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This posting includes flash animations showing the physics discussed.
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Flash animations: |
Keywords: ring resonator, rotating waves, standing waves, cylindrical symmetry, angular momentum, animated rotating waves, Flash animation, technical illustration
| Contents of this posting |
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Waves in a ring resonator - standing and rotating waves:
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| Fig. 11. Microphotograph of a very tiny optical ring resonator. |
1. The resonators and their waves
The circularly symmetric wave system that is equivalent to the one dimensional linear wave systems discussed in the previous posting is a linear system that has been bent around to join itself. Examples include a pipe bent around to join itself which could be excited acoustically, a water channel constructed in a circle supporting water waves, an electrical transmission line bent around to join itself, or an optical fiber bent around and joined. These are generally known as ring resonators. One example is shown in Fig. 11. Generally ring resonators do not need to be strictly circular, but need only cleanly transmit waves around a closed path without reflections.
While the waves in ring resonators do not have beginnings and ends and so have no boundary conditions, they nevertheless have restrictions on the wavelengths of the waves that might be excited to resonate in them. In order to avoid a discontinuity when we move from φ = 2π to φ = 0 and have a nice continuous wave we must have an integer number of wavelengths fit into the circular path (the circumference) of the resonator. Consistent with this requirement, a ring resonator can support two types of waves, standing waves and/or rotating waves. This mix of possibilities is analogous to the possible mix of standing waves and traveling waves in a linear system.
2. Typical pairs of standing wave modes and rotating wave modes
The animations in Figs. 12a,12b,13a and 13b below show both standing waves and rotating waves in a ring resonator, all involving the same mode of oscillation or a shifted version of it.
Mouse over the figures below to see the animations, mouse off to suspend them or click on them to reset them.
| Table 4. Four resonant modes of a ring resonator, all with three wavelengths fitting in the circumference. | |
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↑ Fig. 12a. Standing wave resonance in a ring resonator. The real mathematical expression for this resonant mode is:
while the complex expression is:
where A is the amplitude (real), φ is the position around the resonator measured in radians, ω is the angular frequency of oscillation, and t is time. m = 3 for this standing wave mode and also for the one shown in Fig. 12b. | ↑ Fig. 12b. Second standing wave mode, a shifted version of that in Fig. 12a, realigned a quarter wavelength from that in Fig. 12a. This mode is mathematically independent from the mode in Fig. 12a. The two modes (from Figs. 12a and 12b) can be combined to produce a standing wave mode with three wavelengths having any arbitrary orientation in the resonator. This reference speaks to combining two sinusoids to produce a sinusoid of arbitrary phase.
The real mathematical expression for this resonant mode is:
while the complex expression is:
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↑ Fig. 13a. Positively propagating rotating wave mode in a ring resonator, m = +3.
The real mathematical expression for this resonant mode is:
while the complex expression is:
Rotating waves on a ring resonator are similar to the elementary waves of string theory. | ↑ Fig. 13b. Negatively propagating rotating wave mode in the ring resonator, m = −3.
The real mathematical expression for this resonant mode is:
while the complex expression is:
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3. Mathematical description of waves in a ring resonator
Standing waves in a ring resonator
Standing waves in the ring resonator can be mathematically described by changing the position term, kx, in the linear standing wave expression ( ψ = A cos κx + B sin κx) × cos ωt ) to mφ where m is a positive integer and indicates the number of wavelengths that fit into the circumference for the particular mode in question. Having m = 0 is not common but is allowed in some wave systems.
, (15a)
is the real notation. The complex notation for the same wave is:
. (15b)
Mode pairs
Depending on the constants A and B, Equations (15a) and (15b) can describe two mathematically independent modes: a pair of modes having the same resonant frequency and except for a shifting, the same shape. We can write one pair of modes as:
, (16)
The second mode is mathematically the same as the first if we realign the first by an angle of π/2m .
From linear combinations of these two modes we can form any mode with the same periodicity (same m) but with a different orientation, i.e.:
, (17)
where A, B, C, and θ are constants with C and θ related to A and B by:
and
. (18)
The computer language function atan2 is a more powerful version of the arc tangent (i.e. tan-1), being defined for the whole circle instead of half the circle.
Because of the circular symmetry of the resonator ψ1, ψ2, and ψ3 will all resonate at exactly the same frequency. These equal frequency modes are called degenerate and are hard to independently probe in the laboratory. A tiny amount of asymmetry introduced in the ring resonator will change the frequency of one mode relative to the other and "split" the degeneracy allowing a person to independently probe each mode. Introducing the asymmetry will also fix the axes of the two modes. In the completely symmetric case, the first mode can be picked to be at any arbitrary azimuthal angle θ, while the second mathematically independent mode is then a realigned version of the first, realigned azimuthally by an angle of π/2m.
Rotating waves
The equation for rotating waves is almost the same as that for standing waves except that the mφ and ωt terms occur together in a single sinusoidal function. Below is shown the real and complex forms of the equation for rotating waves, rotating in the positive φ direction.
