The Fourier Transform does not lie - the spectrum really would broaden if you took a CW source and chopped it with ultra-fast speed. Perhaps thinking about it in terms of just wavelengths is not very helpful.

As you narrow the time aperture, you run into the Uncertainty Principle. You are observing some electronic transition. The shorter the time (and hence the uncertainty in timing of the transition), the greater the energy uncertainty (spectral width).

Interestingly, the opposite is also true, and has been experimentally observed. If you take a train of ultra-fast pulses (such as from a mode-locked laser), and resolve it with a diffraction grating, you can observe multiple, CW monochromatic components. Of course, Fourier Transform of a comb of time pulses is a comb of frequencies.

The Fourier Transform does not lie - the spectrum really would broaden if you took a CW source and chopped it with ultra-fast speed. Perhaps thinking about it in terms of just wavelengths is not very helpful.

As you narrow the time aperture, you run into the Uncertainty Principle. You are observing some electronic transition. The shorter the time (and hence the uncertainty in timing of the transition), the greater the energy uncertainty (spectral width).

Interestingly, the opposite is also true, and has been experimentally observed. If you take a train of ultra-fast pulses (such as from a mode-locked laser), and resolve it with a diffraction grating, you can observe multiple, CW monochromatic components. Of course, Fourier Transform of a comb of time pulses is a comb of frequencies.

Such ideas have actually been proposed before as ways of generating short pulses, where the coupling between two energy levels near-resonant with a continuous source are rapidly-varied using population depletion, splitting/shifting of the energy levels with other laser fields, etc. Cross-polarized wave generation (XPW) is also an example of this: the nonlinearity is only induced for a fraction of the duration of the original pulse where the amplitude is high enough, and the XPW pulse is generated with a shorter temporal duration and a broader spectrum than the original. In these cases, one can intuitively blame the new wavelengths after some careful thinking on shifting energy levels or four-wave mixing or something similar. However, I admit that these answers -- or the purely mathematical argument based on Fourier theory, which we know must be right! -- may not be as intellectually satisfying when considering a mechanical shutter generating new wavelengths.

Instead, imagine in the case described above that the mechanical shutter essentially works by providing a near-instantaneous change to the index of refraction, waiting for 30 fs, and then performing another near-instantaneous chancg to the index of refraction. (If the chopper is perhaps made of metal, the refractive index of this material may have a real part less than 1 and a large imaginary part, for example. If the chopper perhaps works based on an instantaneous electro-optic effect, different values of an electro-optic tensor could be in play and alter the refractive index in some way. My point here is simply that it doesn't really matter what the chopper is made of; all choppers work by changing something about the space through which the laser is propagating.) Under such a circumstance, I don't think it's too large of an intellectual leap to imagine that the energy density/flow of this continuous, near-single-frequency electromagnetic field might manifest itself in a new way/with new wavelengths when the medium through which it propagates experiences a near-instantaneous change to its susceptibility/refractive index/etc. Said another way, you could imagine that this is how the electromagnetic field must behave in order to still obey Maxwell's equations/Poynting's theorem/etc. for the situation of extremely time-dependent permittivities/permeabilities/charge¤t densities. (Perhaps if you stare at the equations in the link below and imagine how \epsilon, \mu, \rho, and \vec{J} all depend so strongly on time, you can convince yourself that the solution for \vec{E} and \vec{B} must be very interesting.)

Perhaps someone else has also found their own way to resolve this paradox in their own mind, but this is what works for me, and hopefully it is helpful to someone here too.

http://optics.byu.edu/BYUOpticsBook_2015.pdf#page=62

If you are using a mechanical chopper at the output of HE-Ne laser, how are you relating the optical wavelength to the output of the chopper which has a modulated intensity at the rate of 30 fs? Actually, by chopping the laser light, you are modulating the intensity of the light, whatever may be the wavelength. 

I wonder if I understood your experiment!

Yes, you will measure a broad spectrum. It will be created by modulating your CW sinusoidal wave.

Hi to all of you! Thank you for the answers. It made me see the problem from a different point of view. I will check the literature to check it with maths. Still it is a bit hard to imagine the creation of new wavelengths with just a mechanical chopper!

Hello Parviz, thanks for your comment. I think the answer to your question is yes, you have a 1 ps pulse.

@Parviz Elahi

Actually, this is completely true. The pulse width _will_ increase to 1ps.

Now, about the question.

A lot of good answers have been posted already, so I will just add some examples of real experiments that are about as similar to the situation described in the question as can be :)

While a mechanical chopper with a 30fs temporal window might be difficult to come by, optical "shutters" based on nonlinear optical effects certainly can be tens-of-fs long. And the result (I have done quite a few experiments myself) is always as described by the many answers here: if you gate a long pulse with a much shorter pulse, the spectrum of the gated pulse always gets broader. 

I have included some scenarios of CW lasers gated by fast optical shutters in the links.

https://www.osapublishing.org/oe/abstract.cfm?uri=oe-21-1-730

https://www.osapublishing.org/oe/abstract.cfm?uri=oe-24-19-22296&origin=search

Hi Jorge,

How can one represent the chopped cw laser field after your modulator?

You can consider it as a product of the initial cw field and a window function (e.g. a square window, or a Gaussian window, etc.) of the modulator , right?

According to the convolution theorem, the Fourier transform of the product (chopped field) is the convolution of the Fourier transform of the initial cw field (very narrow distriution) with the Fourier transform of the chopper's window.

If this window is of a femtosecond duration T, the window function has the correspondingly wide spectrum (~1/T). And so will have of couse the convolution - your chopped cw field.

To make it short, when you modulate a very narrow spectrum cw field with a very fast modulator, the resulting modulated field will have the spectrum of the modulator window function.

