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Is the broad spectrum light from a super continuum (white) laser system when filtered for a particular wave length still temporally coherent to a similar degree as the source laser?

i.e. Does the part of the resultant beam of a specific wavelength still have the same temporal characteristics of a laser or is it more like an incoherent source such as monochromatic light filtered from the sun.

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  • $\begingroup$ Are you talking about spectral coherence (single wavelength) or spatial coherence (all traveling in the same direction)? $\endgroup$ – Floris Aug 7 '16 at 0:59
  • $\begingroup$ Ah sorry, I'm talking spatial coherence (constant phase and uniform direction of the photons)... I assume spectral coherence would be as per normal outside of the nonlinear medium. $\endgroup$ – user263399 Aug 7 '16 at 1:32
  • $\begingroup$ Spatial coherence usually increases with distance from the source and does not require any sort of laser light property. It's a purely geometric effect. Spectral (or temporal) coherence is independent of the distance and requires filtering to improve. $\endgroup$ – CuriousOne Aug 7 '16 at 7:55
  • $\begingroup$ So for a beam going past a fixed point (say the location of an electron) - the property that all the wave crests occur a full wave period apart (and so would contribute constructively to oscillating the electron) is down to spectral/temporal coherence? $\endgroup$ – user263399 Aug 7 '16 at 10:53
  • $\begingroup$ If the wave has a single frequency it has spectral (temporal) coherence - meaning that the phase at one point along the beam is a good predictor of the phase at another point. This is usually measured by interfering the beam with a delayed version of itself (think Michelson interferometer with unequal arms). The shorter a pulse duration the greater the spectral broadening and the shorter the temporal coherence. $\endgroup$ – Floris Aug 7 '16 at 11:25
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Temporal coherence$^{\text{1}}$ in supercontinuum generation (SCG) is a very active area of research in ultrafast optics at the moment. This is in part because there are a lot of exciting applications which rely on broadband coherent light, but also because there are so many nonlinear processes involved in supercontinuum generation that a full understanding of how they each interact to influence the coherence will take time. This means your question is a very exciting one, but also that it is extremely broad and difficult to answer.

This answer is largely based on information which can be found in Agrawal, this review article on SCG using anomalous dispersion pumping, and this article on SCG pumping in the normal dispersion regime, and includes simulations covering just four relatively broad example cases: Two where the coherence of the supercontinuum pump source is conserved, and two where it is destroyed. As there are plenty of other sources which go into detail about the underlying physics of each nonlinear process involved, I've left this information out except where it's necessary$^{\text{2}}$. Unfortunately this means assuming a degree of familiarity with SCG, but I will link some references at least. There are also a few extra modes of coherence degradation which I have not described, but are also detailed here.

Firstly, in the SCG community coherence is generally defined in the spectral domain as follows:

$\left|g_{1,2}^{(1)}(\lambda, t_{1}-t_{2})\right| = \left|\frac{\langle A_{1}^{*}(\lambda, t_{1})A_{2}(\lambda, t_{2}) \rangle}{\sqrt{\langle |A_{1}(\lambda, t_{1})|^{2} \rangle\langle |A_{2}(\lambda, t_{2})|^{2} \rangle} } \right|$

$\langle \rangle$ denote an ensemble average over independently generated pairs of spectra $A(\lambda)$, emitted by the supercontinuum source at different times $t_{1}$ and $t_{2}$. As SCG is most often carried out using pretty stable modelocked lasers, I will assume that these pairs of spectra are nearly identical at the input of the optical fibre except for a noise contribution. $|g_{1,2}^{(1)}|$ is a measure of amplitude and$^{\text{3}}$ phase stability as a function of wavelength, defined over the interval [0; 1] (with 1 indicating perfect coherence and 0 complete decoherence), and quantifies the sensitivity of the spectral broadening to the noise on the input signal.

