecting the fourth term, which although “small” is an EH11 component perpendicular to CD and thus, in the present case, nearly parallel with OC.

Solutions via algebra or plane trigonometry (within our approximations and two-mode assumptions) are not very challenging for pencil and paper and 2x2 matrices, but tedious if we include more terms. Care is needed with absolute-value variables, differentiating them (e.g. to find extrema of |λ|2), and placing phasor angles in the correct quadrants – software packages may have different conventions. Here readers might stop and think about the round-trip condition

ME = λE

and how we approximate it. The 2x2 (four-term) model is imperfect, and the three-term model is presumably worse. But we can certainly imagine a resonator in which all the guide modes of order higher than 2 have enormous attenuation losses, so any light coupled into them disappears through attenuation and there is none left to contribute to self-coupling or cross-coupling at the mirror. Then a 2x2 model would be realistic. And our neglect of the fourth term seems reasonable in the limit of small tilt.

Does such a 2x2 resonator possess self-consistent modes? Because the characteristic equation is quadratic, we expect – as said above – two roots or eigenvalues (usually complex in our resonator studies) except in pathological cases.

Does such a 2x2 matrix model, if we set to zero the coupling coefficient for the neglected fourth term (i.e. c12(φ)), give the same results as our vector-triangle approach? The answer is no, and I suggest a pause if this is not understood. If there is no coupling from EH21 to EH11 then any light in EH21 vanishes from the system – that is, coupling from EH11 to EH21 may as well be treated as a separate additional attenuation of EH11. We could adapt our vector definitions accordingly – but, unless we do, the two approaches are not the same.

Now let L be near but not necessarily equal to a mode-coincidence value such as (16/3) a2/λ; again the perturbation is a small tilt that couples EH11 to EH21 with a phase shift of π/2. The corresponding vector diagram might resemble:

Figure 41: Sketch of phasors in similar-triangles configuration for successive round trips in a dual Case I resonator given a slight tilt perturbation, where the guide length is slightly shorter than a mode-coincidence length.

This is only a slight extension of the mode-coincidence case, so more readers may skip ahead, but we will cautiously repeat the definitions and steps in the construction of such diagrams:

OA = |a1|

- A positive real number, the length or magnitude of the EH11 component of the fundamental resonator mode. We may arbitrarily place it along the x-axis (and this is the default output of some software matrix packages).

OC = |a1new| = |exp(i2γ1L).c11.a1| = x. OA

- A somewhat smaller positive real number, the length or magnitude of the EH11 component [1] of the fundamental resonator mode after one more round trip. To help comparisons we rotate a1new so that it too lies along the x-axis, and must therefore similarly rotate any other components of the new eigenvector. We assume that the mirror tilt is so small that the self-coupling losses 1 - |c11|2 and 1 - |c22|2 can be neglected. The attenuation losses are at least ten times larger, and not neglected: that is, x and y are significantly less than unity.

AB = |a2|

- A considerably smaller positive real number (perhaps 0.1 – 0.3 in our examples), the length or magnitude of the EH21 component of the fundamental resonator mode.

CD = the length or magnitude of the EH21 component [2] created when the a2 component propagates through 2L and, reflecting from the tilted mirror, self-couples to EH21.

ϕ1 = the angle between the two components (EH11 and EH21) of the fundamental mode. Because of the rotation of OC above, ϕ1 is also the angle between AB and the x-axis.

ϕ2 = the angle between the two components (OC and CD, self-coupled EH11 and self-coupled EH21) of the fundamental mode after one more round trip. Because of the rotation of OC above, ϕ1 is also the angle between the EH21 component and the x-axis.

The phase shift ϕ2 - ϕ1 (modulo 2π) between EH21 and EH11 is zero by definition at mode coincidence; for the guide parameters in our current tilted-mirror example, it is equal to 0.1188 radians per centimetre of departure from the mode coincidence length (16/3)a2/λ.

DE = |a2extra| = |exp(i2γ1L).c21.a1| = gx.OA

- the length or magnitude of the EH21 component [3] created when the a1 component propagates through 2L and, reflecting from the tilted mirror, cross-couples to EH21. As above, this cross-coupling component is perpendicular to OA, so DE is drawn parallel with the y-axis.

