What's the function of the dielectric resonator in a dielectric resonator oscillator?

To understand why dielectric resonator is used you can think in this direction: we put output of transistor to its input - so called feedback. After powering on there are some
random output of transistor comes to its input and again goes to output connected with 180 phase length microstrip. But frequency cant be defined only with phase, so Frequency is undefined and actually rarely can get any constant oscillation. After adding dielectric resonator we filter transistor output to some value. After powering on first random spikes filtered to fixed weak frequency and then amplified by transistor again and again. So powering on weak signal becomes strong and make itself live again and again, amplified with transistor and filtered with resonator. You can imagine dielectric resonator as an empty cola or beer cylinder resonator. For high frequencies this cylinder will be very small, with tuning screw on top. We can imagine that resonator is empty, and surrounding air is metal. Microstrip of 180 phase delay helps to achieve right amplitude on the input of transistor. If phase is wrong then signal can eat itself or even make zero amplitude. Resonator helps to filter transistor output, so only one (n is 1) harmonic of NxPhase will occur. Tiny Cola (i forgot how is right name of such resonator), microstrip bandpass filter or dielectric resonator all can be used. I am newbie, i hope my answer help you a little. Maybe somebody correct me and give a hint about cola resonator. Cola resonator must be coupled to microstrip different than dielectric resonator.

I don't think so. Oscillation frequency determined by phase and amplitude balance in closed loop. Where the phase shift is 2*pi and transistor gain is greater than feedback loss there is (at this frequency) an oscillation. Feedback elemet may not have amplitude frequency response like bell. For example, in optoelectronic oscillator feedback elemet is delay line with infinite pass band. Otherwise, following your words, a spectrum shape would be similar to frequency response of feedback element.
There is one important characteristic of oscillator - phase noise which depeds on resonator quality, exactly equivalent loaded quality factor. In terms of phase balance in closed loop, stability of oscillation frequency is determined by derivative of phase-frequency characteristic of feedback element (group delay). When the signal going through amplifier becomes random deviation of oscillation frequency (say, increasing), then feedback elements slow down the wave adding some delay - stabilizes the frequency. Stabilization effectivity, obviously, depends on group delay. So, for greater stability you must distort phase-frequency response of feedback element. Most obvious way is to use resonators. Stability will depend on loaded quality of resonator. Concomitant effect is frequency dependend amplitude response of resonator which you may use for noise suppresion at far offsets from carrier.

you could use a ?λ resonator as an oscillator too, but the stray reflections of near field moving objects would affect frequency too much compared to patch antenna. but works well for motion or security sensor. or just for fun
But better to simply use stripline return loss bridge with hot carrier diode.

How about harmonics? Phase and amplitude balance is good, but phase cant determine certain frequency, only harmonics that fit in lambda/4. As we can replace resonator with bandpass microstrip filter i guessed that main function of resonator is still filtering. So even random warming up current will be filtered and provide stable start of oscillator at desired frequency, he he. I like to think this way because i do not know theory. I am interested in frequency personce thing you said: so if we make filter that passes two frequenciees, and put it in feedback loop then we cant do multiple frequency oscillator? For example 2, 4 and 8GHz at the same time? Using triple band passing filter in feedback... In optoelectronics i am a superdummy, but i can imagine only a laser, that have certain frequency by nature, so to build some thing we need only delay lines and mirrors to build some interferometer. And again about main function of resonator, is not it filtering? It resonates, so it filters? No?

Dielectric resonator is just an equivalent parallel RLC circuit with very high quality factor.Resonance frequency is determined by pyhsical and materiel properties of the resonators.
Why we use DR ? In order to obtain very Phase Noise Oscillators with high temperature stability.They are bulky and expensive but also useful for very high frequencies.Their thermal stability,Phase Noise performance,frequency accuracy is pretty good.

anything that resonates is prone to some harmonic resonance. Usually Xtals that run > 50MHz are resonant overtones by design. The same could be true for any technology. The bottom line is Q, stability, Bandwidth, phase noise and cost realization... If it is distorted or nonlinear, it contains harmonics.

It is not necessary to have a feedback element with filtering properties. If you are not interesting about phase noise you can use delay line in feedback for oscillator.
If oscillation conditions are satisfied at two frequensies oscillation will occur at one of them - where gain margin will be greater. If gain margins at these frequencies are equal oscillation will occur at one of them randomly. Second mode will be suppressed dynamically. And there will be random frequency hoping.

BTW. DR has relativelly high Q sufficient to achieve equivalent Q up to 1e5 using combined frequency stabilization implemented in Poseidon Scientific Instruments oscillators www.psi.com.au