Nonlinear Optical Pulse Propagation
This is a continuation from the previous tutorial - silica nanofibers and subwavelength-diameter fibers.
All the propagation phenomena described in the linear pulse propagation tutorial are linear propagation effects, produced by the linear response of the propagating systems.
In this tutorial we will give a brief survey of a few of the most important nonlinear propagation phenomena that occur with optical pulses. These effects include in particular: gain saturation in pulsed amplifiers (which is a relatively weak form of nonlinearity); optical pulse propagation through nonlinear dispersive systems in general; and the especially interesting topic of nonlinear pulse propagation in optical fibers, including the fascinating topic of soliton propagation in optical fibers.
1. Pulse Amplification with Homogeneous Gain Saturation
As we mentioned earlier, laser amplifiers are much more commonly used for amplifying optical pulses than for amplifying cw optical signals. Common examples of pulsed laser amplification include flash-pumped Nd:YAG and Nd:glass amplifiers at 1.06 μm; electron-beam-pumped TEA CO2 laser amplifiers at 10.6 μm; excimer lasers in the visible; and pulsed dye laser amplifiers, which are themselves often pumped by another pulsed laser, and which can amplify across broad bandwidths in the visible and near infrared.
Short pulses passing through laser amplifiers will be broadened and distorted by the effects we discussed in the linear pulse propagation tutorial. These effects are, however, entirely linear effects, and generally require quite short pulses and sizable dispersions to be significant.
Let us now consider an entirely separate form of weakly nonlinear pulse distortion that can arise with much longer pulses, as a result of time-varying gain saturation effects when a higher energy pulse is amplified in a homogeneously saturable laser amplifier.
Pulse Energy Saturation in Amplifiers and Absorbers
In order to obtain efficient energy extraction from a laser amplifier, an amplified pulse must be intense enough to cause significant saturation of the population inversion during its passage through the amplifier. But this means that the amplifier gain must necessarily be reduced from a large initial value to a small residual value during the passage of the pulse; hence this time-dependent saturation during the passage of the pulse must also lead to time-varying gain reduction and pulseshape distortion.
In the same fashion, when a strong enough pulse is sent through a saturable absorber medium, the signal energy in the pulse may partially or completely saturate the atomic absorption and thus increase the energy transmission, leading to an analogous though oppositely directed pulse distortion.
Such pulse propagation through saturable absorbers is widely used to shorten mode-locked pulses in passively mode-locked lasers. Again, the pulse itself must change the transmission of the saturable absorber during the passage of the pulse. The fundamental physics is the same as it is for the saturable amplifier, except for a change of sign in going from saturable amplification to saturable absorption.
In order to explain both of these effects, this section presents an analysis of the population saturation and the pulseshape distortion that results when a sufficiently intense pulse passes through a homogeneously saturable, single-pass laser amplifier, or saturable absorber.
The physical approximations made in this section are thus significantly different from the linear pulse propagation analysis of the linear pulse propagation tutorial. The pulsewidths we are concerned with here are generally long enough, and the propagation lengths short enough, that pulse compression or expansion effects due to finite amplification bandwidths or to group velocity or gain dispersion effects are generally of minor importance. Our emphasis is thus entirely on the time-varying saturation effects in the atomic material.
Homogeneous Saturation Approximations
Two physical approximations help to simplify this analysis.
First, although pulse amplification often involves short pulses with fast time-variation and high intensities, usually the rate-equation approximations are still valid, and a purely rate-equation analysis can be employed.
Second, in most situations of interest for laser pulse amplification, the amplified pulse durations are short enough that we can neglect both any pumping effects and any upper-level relaxation during the transit time of the amplified pulse. Hence in this section we will analyze a short pulse propagating through a prepumped and inverted laser medium, without including any pumping or relaxation effects during the pulse interval.