The three sides of the CDE triangle can now be considered together. The phasor sum or resultant of the two EH21 components must be parallel with AB and of length x.AB if the triangles OAB and OCE are to be similar as required. The third term (cross-coupled amplitude) is shown as a vector that must terminate at E, and be perpendicular to OC because c21 is perpendicular to c11. This perpendicularity of vectors need not hold for the four-term sum, where we must include both self-coupling and c12 cross-coupling amplitudes (not in general with equal phase angles) when forming OC.

Note again the simplifications and the exaggeration for clarity in these sketches. “Slightly” and “near” are vague words, but the results above suggest that very significant changes in the phasor arrangement occur when L departs from mode coincidence by only 1 mm.

Because we are considering values of L at or near a mode-coincidence kink, the angle ϕ2 – ϕ1 is indeed small – certainly less than π/2 for the examples used here (otherwise we might face extra work tracking the signs of quantities in different quadrants). In Figure 37 the orientation of the two EH21 phasors (considered as a nearly parallel pair, and both on the same side of vertical) changes much faster than the angle between them. The changes in x, y and ϕ2-ϕ1 are small, but the changes in AB and ϕ1 may be large.

The diagram can be drawn with AB and CD on the other side of vertical, and with a slight departure from mode coincidence in the other direction (L slightly longer than (16/3)a2/λ. But it is not possible to form the required triangle if AB and CD do not both lie on the same side of vertical (i.e. if ϕ1 – π/2 and ϕ2 – π/2 are of different sign): in that case, as is already clear in Figure 41, the lead (or lag) of the a2new phasor relative to a1new would only increase when we add the vertical third term.

One or more of these assumptions may be inappropriate. Large losses and/or strong coupling may mean there is no pair of EH21 terms (CD and DE in Figure 41) that will lead from point C to a point E on OB such that the triangles OAB and OCE are similar; the inclusion of further higher-order modes (EH31, EH41,…) may be essential.

In the limited cases where our model is reasonable, AB and CE are parallel according to our similar-triangles condition; but AB need not be perpendicular to OA, or parallel with CD. The net phase shift π(22-12)/(8N) between EH21 and EH11 for propagation through 2L – that is, the angle that AB makes with CD, shown as (ϕ2 – ϕ1) – is no longer assumed to be a multiple of 2π. In our example above, this phase angle varies through about +/- 0.12 rad as L varies by +/- 10 mm around (16/3) a2/λ ~ 529 mm. This phase shift is “small” in the above sense (well below π/2), but it is sufficient (given the extra freedom mentioned above) for the vectors to form similar triangles with much weaker EHm1 mode content.

Various approaches are now possible. For example, with CD and DE no longer necessarily collinear, the above result AB/OA = xg/(x-y) is modified:

xg.OA + y.AB sinϕ2 = x.ABsinϕ1

AB/OA = xg/(x.sinϕ1 – y.sinϕ2)

Thus, given the other parameters (x, y, g, L), we can solve for ϕ1 and hence ϕ2 and AB/OA – which define a1 and a2.

Or by the sine rule:

sin(ϕ1 - ϕ2 )/(gx.OA) = sin(π/2 + ϕ2)/(x.AB) = sin(π/2 - ϕ1)

hence

OA/AB = sin(ϕ1 - ϕ2)/(g.cosϕ2) and x/y = cosϕ2 /cosϕ1

Expanding cos(ϕ2) = cos(ϕ1+ p) and using sin2ϕ1 + cos2ϕ1= 1, we obtain

cos2(ϕ1) = sin2(p)[1 + (x/y)2 – 2(x/y)cos(p)]-1

Taking the appropriate sign (i.e. placing ϕ1 in the first quadrant for a small negative ϕ2 – ϕ1, as in Figure 41), we can evaluate ϕ1 as a function of L.

In this case or more complicated cases, and admitting that all “solutions” are approximate, and perhaps interested in how sensitive the eigenvalues and resonator behaviour are to variations in the parameters, we may search for an “optimum”, according to the criterion that the resonator will choose the one fundamental mode (a1 + a2), from all the possible (a1 + a2) combinations, that minimises the round-trip loss. This raises an important point.

These kinks or wiggles imply that the resonator round-trip loss can be higher, sometimes much higher, than the round-trip loss of a pure EH11 waveguide mode. But, near EH11/EH21 coincidence, pure EH11 is not available as a resonator mode. The resonator works with what it is given, which (except in pathological cases) includes other modes whose potential for interference must be accepted. Thus, perhaps counterintuitively, a decrease in loss for higher-order modes (e.g. the inclusion of extra modes with finite losses in a model that previously excluded them or gave them infinite losses) can bring an increase in fundamental resonator mode loss in certain situations.

