Coherency between two waves is that their mutual phase difference

[Niles]

Hi

The way my teacher has defined coherency between two waves is that their mutual phase difference ∆ must be constant. With this in mind, does it make sense to talk about coherence between two waves, who do not share the same frequency?

 

[marcusl]

That's a very good question. The answer is, yes it makes sense because coherence is not a yes-no property but a continuum. Two signals or waves can be coherent (the case you mentioned), completely independent (non-coherent), or something in between.

Two infinitely long sine waves of different frequency are non-coherent. This is a statement of the orthogonality of sines and cosines. But if you look at finite length sines whose frequencies are not too different (we'll define how different they can be), then their phases aren't too different over the signal length and there is partial coherence.

Mathematically, the coherence or correlation between two signals of unit amplitude is

$$\rho_\infty=\int_{-\infty}^\infty f^*_1(t) f_2(t) dt,$$

or the corresponding spatial integral if your problem is spatial instead of temporal. This is also the definition of the inner product of f1 and f2, BTW. Two waves of length T and differing frequencies have correlation

$$\rho=\int_{-\infty}^\infty \sqcap(T/2) f^*_1(t) f_2(t) dt,$$

where the first term is the usual rectangle function of Fourier theory.

How close must the frequencies be to be "coherent"? We can answer by looking at the power spectrum of rho, that is, the absolute value squared of its Fourier transform. As usual, the (amplitude) spectrum of rho is the convolution of the spectra of rect and rho_inf. The spectrum of rect is a sinc function, so

$$\tilde{\rho}(\omega)=sinc(\omega T/2) \ast \tilde{\rho}_\infty.$$

The half-power bandwidth of a sinc is

$$\Delta\omega=0.89 \frac{2\pi}{T}$$

(look in your Fourier transform text) so the frequencies can differ by plus or minus $$\Delta\omega /2$$ and still have significant correlation. Furthermore, although we started talking about sines we kept the equations quite general. Accordingly, we can generalize to say that two signals are significantly correlated if the bandwidth of their cross-spectrum is less than or equal to $$\Delta\omega$$.

 

[Cthugha]

Two infinitely long sine waves of different frequency are non-coherent.

The explanation is very good, but it should be noted that the situation is different if optical meaning of coherence is considered. Here two infinitely long sine waves of different frequency are mutually coherent. Although it is true that the two sines are orthogonal and the mathematical cross correlation vanishes, these are not the quantities defining optical coherence. The most basic notion of coherence of an optical field E is given by first order coherence:

$$g^{(1)}(t_1,r_1,t_2,r_2)=\frac{\langle E^*(t_1,r_1)E(t_2,r_2)\rangle}{\sqrt{\left|E(t_1,r_1)\right|^2\left|E(t_2,r_2)\right|^2}}$$.

Although this looks similar to the expression given before, it is not the cross-correlation of the two sines, but the autocorrelation of the sum of the two waves. Assuming the easy case where just one position is considered ($$r_1=r_2$$), this function will only depend on the delay between the times $$t_1$$ and $$t_2$$ and is equal to the time-delay-dependent visibility of the interference pattern in a Michelson interferometer. It will start out at one and decrease towards zero as the fixed phase relationship of the field gets lost for large delays for real light sources. For monochromatic light it stays at 1. For a superposition of two monochromatic waves, the superposition will show a beating. In a Michelson interferometer you will see a time-delay-dependent interference pattern which will also show this beating, but the visibility of the interference pattern will not decrease over time. Therefore two monochromatic waves are mutually coherent in terms of optics.

 

[marcusl]

The most basic notion of coherence of an optical field E is given by first order coherence:

$$g^{(1)}(t_1,r_1,t_2,r_2)=\frac{\langle E^*(t_1,r_1)E(t_2,r_2)\rangle}{\sqrt{\left|E(t_1,r_1)\right|^2\left|E(t_2,r_2)\right|^2}}$$.

Although this looks similar to the expression given before, it is not the cross-correlation of the two sines, but the autocorrelation of the sum of the two waves.

Actually, this is not an autocorrelation nor is there a sum. It is a mutual coherence function (the brackets indicate ensemble average), which is a type of cross-correlation. Furthermore, you have chosen an expression with no explicit frequency dependence so it does not directly address the OP's question about coherence of waves of differing frequencies. The two-frequency mutual coherence function is more appropriate for this purpose, but is rather complicated--hence my decision to discuss the simpler time-domain case in my first post.

