# Can the phase of a light wave be measured experimentally?

The phase difference can be measured but is it possible to measure the instantaneous phase of the electric field in light wave at one position?

• If you can define it, you can measure it (in principle). However, to actually define phase, you need a temporal reference, which one can argue ultimately means that you're just defining the phase difference between your system of interest and your reference oscillator. So: define "phase" without that caveat, or you're just running in circles. Commented Jun 1, 2017 at 11:24

Yes, for a classical field (e.g. a large photon number, coherent state), in principle the phase can be measured in the sense that one can in principle observe that the electric (magnetic) field at a given point oscillates sinusoidally with time and one can measure at what times the field is zero and when it reaches its maximums.

If you don't believe this for light, simply imagine the experiment with microwaves, or radiowaves. Phase locked loops have been synchronizing oscillator circuits with incoming radio waves for the best part of a hundred years.

So at optical frequencies, the difference is one of technology only, not physics. Optical phase locked loops now exist and realize coherent modulation schemes over optical fiber communication links.

If you mean by your question the phase of a quantum state of light, then, of course, the state is a ray in projective Hilbert space and is invariant with respect to multiplication by a global phase. No meaning can therefore be given to the absolute phase of a quantum state. Phase differences between the complex amplitudes of base states in superposition are meaningful, and these determine interference effects and the evolution of the quantum state.

Classical and quantum phase are quite distinct, however, and don't simply merge through a "correspondence principle" in a large photon number limit. To understand the difference between them, and the lack of a "correspondence principle", consider a photon number coherent state $\psi(\alpha,\,t)$ of the quantum harmonic oscillator:

$$\begin{array}{lcl}\psi(\alpha,\,t) &=& e^{-i\,\frac{\hbar}{2}\,\omega_0\,t}\,\exp\left(\alpha\,e^{-i\,\hbar\,\omega_0\,t}\,a^\dagger - \alpha^\ast\,e^{+i\,\hbar\,\omega_0\,t}\,a\right)\,|0\rangle \\&=& e^{-i\,\frac{\hbar}{2}\,\omega_0\,t}\,e^{-\frac{|\alpha|^2}{2}}\,\sum\limits_{k=0}^\infty\,\frac{\alpha^k}{\sqrt{k!}}\,e^{-i\,k\,\hbar\,\omega_0\,t}\,|k\rangle\end{array}\tag{1}$$

which is a superposition of Fock (number) states beating together as their relative phases oscillate at different frequencies. Suppose we measure the photon number with the number observable $\hat{n}=a^\dagger\,a$; its mean $\langle\psi|\hat{n}|\psi\rangle$ is constant at $|\alpha|^2$ (indeed, so are all the moments $\langle\psi|\hat{n}^k|\psi\rangle;\,k\in\mathbb{N}$ and by calculating them, one can show that the photon number is Poisson distributed with mean $|\alpha|^2$). However, suppose we measure the position with the position observable $\hat{x}= \sqrt{\frac{\hbar}{2}\frac{1}{m\omega}}\,(a^\dagger + a)$ or the momentum with observable $\hat {p}= i\sqrt{\frac{\hbar}{2}m\omega}\,(a^\dagger - a)$; the means of these measurements $\langle \psi|\hat{x}|\psi\rangle$, $\langle \psi|\hat{p}|\psi\rangle$ are phase-quadrature, time harmonic functions:

$$\langle \hat{x}(t) \rangle = |\alpha| \sqrt{\frac{2\,\hbar}{m\,\omega}} \cos (\arg(\alpha) - \omega_0 t)$$ $$\langle \hat{p}(t) \rangle = |\alpha| \sqrt{\frac{2\,m}{\hbar\,\omega_0}} \sin (\arg(\alpha) - \omega_0 t)\tag{2}$$

and the phase of these quantites, i.e. $\arg(\alpha)$, is the classical phase. Take heed that we can multiply the state $\psi$ in (1) by any wildly time-varying quantum phase $e^{i\,\varphi(t)}$ we like and the results of the above calculations will be unchanged; the quantum phase disappears under the multiplication of the state and its complex conjugate in the calculation $\langle \psi|\hat{X}^k|\psi\rangle$ of any moment of any observable $\hat{X}$. Note that the classical phase is perfectly well defined for any, arbitrarily small mean photon number coherent state (it's simply that for $\alpha$ very small, you'll need many measurements on the same state to measure it, whereas a single measurement will do for laser-sized values of $\alpha$).

For measurement of classical phase of light, see Emilio Pisanty's answer here describing the relevant experimental technology in the Physics SE thread Have we directly observed the electric component to EM waves?.

Also, one can indeed define an observable for the classical phase of a quantum state, although it's tricky; an influential quantum optics researcher has written a whole book on the subject:

Stephen M. Barnett & John A. Vaccaro, "The Quantum Phase Operator: A Review"

(Stephen M. Barnett, along with Bruno Huttner, was one of the first people to work out a fully quantized theory of the electomagnetic field in a dielectric (in this paper here)).