PCF is widely used for supercontinuum generation because the fibre structure can be adjusted to customize the dispersion curve, and because the mode confinement can be very small allowing for extremely high optical intensities which provide a lot of nonlinearity. I will also assume single-mode silica PCF here. Propagation in these fibres is usually modelled using the generalized nonlinear Schrödinger equation. In short, this is a partial differential equation including two terms, one for loss and dispersion (linear term), and the other for the intensity-dependent material response (nonlinear term). Both terms contribute to whether the coherence is preserved or destroyed. Although the mechanism responsible for this can be loosely interpreted as a competition between the following nonlinear processes (not exhaustive):

  1. Self-phase modulation
  2. Soliton fission
  3. The Raman effect
  4. Modulation instability

it is arguably better to categorize it by the dispersion regime, i.e., whether the input optical signal is launched in the anomalous side of the PCF dispersion slope, or the normal side. Anomalous dispersion is characterised by short wavelengths having a higher group velocity than longer wavelengths, and normal dispersion is characterised by long wavelengths having a higher group velocity than shorter wavelengths.

Some useful quantities which I will refer to:

  1. $L_{NL} = \frac{2n_{2}}{P_{0}\lambda(\text{MFD}/2)^{2}}$ is the nonlinear length, and is the propagation distance over which the accumulated nonlinear phase reaches $2\pi$. $n_{2}=2.7\times 10^{-20}$ m$^{2}$/W is the nonlinear refractive index for silica, MFD is the mode field diameter, $P_{0}$ is the peak power of the pulse which has a wavelength $\lambda$.
  2. $L_{D} = \frac{T_{0}^{2}}{|\beta_{2}|}$ is the dispersion length, and is the propagation distance over which the accumulated dispersive phase reaches $2\pi$. $T_{0}$ is the FWHM duration of the pulse, and $\beta_{2}=\frac{-\lambda^{2}D(\lambda)}{2\pi c}$ is the group velocity dispersion in units of fs$^{2}$/m ($D(\lambda)$ is equivalent and has units of ps/(nmkm)).

Dispersion regimes and (very brief) summary of the dynamics.

Here's a couple of typical PCF dispersion curves designed for pumping around 1 $\mu$m (e.g., using an Yb-doped fibre laser):

enter image description hereenter image description here

The one on the left has both normal and anomalous dispersion regions, with a zero-dispersion wavelength at 1 $\mu$m inbetween. For this dispersion curve, supercontinuum is usually generated by pumping on the anomalous dispersion side, close to the zero-dispersion wavelength. Both coherent and incoherent spectral broadening are then driven by soliton propagation and dispersive wave generation in the fibre (ref). Whether the output spectral coherence is high or low depends on how the initial pulse divides into fundamental solitons, and this will be described in the first section below. In reference to the competition between nonlinear processes outlined above, the input pulse bandwidth and duration determine whether it divides by soliton fission (coherent), or due to modulation instability (MI, incoherent).

The dispersion curve on the right has only normal dispersion, and fibres with this kind of dispersion profile are referred to as all-normal dispersion (ANDi) fibre. Supercontinuum is usually generated by pumping at the dispersion minimum to maximize the peak intensity over the length of the fibre. In reference to the competition between nonlinear processes, coherence is either preserved or destroyed depending on whether optical wave breaking occurs (coherent), or whether there is significant Raman gain over the fibre length (incoherent). This process is described in the second section below.

Example 1: Incoherent and coherent broadening when pumping in the anomalous dispersion regime.

The image below shows incoherent supercontinuum generation over a 6 cm fibre length with a dispersion curve similar to that shown in the left-hand dispersion figure above. The input pulse was transform limited with a duration of 1 ps, central wavelength of 1040 nm, and energy of 30 nJ. The top figure shows the spectral development as a function of fibre length (spectral power density units of dBm/nm), and the bottom figure shows the coherence as a function of fibre length for the same simulation.

enter image description here

Initially the pulse undergoes SPM, which broadens the spectrum coherently with a linear dependence on fibre length. Modulation instability (MI) quickly follows, characterized by two symmetric sidebands appearing either side of the SPM-broadened pump. MI is an amplification process, and goes like $\text{exp}(g_{\text{MI}}z)$ causing rapid amplification of the sidebands after ~1 cm. After around 2.5 cm, four-wave mixing causes the MI process to cascade leading to very rapid bandwidth expansion.