. (19a)
. (19b)
Next we write the real and complex equations for the rotating wave propagating in the negative φ direction.
. (20a)
. (20b)
Mathematical relationship between standing waves and rotating waves in a ring resonator.
A rotating wave can be constructed from the two standing wave modes that are identical except that one is realigned by a quarter wavelength in the φ direction (we see this in Figs. 12a and 12b above). We also need to shift the phase of one mode in time by a quarter cycle, by π/2, which we will do by using the sine function instead of the cosine for the second mode.
. (21a)
We used trig identities (i.e. cos(u − v) = cosu cosv + sinu sinv ) to manipulate the sinusoids in the middle expression to arrive at the final right-most expression. In the middle expression we see the two standing wave modes and in the right-most expression we see an expression for a mode rotating in the positive φ direction.
If we subtract the two standing wave modes instead of adding them, we get a mode rotating in the negative φ direction.
. (21b)
One nice aspect of the rotating wave description of the resonant fields in a ring resonator is that the index m distinguishes between the two mathematically independent degenerate modes. +m is reserved for the mode rotating in the positive φ direction, whereas −m is used for the mode rotating in the opposite direction. In the standing wave description, both mathematically independent degenerate modes have the same m index and we just need to remember that this index represents two modes.
4. Ring resonator frequencies.
As we have seen above, in ring resonators we can describe the modes of oscillation in terms of standing waves or rotating waves. Rotating waves offer the clearest insight into circular wave propagation and angular momentum in these resonators.
Below we see the first three standing wave pairs, then we see the first three rotating wave pairs.
Mouse over these figures to see the animations, mouse off to suspend them or click on them to reset them.
| Table 5. Standing wave modes of ring resonator, m = 1 through m = 3. | ||
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⇐ Fig. 15. The left most mode is described by Equation (16a) while the right mode is describe by (16b), both with m = 1.
Note that for both, there is one complete wave cycle in the φ direction. | ||
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⇐ Fig. 16. The left most mode is described by Equation (16a) while the right mode is describe by (16b), both with m = 2.
Note that for both, there are two complete wave cycles in the φ direction. | ||
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⇐ Fig. 17. The left most mode is described by Equation (16a) while the right mode is describe by (16b), both with m = 3 substituted into the equation.
Note that for both, there are three complete wave cycles in the φ direction. | ||
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⇐ Fig. 18. The line spectra for the first 5 standing wave resonant modes of a ring resonator. Each line represents two degenerate modes (modes having the same resonant frequency). For a non-dispersive wave medium, such as waves on a string, acoustical waves, or electromagnetic waves, the higher frequency modes are integer multiples of the lowest frequency mode (the fundamental) making this resonator harmonic.
Note also that the rotating wave modes shown in Table 6 below have the same spectra, also with each mode on the frequency scale being doubly degenerate. | |
Next, we show the lowest three modes of a rotating wave in a ring resonator (as opposed to standing wave modes shown just above). The line spectra of the rotating wave modes is identical to that of the standing wave modes (see Fig. 18).
As for the above animations, mouse over these figures to see the animations, mouse off to suspend them or click on them to reset them.
| Table 6. Rotating wave modes of ring resonator, m = ±1 through m = ±3. | ||
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⇐ Fig. 19. The left most mode is described by Equation (20a) with m = −1 (substitute +1 into (20a) since there is already a − sign there). The right most mode is described by (19a) with m = +1.
Note that for both, there is one complete wave cycle chasing its tail inside the resonator. | ||
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⇐ Fig. 20. The left most mode is described by Equation (20a) with m = −2 (substitute +2 into (20a) since there is already a − sign there). The right most mode is described by (19a) with m = +2.
Note that for both, there are two complete wave cycles chasing their collective tails inside the resonator. | ||
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⇐ Fig. 21. The left most mode is described by Equation (20a) with m = −3 (substitute +3 into (20a) since there is already a − sign there). The right most mode is described by (19a) with m = +3.
Note that for both, there are three complete wave cycles chasing their collective tails inside the resonator. | ||
5. Relationship between energy and angular momentum for rotating waves.
If we consider a rotating wave to basically be a traveling wave going around a circular track, then its linear speed will be c = 2πr f/m = rω/m because it travels a distance of 2πr /m each cycle (observe the progress of the wave in the above rotating wave animations). f is the frequency.
In a waveguide, some types of waves carry a momentum per length equal to the energy in the wave per length divided by the group velocity, gx = E1/c . These include electromagnetic waves such as microwaves. Incidentally, transverse waves on a string and sound waves do not carry momentum, contrary to some references (see one of my postings.) Since the angular momentum L per circumference length is just the linear momentum per length times the moment arm, r , we can write:
L/circumference = r gx = rE1/c , (22)
where gx is the momentum per length around the circumference, E1 is the wave energy per length around the circumference. We have used the fact that the wave energy in most simple media equals the wave momentum times the wave velocity, E = g c .
We can multiply both sides of (22) by the circumference to get the proportionality between the total angular momentum of the waves and the total energy around the ring, U:
L = r U/c . (23)
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