# How to compress a range of wavelengths into a single wavelength?

Given the light spectrum from 400 to 500 nano-meters, How can I convert all wavelengths in the range to one wavelength of say 450nm?

• Question is: why? – lalala May 20 '17 at 19:58
• Use it to charge up a solar cell, then use that to power a laser. – knzhou May 20 '17 at 20:23
• Split it by a prism and reflect of a mirrored cone rotating at about 20% of the speed of light. The redshift (or blueshift) can compensate for the wavelength difference. Alternatively split the light by a prism and run it near a neutron star for the gravitational redshift to work the same way. – safesphere Oct 3 '17 at 5:09

If you want to convert your wavelengths to a single, mathematical, real-number, point-like wavelength at exactly $\lambda=450\:\mathrm{nm}$, it isn't possible, unless you're willing to wait for a literal eternity (details).

If you just want to compress light in some bandwidth $\Delta\lambda=50\: \mathrm{nm}$ down to a very thin sliver of spectrum at, say, $\delta\lambda=1\:\mathrm{pm}$, centered at $\lambda=450\:\mathrm{nm}$, then you're still looking at some very major difficulties, and frankly your best bet is to simply filter the light you don't want, or find a better light source.

The reason for this is that linear processes are completely incapable of altering the frequency of the radiation they operate on, so as far as linear optics is concerned, $400\:\mathrm{nm}$ light is going to stay at $400\:\mathrm{nm}$ no matter what.

That immediately tells you that do do your compression scheme you're going to need a nonlinear process, and those can be fiddly beasts. In principle, it is possible to use third-order nonlinearities to, say, take a pair of photons* at $440\:\mathrm{nm}$ and $460\:\mathrm{nm}$ and convert them into two degenerate photons at $450\:\mathrm{nm}$, and if you did this carefully enough, and your initial spectrum was coherent enough (and completely symmetric about your target wavelength), then you could hope for this to be doable. As an example, the reverse process, called supercontinuum generation, is perfectly possible (link, link), and it is an integral part of many technological breakthroughs of the past 20 years.

However, to arrange those processes in a way that narrows the bandwidth instead of widening it would require some very special preconditions, and it would require a lot of care in how you did this. More importantly, I don't see this working over a range as broad as $100\:\mathrm{nm}$ without some very special conditions in place.

Bottom line: get a new light source.

* I'm oversimplifying the math - it's the sum of frequencies that gets conserved, not the sum of wavelengths.

As Emilio Pisanty said it's not possible to reduce the input spectral width to a single wavelength, and I agree that the best path to reducing the bandwidth is likely to be either spectral filtering or abandoning your current light source for a more appropriate one. However, if your broadband signal has a relatively high-intensity time domain distribution (order of kW) then a lot of spectral compression can be achieved with a neat trick that doesn't require too much practical effort or appropriately balanced phase matching constraints and doesn't result in the heavy losses associated with spectral filtering.

Spectral compression can be achieved nonlinearly by inducing self-phase modulation on a negatively chirped pulse. Self-phase modulation changes the instantaneous frequency of an optical signal as a function of its intensity gradient and propagation distance, which for a Gaussian-shaped pulse looks like this:

$\omega(t,L)=\omega_{0}+\frac{4\pi Ln_{2}I_{0}}{\lambda_{0}\tau^{2}}t\times\text{exp}\left( \frac{-t^{2}}{\tau^{2}} \right)$

$\omega_{0}$ is the central angular frequency, $L$ is the propagation distance, $n_{2}$ the nonlinear refractive index of the propagation medium, $I_{0}$ the peak intensity, $\lambda_{0}$ the central wavelength, and $\tau$ the pulse duration. If we take the difference between the maximum and minimum values of $\omega(t,L)$ to be the signal bandwidth, we see that self-phase modulation causes the bandwidth to change with $L$. Using the example of a Gaussian pulse, the minimum and maximum instantaneous frequencies are given by:

$\omega_{\text{min, max}}(t,L)=\omega\left( t=\pm \tau/\sqrt{2}, L\right)$

and the bandwidth by:

$\Delta \omega(t,L) = \omega\left( \tau/\sqrt{2}, L \right) - \omega\left(-\tau/\sqrt{2}, L \right)$

We also see that self-phase modulation gives the instantaneous frequency a positive and largely linear variation, or chirp, about $t=0$ with a gradient given by $(4\pi Ln_{2}I_{0})/(\lambda_{0}\tau^{2})$. If we introduce a negative linear chirp at the input, we can force $\Delta\omega(t,L)$ to pass through a minimum at some propagation distance, compressing the optical spectrum from a broadband input to a narrowband output. If the propagation medium is truncated at this length, the broadband input signal will be spectrally compressed to a narrowband output.

Example:

Input: Gaussian pulse with a duration of 1 ps and a chirp parameter of -20 (meaning that the input optical bandwidth of the 1 ps pulse could support 50 fs), and a central wavelength of 1040 nm (288.3 THz). The pulse energy was 1 nJ giving an input peak power of ~0.94 kW. The nonlinear propagation medium was standard single-mode silica fibre. The figure above shows the spectral evolution as a function of fibre length (linear scale, colourmap normalized to peak power spectral density). The compression is smooth and roughly linear with propagation distance away from the minimum. Overlaying the input and output pulse profiles shows the extent of the nonlinear spectral compression quite nicely: The input and output time domain profiles are shown on the left, and the spectra on the right in wavelength. The FWHM of the output spectrum is smaller than the input by a factor of 7.2. By optimizing the fibre length, intensity, and chirp, higher compression factors will be possible. Other than the small propagation loss from the fibre (~0.001 dB here) the input average power is transferred directly to the output, meaning a much higher spectral power density for the compressed spectrum than for the input. This isn't possible with spectral filtering.

I've chosen different wavelengths to the question for my own convenience, mainly, but this choice is kind of arbitrary. Given an appropriate propagation medium this will work in the 400 -- 500 nm range too.

Regarding the "why" question which has appeared in the comments: There are a lot of reasons why you might want to induce spectral compression. The exact method described above is used in many so-called 'breathing-pulse' and soliton modelocked oscillators as a form of bandwidth control without having to introduce excessive cavity loss with a spectral filter. Similarly, if you want to seed a narrow-band amplifier adequately it can be a good idea to squeeze as much of the spectrum into the gain bandwidth as possible, and this method works well for doing that.

Some lasers are tunable. E.g., diode lasers can be tuned via the current and temperature. If you can find a diode that can be tuned around the wavelength you're interested in, then the linewidth can be dramatically reduced by locking the laser to a stable cavity. A commonly used technique (which was crucial in the experimental achievement of BEC) is Pound-Drever-Hall:

https://en.wikipedia.org/wiki/Pound%E2%80%93Drever%E2%80%93Hall_technique