Obviously, for guide lengths not near EH11/EH21 coincidence – which means almost all lengths – AB and CD are far from parallel so that the formation of the similar triangle would need a large restoring component DE. Since no such component is available in our 2x2 picture, the resonator cannot do much better than pure EH11.

If guide attenuation is the only significant loss (as in our simple model for a dual Case I resonator with a very small tilt), the resonator chooses the smallest |a2|/|a1|from the available set – since EH21 or any other mode is lossier than EH11.

A specific guide length L fixes ϕ2 – ϕ1, but ϕ1 and ϕ2 may be imagined as not fixed until the resonator “explores” all the possible initial angles for ϕ1 or ϕ2. Sometimes we imagine a resonator “seeded” with all possible random fields, which are left to propagate round and round; only some of these are eigenvectors; only one (except in pathological cases) is the lowest-loss eigenvector, and it is the last one standing in the long term.

Thus for a general value of L we can explicitly evaluate |a2|/|a1| for all possible angles ϕ1, subject to one of the similar-triangles constraints for each value of ϕ1. For example, we set to zero the derivative of OA/AB with respect to ϕ1:

0 = cosϕ1 - (y/x) cos(ϕ1+ p)

where g has been removed as common to both terms. Again expanding cos(ϕ1+ p) and using sin2ϕ1 + cos2ϕ1= 1, we see again:

cos2(ϕ1) = sin2(p)[1 + (x/y)2 – 2(x/y)cos(p)]-1

So one explicit procedure (with x, y, g and L given) would be:

- Define the phase shift p, e.g. p = 11.88 [L – (16/3)a2/λ] radians

- Find ϕ1 = cos-1[sin(p)[1 + (x/y)2 – 2(x/y)cos(p)]-1/2] (in the correct quadrant)

- ϕ2 = ϕ1 + p

- |a1|2 = 1/(1 + f2), |a2|2 = f2/(1 + f2), where f = x.g/(x.sinϕ1 – y.sinϕ2 )

- Loss = 1 – x2|a1|2 – y2|a2|2

These sample results suggest, again, fair agreement with the matrix method:

Figure 42: Top: 2x2 matrix results. Bottom: Comparison of 2x2 matrix results and results from the “similar triangles” procedure above.

Our three-term model is speculative and simplified. Again, the assumed losses and coupling coefficients may mean that the “available set” is empty, i.e. there is no pair of EHm1 terms (CD and DE in Figure 41) that will lead from point C to a point E on OB such that the triangles OCE and OAB are similar. But, when such solutions exist, they may offer interesting and novel insight. If there are further sources of loss, we do not know in advance that minimising |a2|/|a1| will minimise the round-trip loss. Thus, although the similar-triangles argument used for the tilted-mirror resonator applies to a general small perturbation once we know the mode most strongly coupled to EH11 (usually EH21 or EH31) and the phases of the coupling coefficients, more work may be needed if the guide attenuation loss 1 – x2|a1|2 – y2|a2|2 is not the total round-trip loss.

Notably, a waveguide resonator that is well-aligned but no longer dual Case I (that is, with no mirror tilt, but at least one of z1 and z2 nonzero) has loss kinks of the form shown in Hill (1988) and for instance in our Figures 5, 12 and 18 above, and the two participating modes in a three-term argument are now EH11 and EH31; EH21 does not interact with them if there is no misalignment. Some light spills over the guide entrance after reflection from the distant mirror; Henningsen et al. noted that the relative guide mode phases affect this coupling loss, but did not quantify how their resonators found the balance that minimised the total (coupling plus attenuation) loss.

With EH31 replacing EH21 and c31(z2) replacing c21(φ), and with a corresponding change in the guide propagation parameters, recoupling from the distant well-aligned mirror provides the four terms

EH11 self-coupling exp(i2γ1L).c11(z2).a1

EH31 self-coupling exp(i2γ3L).c33(z2).a3

EH11 to EH31 cross-coupling exp(i2γ1L).c31(z2).a1

EH31 to EH11 cross-coupling exp(i2γ3 L).c13(z2).a3

The fourth term is neglected for the same reason as before, and we examine the remaining three-term vector sum. The phase of the third term depends on z2, and for the small z2 considered here may be taken as π/4. The round-trip phase shift between EH31 and EH11 increases by 31.7 radians per metre of guide length (8/3 times the previous value 11.88).

