laseroptics
In laser resonators, higher order modes (i.e. TEM01, etc) accumulate phase faster than the fundamental TEM00 mode. This extra phase is called Gouy phase. What is an intuitive explanation of this effect?
Gouy predicted and then experimentally verified the existence of this effect long before the existence of lasers. How did he do it, and what motivated him to think about it?
To my mind, the core experimental benchmark that really requires a quantized electromagnetic field to be explained is, as mentioned tangentially in the question, the Hong-Ou-Mandel effect.
The wavelike nature of photons, as displayed in the Hong-Ou-Mandel effect, is rather subtle ground, but let me start off with a statement that sounds controversial but really isn't:
Interference effects such as double-slit interference are not representative of the quantum-wave nature of photons.
This is somewhat amplified when seen from a quantum-optics perspective, but it's important to keep in mind that photons aren't "particles", much: what they really are is discrete excitations of the classical modes of the electromagnetic field. (You can then go full-circle and argue that all particles, from electrons to atoms in a BEC, are also excitations in a matter field, but that's a separate argument.)
As such, you only really start to be able to talk about photons in a way that is quantum-mechanically meaningful when you talk about counting statistics of a given state of light. Those counting statistics are, in a sense, 'riding' on top of the classical modes that they inhabit, and any interference features that those modes exhibit (like, say double-slit fringes or diffraction rings) are not really "the photon interfering with itself", they're just a geometrical feature of the mode that's being excited.
The reason I discount those interference features as unrepresentative of the 'true' quantum wave of photons is that there's yet another layer of interesting interference, and it is when you make those excitations themselves interfere with each other, in both constructive and destructive ways. This is what the Hong-Ou-Mandel experiment does: it combines the probability amplitudes of different combinations of possible excitations to rule out some of them,
in such a way that the light emerging from the beam splitter has a 50:50 split of the energy, on average, but this only ever comes in joint pairs of photons on each arm, and never as coincident photons on both arms (an outcome whose probability amplitude has vanished through destructive interference). There is simply no semiclassical model that can account for this.
Now, as OON has noted, you can also get experimental observations of photon-counting statistics that are not explainable by any semiclassical model by simpler configurations of sub-poissonian light, but for me the Mandel dip is much more striking, much more clearly recognizable, and not that much more recent historically speaking.
Also, I would like to address some the comments in anna v's answer. All light is quantum; we know this because we've tried repeatedly to find cracks in quantum mechanics, including its treatment of light, and we haven't found any. Light often 'looks' classical, but that is only because quantum mechanics, in its classical limit, looks like classical mechanics.
However, there is still a lot of value of which experiments can still be explained by having a classical electromagnetic field (so, e.g. replacing coherent states with just classical states, possibly with some shot noise) interacting with quantized matter, and this includes things from atomic absorption and emission spectra to the photoelectric effect, as well as the pointwise response of film in double-slit experiments (whose interference features, again, are just a classical-optics feature of the mode that photons are riding on).
Given what we know of the fundamentally quantum nature of the electromagnetic field (through experiments like the Hong-Ou-Mandel dips), we know that these semiclassical descriptions are just effective models that don't fully describe the core aspects of nature, but it is those experiments that are undescribable without quantized fields that really force us to adopt that perspective. Take those away, and saying "the EM field is quantized" becomes just an opinion without meaningful experimental support.
Best Answer
Several sources link to this paper: S. Feng, H. G. Winful, Physical origin of the Gouy phase shift, Optics Letters, 26, 485 (2001), which tries to give an intuitive explanation of the Gouy phase. Briefly, the point is that convergent waves going through the focus have finite spatial extent in the transverse plane. The uncertainty relation induces then some distribution over the transverse and consequently longitudinal wave vectors. It is claimed that the net effect of this distribution over wave vectors is an overall phase shift, which is larger for higher modes. However to see that one really needs to look into the formulas.