MI arises because high peak power pulse propagation in the anomalous dispersion regime is unstable under the right conditions, and a perturbation analysis of the nonlinear Schrödinger equation shows that the SCG output is highly sensitive to intensity modulations (e.g., from input noise), which is amplified according to the following gain curve:

$g=2\sqrt{-\beta_{2}^{2}\omega_{m}-2\gamma P\beta_{2}\omega_{m}^{2}}$ for $\omega_{m}<\frac{-2\gamma P}{\beta_{2}}$

$g=0$ for $\omega_{m}\ge \frac{-2\gamma P}{\beta_{2}}$

where $\omega_m$ is the difference between the modulation angular frequency and the pulse central angular frequency, $P$ is the peak power, and $\gamma=(2\pi n_{2})/(\lambda (\text{MFD}/2)^{2})$. This gain curve shapes the MI sidebands shown in the figure.

This amplification of out-of-band noise results in interference with the residual input signal to create noisy, ultrafast and high peak power modulations in the time domain. The cascaded MI increases the time-domain interference and noise, resulting in modulations which can evolve into fundamental solitons. These then shed energy to high-frequency dispersive waves as they propagate and also undergo soliton self-frequency shifting to longer wavelengths, giving extended spectral broadening beyond that from MI. This latter process isn't shown in the figure, but would occur if the fibre length was extended. As this process is seeded by noise, the supercontinuum output will be incoherent. The coherence figure shows this, as the MI sidebands have zero coherence. The coherence of the residual pump also degrades rapidly as MI randomly strips energy from the signal.

The figure below shows the output time and wavelength domain ensemble distributions. The bold line is the ensemble average, and the shaded areas show the individual simulations in the ensemble. Although each pulse had the same time and spectral domain shapes, the different input noise has had a drastic influence on the propagation dynamics, with no two output pulses being alike. This is the consequence of low coherence, and filtering this spectrum at any particular wavelength would not give a stable time domain output.

enter image description here

MI can be suppressed to give highly coherent spectral broadening by using shorter pump pulses, which have a correspondingly broader bandwidth. This suppression occurs partly because the broad, coherent input bandwidth seeds the MI gain spectrum, reducing the influence of noise, and also because the broader bandwidth is more susceptible to higher-order dispersion and inter-pulse Raman scattering, which perturb the input pulse and cause it to fission into fundamental solitons more quickly than MI can amplify noise. This happens over a length scale given by:

$L_{\text{fission}}=\sqrt{L_{D}L_{NL}}$

As these two perturbations are insensitive to noise, the resulting SCG is coherent. If $L_{\text{fission}}$ is smaller than the length required for significant MI buildup, the output spectral coherence will be high.

enter image description here

This is precisely what is happening in the figure above, where the input pulse was 80 fs long, had a 1040 nm central wavelength, and energy of 1 nJ. The broad input bandwidth of the short pulse undergoes a small amount of SPM before soliton fission occurs, causing explosive spectral broadening just millimetres along the fibre length. The pulse gradually fissions into its constituent fundamental soliton parts, shedding excess energy as a high-frequency dispersive wave on the other side of the zero dispersion wavelength as it does so. Inter-pulse Raman scattering causes these solitons to gradually shift to lower frequencies which gradually broadens the spectrum with propagation distance. This process is called soliton self-frequency shifting (SSFS in the figure). The lower figure shows that these processes preserve coherence, which is very high over the whole supercontinuum bandwidth. Filtering this spectrum over a smaller wavelength band will give a stable output in the time domain.

enter image description here

The time and wavelength domain supercontinuum shapes are shown in the figure above. Unlike the case where MI was dominant, the individuals in the ensemble are indistinguishable indicating that the broadening process was insensitive to input noise.

Example 2: Incoherent and coherent broadening when pumping in ANDi fibre.

As with anomalous dispersion pumping, longer pulses more readily lead to incoherent spectral broadening in the ANDi regime. The figure below shows how a 7 ps input pulse with 60 nJ energy and central wavelength of 1040 nm propagates in the ANDi fibre shown in the right-hand dispersion figure. Again, SPM acts first to coherently broaden the input spectrum linearly with propagation distance. However, as ANDi fibres generally support dissipative high-intensity solutions, MI doesn't play a role in the spectral broadening in this dispersion regime and the Raman effect becomes significant instead. This is evident in the spectral figure, where a broad Stokes peak develops at the low-frequency side of the input pulse spectrum. As was the case with MI, the Raman effect in the ANDi regime generally leads to the amplification of out-of-band noise, with a gain bandwidth set by the material Raman response (gain peak at 13.2 THz detuning below the peak frequency of the input spectrum for silica). The Stokes wave is quickly followed by an anti-Stokes wave at the high-frequency side of the input spectrum. After approximately 60 cm of propagation, four-wave mixing causes the Raman effect to cascade, forming higher-order (anti-)Stokes peaks.