For a dual Case I resonator (z1 = z2 = 0), there is exact EH11/EH31 mode coincidence if L = 2a2/λ (~198 mm). If z2 is nonzero but small, we expect coincidence for roughly L = 2a2/λ – z2 in the sense that after one further round trip the recoupled EH11 and the main recoupled EH31 phasors retain the previous angle (modulo 2π) between them. The secondary EH31 component created by cross-coupling from EH11 to EH31 now makes an angle of roughly π/4 with EH11. By the same argument as before it is parallel with the main EH31 component at mode coincidence. The free-space propagation through z2 will itself bring small mode-dependent phase shifts, and 5 mm is not very small in this context, so we might say only that we expect nearly parallel components for small z2 near coincidence. But the argument is based on the phasors for the guide EH modes, and any on-axis relative phase shift is included in the complex coupling coefficients e.g. c13(z2). To this extent the argument is unchanged from the tilted-Case I argument above; the new complication is the mirror coupling loss.

Figure 43: 3x3 matrix results for a resonator with one Case I plane mirror (z1 = 0) and one distant plane mirror (z2 = 5 mm); no mirror tilt. Top: round-trip resonator mode losses. Middle: guide mode content and relative phase; these curves are at first sight similar to those in Figure 42, but the phase difference shows an offset. Bottom: resonator mode frequency coincidence near guide length 2a2/λ – z2 ~ 194 mm. The “mode 2” is shown, but is not involved because it is the lowest-loss mode of odd symmetry (mainly EH21) and with φ = 0 it does not interact with EH11 and EH31.

Readers who have skipped the step-by-step exposition could start at this more general case (Figure 44) where the problem is to find ϕ1, given x, y, g, ϕ and ϕ2 – ϕ1:

Figure 44: Top: Sketch of phasors in similar-triangles configuration for successive round trips in a dual Case I resonator given a slight perturbation. The perturbation need not be a tilt so DE need not be perpendicular to OC. Bottom: Examples of phasors (EH11 and EH31 mode amplitudes) near a mode coincidence length L = 2a2/λ – z2 ~ 194 mm as for Figure 43. The short light blue phasors represent DE. In our current example this is the EH11-to-EH31 coupling with its relative phase ϕ of roughly π/4. In general ϕ may take any value, so the (positive) length gx.OA.|sin(ϕ)| and the solutions for ϕ1 and OA/AB are appropriately modified. The angles ϕ1 and ϕ2 between the dark blue phasors (EH31 amplitudes) and the superimposed red phasors (EH11 amplitudes) change rapidly with L; the difference ϕ2 – ϕ1 changes much more slowly, at ~ 31.7 radians per metre of guide length.

The component DE (the cross-coupled EH31 component in this case), which makes the angle ϕ (~ π/4 in this case) with the x-axis OCA, must connect CD with the point E if the triangles are similar. When ϕ is varied, D can lie inside/on/outside the triangle OCE. As ϕ approaches π/2 (the case considered above), the solutions are forced apart as ∠COE must be either near 0 (as in the phasor plots above) or near π/2 (involving a much larger |a2|/|a1| and thus not favoured). Our present example has been chosen so that ϕ is intermediate (~ π/4) and the mirror loss (with z2 = 5 mm) is comparable with the guide loss. The resonator is juggling different functions of the a1 and a2 phasors:

- they must fit the similar-triangles condition;

- they imply a guide loss 1 – x2|a1|2 – y2|a2|2;

- and they imply a coupling loss which may be larger or smaller than the pure EH11 coupling loss. We can recall the part-explanation in Henningsen et al. (referring in turn to Gerlach et al.): “the relative phases [of, in these papers, circular-symmetry modes] can adjust such that each time the two modes reenter the waveguide, they interfere destructively at the edge, and constructively at the center, leading to a resulting field which is more confined along the axis than the pure EH11 mode, and hence couples better into the waveguide”. The coupling efficiency |c|2 and loss 1-|c|2 for the resonator mode depend on the exact a1 and a2.

This trigonometric treatment has no startling new results, but may help students to appreciate what the resonator is doing with its constant search for eigenvectors and its many phase flips and loss kinks.

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