We calculate coherence as the degree of first-order coherence, defined in Eq. 6, Dudley et al., Rev. Mod. Phys. 78, (2006), where input pulse shot-noise is modeled semi-classically through the addition of a noise seed of one photon per mode. From this model, we expect that the coherence depends on the repetition rate, and the coherence will be higher for a high repetition rate. Is that true from a physical point of view?
Well, this is a basic question. You need to understand where pulses come from: they do not exist at all! All there is is a sum of many completely benign, constant-amplitude sine waves whose freqienciees and phases are so cleverly designed that they cancel out most of the time while at some monents they do not cancel, because of which it appears there are a large bursts of energy - pulses. Of course that repetition frequency is highly dependent on how yo arrange the sinewaves. So yes, it is an integral part of the story.
If the repetition rate of the pump laser is constant, then it does not affect the coherence of the generated supercontinuum. The first-order coherence (also known as Dudley-Coen degree of coherence) takes two subsequent pulses, and the separation between the pulses (inverse of repetition rate) has to be larger than the pulse length for it to make sense (and to actually have pulses at all!). If the repetition rate varies, then there is a variable shear between any two subsequent pulses, leading to decreased coherence in this definition.
In general, second-order coherence (also known as two-coordinate coherence, not to be confused with intensity correlations) is preferred when quantifying the correlation properties of supercontinuum (see for example: https://www.osapublishing.org/josab/fulltext.cfm?uri=josab-28-9-2301&id=222230 ), but it is much more difficult to measure (Article Experimental Measurement of the Second-Order Coherence of Su...
).
Thank you all for participating in the discussion and for your answers. Thanks to Mr. Koivurova for the article on coherence characterization.
Any variation in pulse parameters, including the variation in the repetition rate, will introduce noise and degrade the generated SC's coherence.
However, I am interested in the SC's coherence of the two scenarios where all pump pulse parameters are the same except for the repetition rate. Does the variance in any pulse parameter (added noise) depend on the repetition rate? Will be added more noise if the time duration between two subsequent pulses (inverse of the repetition rate) is longer.
The repetition rate directly affects only the spacing of the spectral components: a high repetition rate leads to a loosely spaced frequency comb, whereas low repetition rate will cause the frequency components to become more tightly packed.
Now, if the repetition rate varies with time, then the resulting frequency comb will have components that move back and forth - like an accordion! Note that the overall shape (or the envelope) of the spectrum remains the same, just the frequency spacing varies.
You can read more about frequency combs here: https://www.rp-photonics.com/frequency_combs.html
I hope this answers your question!
Hello Matias, thank you for your answer. However, this is still not an answer to my question. I apologize if my question is not straightforward and I will try to explain it better by using an example. Let's assume we are generating two SCs using two different pumps. Those two pumps deliver the pulses at the same central wavelength, with the same pulse duration (at FWHM), pulse shape, and peak power. The first pump delivers pulses at the repetition rate fr1 while the second provides pulses at the repetition rate fr2 (fr1 > fr2). Is the coherence of the two generated SCs the same?
As I said earlier, the repetition rate affects only the spacing of the frequency components. So yes, the two should have similar coherence.
Thank you for your answer. In my opinion, the coherence will be different. If all pulse parameters but the repetition rate are the same, the measured average (output) power or signal-to-noise ratio will differ. More noise will result in less coherence. When thinking in the frequency domain, we should get the same conclusion. In simulations, noise is added as one photon per mode with random phase and amplitude. Therefore, less frequency spacing between frequency components (comb lines or modes) will result in more added photons to model noise. Is this correct?
Let's think about this some more:
What you propose requires multiplicative noise, i.e. noise that exists only where the frequency components exist. Usually, noise is additive, and it is present also where there are no frequency comb components. So let's say we have additive photon noise.
If the peak power and the shape of the pump pulse remains the same (i.e. pulse energy remains the same), then increasing the repetition rate causes the average power to increase. For example, 1 mJ pulses at a repetition rate of 1 kHz yields 1 W of average power, whereas if the repetition rate increases to 10 kHz, then the average power will be 10 W.
If you add a single photon of noise per each frequency bin in both cases, then the signal to noise ratio will decrease as you increase the repetition rate. But this is only because the average power increases in relation to the noise power. Of course, you could argue that it does become more coherent, in which case it is a kind of a "quantum limit" to the degree of coherence. So strictly speaking, in such setting the coherence is affected.
In practice, the average power is usually much larger than the photon noise power, and the coherence is (essentially) unaffected by this. In an experimental setting, the average power generally remains the same when repetition rate changes, which causes the pulse energy to change.
Thank you Matias Koivurova for your answer. We are modeling SC generation, and we are interested in both, the technical noise to reproduce our experiments and the "quantum noise" limit to see how far we can go. Thank you for participating in this discussion.
For a prefect pulsed laser, the repetition rate will not affect the first-order coherence. Happenly, I have also worked on this noise model of SC generation several month ago. In fact, I have tried to model several kinds of noise such as the quantum noise, technical noise, as well as the amplified spontaneous emission noise. The last one makes us publish some papers. see " Spontaneous emission noise in mode-locked lasers and frequency combs" and " Tailoring the waveguide dispersion of nonlinear fibers for supercontinuum generation with superior intrapulse coherence ". I also wonder what mathematical models you will adopted for noises?
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