We can make some observations, however, without digging into the equations. Assume for simplicity propagation in a vacuum. If the two frequencies are not too different, there will be macroscopic regions where the two waves are largely coherent and others where they interfere (as you described for same-frequency waves). The average size of the regions is characterized by a coherence length. Because the waves are propagating, the scene is time-varying as well, characterized by a coherence time. They both depend on the frequency spread, which we can include in an intuitive way through the relations

$$\Delta t \propto \frac{1}{\Delta\nu}$$

(you can see a similar equation in my first post) and, with $$\Delta l=c\Delta t$$,

$$\Delta l \propto \frac{c}{\Delta\nu}.$$

This gives a general sense of the time and distance over which waves of different frequencies are significantly coherent.

 

[Cthugha]

Actually, this is not an autocorrelation nor is there a sum. It is a mutual coherence function (the brackets indicate ensemble average), which is a type of cross-correlation.

Indeed, from a formal point of view $$g^{(1)}$$ is the cross correlation of some signal at two points/times. However, in most branches of spectroscopy it is common to call it an autocorrelation, too, in the most relevant case of $$r_1=r_2$$. The terminology gets washed out even more when going from field to intensity correlation functions.

However, in optics the experimentally accessible quantity is the total field E. In the mentioned case of two sines this corresponds to E being the superposition of two monochromatic waves.

Furthermore, you have chosen an expression with no explicit frequency dependence so it does not directly address the OP's question about coherence of waves of differing frequencies.

This is why E is defined as the sum of the two monochromatic waves in the above example. It is always the total field present in optics. I thought that goes without saying it explicitly.

We can make some observations, however, without digging into the equations. Assume for simplicity propagation in a vacuum. If the two frequencies are not too different, there will be macroscopic regions where the two waves are largely coherent and others where they interfere (as you described for same-frequency waves). The average size of the regions is characterized by a coherence length. Because the waves are propagating, the scene is time-varying as well, characterized by a coherence time. They both depend on the frequency spread, which we can include in an intuitive way through the relations

$$\Delta t \propto \frac{1}{\Delta\nu}$$

(you can see a similar equation in my first post) and, with $$\Delta l=c\Delta t$$,

$$\Delta l \propto \frac{c}{\Delta\nu}.$$

This gives a general sense of the time and distance over which waves of different frequencies are significantly coherent.

This is not necessarily so and exactly the reason why I chose a different definition of coherence. Often, optical coherence time and length are defined as the time and distance over which where is not a fixed, but a well defined phase difference. The difference becomes clear when one compares a broadened emission line to a superposition of monochromatic emission lines. Simple spontaneous two-level emitters have some characteristic upper level lifetime, which manifests in the linewidth of the emission leading to the coherence times and lengths you described. Those describe the time delay over which the phase becomes randomized. If you somehow managed to realize such a source with few discrete frequencies involved, you would see some characteristic beating in interferometric experiments which vanishes wi on the timescale of the coherence time.

However, you could also take an ensemble of perfectly monochromatic emitters and superpose them to realize light with the same spectral width. As all of them are perfectly monochromatic the long time phase relationship does not randomize and you will see the characteristic beatings (although it will be very difficult to see them for lots of superimposed frequencies) in interferometric experiments for arbitrarily large time delays.

Therefore it is possible to distinguish between multimode coherent light and a source having the same spectral width, but shorter coherence time by means of $$g^{(1)}$$. Although I agree that this is a very specific difference, I wanted to mention it as the question opening this post was rather general.

 

[marcusl]

However, you could also take an ensemble of perfectly monochromatic emitters and superpose them to realize light with the same spectral width. As all of them are perfectly monochromatic the long time phase relationship does not randomize and you will see the characteristic beatings (although it will be very difficult to see them for lots of superimposed frequencies) in interferometric experiments for arbitrarily large time delays.

To the extent that the beats have periods of non-chaotic behavior where the waves are coherent with appreciable intensity at one position, the periods last a time that is on average inversely proportional to the spectral spread. Cannot this too can be thought of as a coherence time?

 

[Cthugha]

Good question. I suppose that you could do so. The list of definitions of coherence is long anyway. ;)

However I am not sure. I am a bit more experienced in higher order coherence, but a bit rusty when it comes to first order. Having reread the topic I am actually not sure whether my remark was helpful for the OP or whether it confused him...