enter image description here

As the (anti-)Stokes peaks are seeded by noise, they have zero coherence. They are amplified during propagation and use the input pulse as a pump source, so the input pulse is randomly depleted over the fibre length causing its coherence to degrade as well. This is evident in the coherence figure at approximately 90 cm, where the coherence is beginning to degrade at the low-frequency side of the SPM-broadened input spectrum.

enter image description here

Looking at the time and wavelength domain output also shows how sensitive the propagation dynamics are to the input noise. The output stability degrades at the leading edge of the pulse in the time domain because the dominant red-shifted Stokes peaks in the output spectrum are brought to the front of the pulse by the normal dispersion. The Stokes and anti-Stokes portions of the spectrum show large-scale instability as well at the long and short wavelength sides of the spectrum. Filtering a smaller wavelength band would not produce a stable output.

It is possible to define a length over which Raman gain becomes significant, and this requires a gain coefficient with a dependence on the ratio of chromatic dispersion to nonlinearity, and describes the coupling between Raman and four wave mixing:

$g=2\gamma \text{Re}\left[ \sqrt{K(2q-K)} \right]$

where $K = -(\beta_{2}\Omega_{R}^{2})/(2\gamma P)$ is the linear phase mismatch between the pump and (anti-)Stokes wave with respect to the nonlinear contribution to the mismatch. $q$ is the (anti-)Stokes order, $\Omega_{R}$ is the frequency detuning which maximizes the Raman gain (13.2 THz for silica). For low phase mismatches ($|K|<1$), Raman is suppressed. As $|K|$ approaches infinity, Raman can contribute more. This is interesting in its own right, because it shows that the four wave mixing which cascades the Raman effect in the example above can also be very effective in suppressing it entirely if the peak power is very high. Using this, the Raman length is given by:

$L_{R}=1/(gP)$.

By pumping with shorter pulses, coherence degradation in the ANDi regime can largely be prevented. As the pulse duration is decreased, dispersion and SPM have a greater influence and can lead to optical wave breaking (OWB), shown in the figure below.

enter image description here

The output spectrum is extremely flat, and has very high coherence over the entire bandwidth. Filtering the spectrum for a small wavelength band will produce a stable time domain output. Looking at each simulation in the ensemble shows not only that the broadening is completely insensitive to input noise, but also that the process preserves the pulse time domain distribution, which is single-peaked.

enter image description here

Wave breaking occurs over a length scale approximated by:

$L_{\text{WB}}\approx 1.1T\sqrt{\frac{1}{\gamma P\beta_{2}}}=1.1\sqrt{L_{D}L_{NL}}$

If the pulse duration at the input is gradually increased, there will be a point where wave breaking requires a longer length of fibre than is required for significant Raman gain. As the competition between these two processes is central to the coherence conservation in this example, it is important to find which combination of input pulse duration and fibre length will give incoherent/coherent output. A coherence length will do this, and is given by taking the ratio of $L_{R}$ to $L_{WB}$:

$L_{c}\propto \frac{L_{R}}{L_{WB}}\propto \frac{1}{f_{R}\Omega_{R}T}$, where $f_{R}=0.18$ for fused silica.

So, if the pulse duration is reduced such that $L_{WB}$ is less than $L_{R}$, the output spectral coherence will be high. Note also that the parameter $K$ depends inversely on the peak power. High peak powers are commonly associated with short pulse durations, so when the input pulse duration is short Raman will be suppressed at the start of the propagation before dispersion becomes significant. This restriction doesn't apply to wave breaking, so this becomes the favourable route for spectral broadening in ANDi fiber at high peak powers and short pulse durations. This holds for peak powers in excess of 100 kW, and even if the peak power is only reduced by a factor of ~2 by dispersion after wave breaking takes place the peak power can still be high enough to suppress Raman very efficiently, giving excellent output spectral coherence.


1 Shortened to "coherence" from now on.

2 An explanation of supercontinuum generation is out of the scope of the question, which is about coherence.

3 I have seen a few instances of spectral amplitude being plugged into this equation with no phase information, and this is inappropriate.

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