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In classical wave optics, if human sensor is able to detect amplitude changes at a frequency of visible light (i.e. at $10^{14}$ Hz order), for wave $$u(x,z) = A(x) e^{j (k_1 x + k_2 z + \omega t)}$$ can we observe:

  • the sinusoid amplitude oscillation?
  • the time-varying amplitude changes ($\cos \omega t$)?
  • the space-varying amplitude changes ($\cos k_1 x$ and $\cos k_2 z$)?

If yes, in this sense, is the complex number representation still a valid and useful approach?


Some clarifications:

  1. By "human sensors", I mean any photodetectors that are able to detect light, some examples are human eyes, CCD etc. But please note the "If", I am referring to a ultra-fast sensor that we human do not have yet, like an optical version of electronic oscilloscopes.
  2. For human sensors we know, the reason why you only see the intensity $|A(x)|^2$ is because the electronic sensors are slow (round $10^9$ Hz): so you can only measure the average of the light energy (in free space propagation Poynting vector reduces to $|E|^2$); phase changes (the exponential terms) are average to be zero.
  3. My question is, if such a ultra-fast sensor do exists, can we see the amplitude changes as a result of the phase directly, by either space-sweeping over $x$ or $z$, or time-sweeping over $t$.
  4. One consequence of Point 3 is, if the phase changes are now visible, for complex number representations, we now need to take care of phase when do the measurements. It might introduce inconvenience, but I am not sure yet.
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    $\begingroup$ Streak camera ? $\endgroup$ – MJC Apr 11 '18 at 14:16
  • $\begingroup$ @MattCliffe Kind of similar ... But instead of measuring actual pulses, I wonder if it is able to detect the amplitude oscillation of a continuous sinusoid wave. I mean, just like an oscilloscope. $\endgroup$ – WDC Apr 11 '18 at 14:22
  • $\begingroup$ I believe you can push this down into the fs regime. Let me have a google. $\endgroup$ – MJC Apr 11 '18 at 14:25
  • $\begingroup$ just to be clear streak cameras do see the electric field oscillations not the envelopes $\endgroup$ – MJC Apr 13 '18 at 12:50
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This question is basically a duplicate of Have we directly observed the electric component to EM waves? and What is the physical significance of the imaginary part when plane waves are represented as $e^{i(kx-\omega t)}$? in about equal measure.

In short: we do have, since about 2004, the capability to directly trace in real time the amplitude of an oscillating electromagnetic field at optical frequencies; see the first link above for details, but basically what you end up getting is data (and then reconstructions) like these:

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Image source: Science 305, 1267 (2004)

The terms you really want to google for are attosecond streaking and petahertz optical oscilloscope, though they're by no means the only available techniques.

These techniques can, in principle, be extended to map the transverse and longitudinal spatial dependence of the beam in conjunction with its temporal structure, but that's a hell of a lot of technology you'd need to develop and you need a clear goal that will make it worthwhile.

As for directly measuring the amplitude of an oscillating field, i.e. measuring the phase on top of the intensity, without caring so much about a direct measurement of the temporal structure, that's been a standard component of microscopy for many decades now; a representative technique is phase-contrast microscopy but there's plenty of others.


With that said, the answer to your next question,

is the complex number representation still a valid and useful approach?

is twofold:

  • On one hand, your understanding of the complex-number representation is deeply flawed, as I explain in detail here. Basically, you never use $u(x,z) = A(x) e^{j (k_1 x + k_2 z + \omega t)}$ as a serious candidate for any physical quantity; instead, you use $u(x,z) = \mathrm{Re}\mathopen{}\left( A(x) e^{j (k_1 x + k_2 z + \omega t)}\right)$.

  • Once you do that, though, then the answer is mostly yes, it is very common to use representations of the form $$ u(\mathbf r,t) = \mathrm{Re}\mathopen{}\left( A(x,y,z-ct) e^{i(kz-\omega t)}\right) $$ to represent the light fields of interest, and you can indeed get a very good idea of what $A(x,y,z)$ is for a given pulse.

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  • $\begingroup$ Thanks for the very helpful links. One small question: in phase microscopy techniques people only measure the "ray deflections" as the phase. How can we know it is really the sinusoid phase of the light? I am having the difficulty seeing the identity between "ray deflections" and "phase", though I am told they are the same thing. $\endgroup$ – WDC Apr 18 '18 at 17:40
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    $\begingroup$ @WDC That's a very broad question simply because phase-sensitive microscopy has by now become a zoo of different techniques, and I'm not particularly well acquainted with any one in particular. It is certainly possible to measure space-dependent phases in very high resolution, but the details will depend on the technique. $\endgroup$ – Emilio Pisanty Apr 18 '18 at 17:43
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It's not obvious what you mean by "human sensor". If you mean human-created photodetectors and photosensors, this is an answer:

There are no sensors (yet) that can directly respond to optical frequencies. However, heterodyne detection is possible. This means that the phase, frequency, and amplitude of a light wave is detected by comparing it with a reference light wave.

Here's a good example: A laser can be frequency stabilized by "phase locking" it to an atomic or molecular resonance. That is then a reference laser. If light from the reference laser and light from another laser whose frequency is close to the reference laser's frequency are both used to illuminate a light sensor, the sensor will detect the beat frequency between the two lasers. The beat frequency is a direct measurement of the frequency difference between the two lasers, and the phase of the sensor's output corresponds to the instantaneous phase difference between the two lasers. The amplitude of the light from the second laser is slightly more difficult to measure, but we know it is just the square root of the intensity; and it's easy to measure the intensity of the laser.

A complex representation is never necessary in electrodynamics, it is only a convenient and compact way to do the necessary math.

If by "human sensor" you mean the human eye, then the answer is "no, it's not possible for the human eye "to detect amplitude changes at a frequency of visible light". Heterodyne detection by eye would be possible in principle, but because the human eye and neural system can detect light flicker at no more than a few tens of Hz, the reference and other light sources would need to have the same frequency within a few tens of Hz.

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  • $\begingroup$ Do you notice your LED lights flickering at 120 Hz (100 Hz in Europe)? No. But your iPhone on slo-mo can see it, sort of. $\endgroup$ – JEB Apr 11 '18 at 3:00
  • $\begingroup$ And you can observe LED flicker if you transform the temporal flicker into spatial "dots" by, e.g., sweeping your eyes from side to side. $\endgroup$ – S. McGrew Apr 11 '18 at 3:07
  • $\begingroup$ Thanks for the intro for frequency comb, or the more general correlation interferogram methods. But that is not what really I mean. See the edits, thanks! $\endgroup$ – WDC Apr 11 '18 at 4:29
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    $\begingroup$ Complex numbers are useful for simplifying calculations that involve waves, but they are definitely not necessary. They are a very nice mathematical trick. $\endgroup$ – S. McGrew Apr 11 '18 at 5:03

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