Hi...

"If a He-Ne continuous laser beam passes through this chopper, I would have a 30 fs signal at the output of the chopper (at a certain repetition rate). If I do the Fourier transform of this temporal signal (pulse) then I would have a broad spectrum."

True !!!

"Can I get a single wavelength 30 femtoseconds pulse by chopping a continuous laser?"

YES YOU CAN !!!  provided the hardware (i.e., chopper or any other imaginary instrument - nothing works this fast till now) is available

"If I measure with a spectrometer would I measure a single wavelength or a broad spectrum?"

You will measure a single wavelength (I shall explain this after addressing your next point)

"Fourier analysis points that I need a broad spectrum. How are these new wavelengths created after passing through the chopper ?"

Here is the catch !!!! First, you have to think about what is being Fourier transformed.

In the case of a chopped He-Ne laser, you are talking about the FT of intensity modulation (of course, by the chopper). [Assuming that you are talking about the FT of INTENSITY detected by a photo diode or any other detector - But you are detecting the intensity - not the wavelength]

This would, of course, give you a broad spectrum and we have to think about what this spectrum tells you !

If I chop a He-Ne laser, I would assume a square pulse pattern, because the intensity given out by the laser is the same all the time.  You are just blocking it and opening it at various intervals.  So, effectively, you have a ON-OFF situation.

You are taking the FT of this intensity modulation.  Now we have to think about what the FT tells you !!

If you modulate the intensity like a square wave (or any other way), Its FT will tell you in what all sinusoidal frequencies you have actually modulated the intensity.  In other words, instead of a chopper, you can use several neutral density filters with their OD graded in a sinusoidal manner.  Several such sinusoidal intensity modulations will eventually give you a square intensity modulated pulses. But the wavelength will remain the same

The scene is totally different in the case of the spectrum of an actual 30 fs pulse. The Fourier transform done by the spectrometer is a totally different kind of FT.  Here you are looking at the wavelengths, or in other words, the monochromatic components that made your actual femtosecond pulse. 

Basically, it is the confusion regarding what is being Fourier transformed is  the reason behind this doubt.

Best

Dear Vinu,

What I understood from your experiment is that you are chopping the intensity of the laser light. And whatever you get is the frequency of chopping process. How are you relating it to the wavelength of light?  The wavelength of the laser light is the intrinsic parameter of the emitted photons, which is special to every laser. The wavelength of your chopped light is the wavelength of the laser light before chopping. That means chopping does not change the wavelength of light.

Good Luck! 

Dear Peter Kapusta,

Is it possible to find a mechanical chopper with a chopping frequency equalling to an optical wave frequency?  I wonder, may be possible!

"A stream of photons with some fixed wavelength" is not correct. An emitted photon in an ideal case has one wavelength, whereas in practical laser light such a condition is not possible with the present technology. Suppose in one cycle of the mechanical chopper, how many wavelength components of light would pass through it? One or many? 

Please give a standard reference on a mechanical chopper frequency. 

I don't mind withdraw my answer. It would be my pleasure to learn something new.

@ Faramarz E. Seraji

The frequency of the chopping process has no relation to the wavelength of light.  That is why I said the wavelength of the He-Ne Laser will remain the same without any change !!  You would still get the same He-Ne wavelength when measured by a spectrometer even when its intensity is modulated @ 30 fs intervals.  Simple intensity modulations cannot alter the wavelength of light.

Dear Peter Kapusta.....

with a mechanical chopper I am modulating the intensity of the light.  I am not cutting down its wavelength !!!

The situation when using a chopper is similar to the case when you switch ON and swith OFF a source emitting 632.8 nm light at 30 fs intervals..... The time corresponding to the wavelength of the light can still be 70 times longer.... But I still can switch ON and OFF at 30 fs intervals... What stops me from doing this ?

Sure, if the chopper is trying to cut down the wavelength (or in other words time corresponding to 632.8 to 30 fs..... ) what you said is a valid point.  But a chopper cannot limit the wavelength of the light. It is just like a switch.

@ Peter,

Don't you know talking decently?

For your info, we are talking of the photons which are emitted from the same laser He-Ne.  Which photon you are referring to that travels 9 nm distance.

Dear Vinu,

I prefer to discuss the matter in private, If  you like.

I have no issues... But How to discuss in private ?

Dear Vincent,

Whatever you said I agree with it. 

A perfectly monochromatic light would broaden to 10Hz...

In the above statement, my question is whether the wavelength of the laser source also changes? As you know, when a laser source is amplitude-modulated, the resulting wave will contain the spectrum of the source and the modulating wave, where the modulating wave appears as an envelope to the carrier wave. Needless to say, if a direct modulation is used, of course, there will be some chirps of the source central wavelength and light intensity. 

The broad spectrum at the output is due to wavelength components of the carrier wave and the envelope signal. As you know, the laser light itself also has a linewidth containing different wavelength components. Therefore, no wonder, the broadened wave consists of components of the source and the modulating wave. In the case of an ideal monochromatic laser source, still, the broadening will exist due to the modulating wave. 

Now, let us look at the question:

The question asks: Imagine that we have a chopper able to open and close to a 30 fs rate (for example) . If a He-Ne continuous laser beam passes through this chopper,...

My answer pointed to the practical aspect of the issue that which type of chopper can be used to have a rate of 30 fs?

Let us assume such a chopper is available. If by chopping I am going to change the laser wavelength, then all optical communication will be in trouble, specifically in the case of WDM optical systems.

Dear Peter,

Whatever you summarized here are what I was trying to say. Not convinced, read the original question raised by Jorge (last para).

I suggest you need not withdraw your comments.