• On the technological side, attosecond streaking (as I explained here) can directly measure the waveform in the IR regime, though you'd struggle if you wanted to trace out more than ten-twenty cycles or so. However, it is still, arguably, an interference experiment, as you always need a phase-locked reference - including the microwave and radio experiments you describe. Commented Jun 1, 2017 at 9:48
• On the quantum side, that depends on what states you're measuring. For Fock states the phase is obviously completely undefined, but not all states are Fock states - and there are indeed phase 'eigenstates' (modulo some strong difficulties in defining them). Phase is (almost) as legitimate an observable as anything else - or, at the very least, there is a perfectly workable POVM for its measurement. (More details here.) Commented Jun 1, 2017 at 9:51
• ... though re-reading your post, you're talking about Hilbert-space phases, which are not the correct analogs to the classical phase of an oscillator. Instead, the quantity of interest is the quantum phase as described in my last link. Commented Jun 1, 2017 at 11:22
• @EmilioPisanty could you possibly summarize for those of us who don't have access to the book? Maybe you could write your own answer? Commented Jun 1, 2017 at 19:25
• @BrianBi To be honest, the quantum perspective is a good deal of a sidetrack w.r.t. the OP, so it wouldn't be all that appropriate to add an answer here. On-site there's this question but it's not a great fit and I'm not thrilled with the answers; it would make a good separate question - though you'd need to demarcate it clearly from the existing one, maybe along the lines of "given that the naive view won't work, what can you do?". Commented Jun 1, 2017 at 19:33

Yes, of course it is possible to measure phase. Holography works by combining a source wave with a reflected-light wave and where the phases match (zero degrees) the light intensity is high (and darkens the film), and where the phases mismatch (180 degrees) the light intensity is minimum, and the film stays transparent. That hologram image, then, captures the reflected-light phase (and reconstruction of the image afterward is therefore possible by illuminating the film with a copy of that original source wave).

• "combining a source wave with a reflected-light wave" is probably the kind of thing OP meant when they said they know "the phase difference can be measured", so this doesn't really answer their question. Commented Jun 1, 2017 at 14:35
• The meaning of 'phase' that makes most sense to me in this context, is the value (phi) in the oscillatory expression "sin(omega *t + phi)" for the wave at a point in space. So, an arbitrary choice of the zero of the time elapsed is intrinsically required, i.e. there must be a reference. So, the physical situation of creating a hologram defines that reference with the 'reference beam'. The reference beam can take another absolute phase and the hologram will reconstruct because only phase differences are important. Commented Jun 1, 2017 at 20:58
• I agree with you that that's the definition that makes sense. But based on the question asked, I'd guess that OP doesn't understand that, or maybe even why it's an issue. So a complete answer should explain why what OP asked for is not sensible. Commented Jun 1, 2017 at 21:49
• Holography measures phase difference as amplitude. There is no phase information in hologram, only amplitude. It means we can't reconstruct original wave with original phase. A question is about direct phase measurement. We can this on slower frequency waves such as radio waves. Visible light is THz waves. But today's fastest transistor is about 400 GHz. It's to slow to directly measure phase of visible light. Commented Oct 8, 2021 at 7:57

Yes, since 2003 we can measure the electric field wave of (visible) light pulses using attosecond light pulses. You may have heard that the 2023 Nobel Prize for physics was awarded "for experimental methods that generate attosecond pulses of light". One of the first applications was to measure the electric field wave of a very short (sub-5fs) light pulse published as "Direct Measurement of Light Waves", see References below.

What we measure is the phase of a (short) light pulse with respect to the pulse envelope (it is called the carrier-envelope phase).

## Requirements

This technique requires that the light wave to be measured is

1. a pulse that
2. is very short, and
3. intense, and
4. can be repeatedly generated with precisely the same wave form (the difficult part).

Requirement #1 is because the pulse envelope serves as the reference against which we measure the phase (of the carrier wave).

Requirement #2 is because we use the pulse to generate a single (isolated) attosecond "probe" pulse and to get an isolated probe pulse (as opposed to a train of probe pulses which would generate ghost images) we need the to-be-measured pulse to have only a "few" optical wave cycles (<3).

Requirement #3 is because we need sufficient field strength to streak electrons (see next chapter)

Requirement #4 is because the technique is not a single-shot technique but requires scanning of a probe pulse across the to-be-measured pulse wave form.

## Technique

Basically we use the to-be-measured light wave to "streak" (accelerate or decelerate) electrons that were previously emitted by the attosecond pulse. We measure the (streaked) kinetic energy of those electrons as we scan the short attosecond pulse over the longer (but still ~5fs short) to-be-measured pulse to reproduce its electric field waveform.

To make this work, the probing attosecond pulse must be shorter than the period of the to-be-measured pulse, otherwise we would not be able to resolve its waveform (it would "wash out").

The probing attosecond pulse must be precisely synchronized to the to-be-measured pulse. This is fulfilled by generating the attosecond pulse from a copy of the to-be-measured pulse (using a beam splitter).