Analysis of Homogeneous Pulse Amplification
We consider therefore a short pulse with signal intensity \(\hat{I}(\hat{z},\hat{t})\) traveling in the \(+\hat{z}\) direction through a laser medium with inverted population difference \(\Delta\hat{N}(\hat{z},\hat{t})\), where \(\hat{z}\) and \(\hat{t}\) are the usual laboratory coordinates. (The reason for the "hats" on all these quantities will become apparent in a moment.)
We neglect any transverse intensity variations to simplify the analysis. The basic differential equations for this situation can then be developed as follows. Let us denote the electromagnetic energy density in the optical signal pulse, measured in \(\text{J/m}^3\), by \(\hat{\rho}_\text{em}(\hat{z},\hat{t})\). The instantaneous intensity \(\hat{I}(\hat{z},\hat{t})\) in \(\text{W/m}^2\) being carried by the pulse through any plane \(\hat{z}\) at time \(\hat{t}\) is then given by \(\hat{I}(\hat{z},\hat{t})=\hat{\rho}_\text{em}(\hat{z},\hat{t})\times{v_g}\), where \(v_g\) is the group velocity in the laser medium. This velocity is normally very close to the phase velocity \(c\) in most laser media; so for simplicity we will write \(v_g=c\) in the following analysis.
Consider then a short segment of length \(\Delta\hat{z}\) in the laser medium, as shown in Figure 10.1. The rate of change of stored signal energy in the length \(\Delta\hat{z}\) is given by the energy flux into one end minus the energy flux out the other end of the segment, plus the net rate of stimulated emission within the segment, or
\[\tag{1}\frac{\partial}{\partial\hat{t}}[\hat{\rho}_\text{em}(\hat{z},\hat{t})\Delta\hat{z}]=\hat{I}(\hat{z},\hat{t})-\hat{I}(\hat{z}+\Delta\hat{z},\hat{t})+\sigma\Delta\hat{N}(\hat{z},\hat{t})\hat{I}(\hat{z},\hat{t})\Delta\hat{z}\]
where \(\sigma\) is the stimulated-transition cross section of the laser medium. Using \(\hat{I}(\hat{z},\hat{t})=c\hat{\rho}_\text{em}(\hat{z},\hat{t})\) and combining this with the rate equation for the inverted population inside the same segment then gives the two basic equations of this pulse saturation analysis, namely,
\[\tag{2}\frac{\partial\hat{I}(\hat{z},\hat{t})}{\partial\hat{t}}+c\frac{\partial\hat{I}(\hat{z},\hat{t})}{\partial\hat{z}}=\sigma{c}\Delta\hat{N}(\hat{z},\hat{t})\hat{I}(\hat{z},\hat{t})\]
and
\[\tag{3}\frac{\partial}{\partial\hat{t}}\Delta\hat{N}(\hat{z},\hat{t})=-\left(\frac{2^*\sigma}{\hbar\omega}\right)\Delta\hat{N}(\hat{z},\hat{t})\hat{I}(\hat{z},\hat{t})\]
Note that no pumping or relaxation terms are included in the atomic rate equation. We use the convention mentioned earlier that the "saturation factor" \(2^*\equiv1\) if the lower laser level empties out rapidly compared to the pulse duration; but \(2^*\equiv2\) if the lower-level population accumulates or "bottlenecks" during the pulse.
Transformation to Moving Coordinates
These two basic equations can then be solved with the aid of a few minor tricks, as follows. We first make a change of variables to a coordinate system that moves with the forward-traveling pulse, as defined by the transformation
\[\tag{4}z\equiv\hat{z}\qquad\text{and}\qquad{t}\equiv\hat{t}-\hat{z}/c\]
That is, whereas \(\hat{z}\) and \(\hat{t}\) are ordinary laboratory coordinates, for the remainder of this section \(z\) and \(t\) refer to coordinates in the moving pulse frame. Note that the delayed time coordinate \(t\) is essentially centered on the pulse's arrival time at each plane \(z\). For example, if the pulse starts out at an input plane \(\hat{z}=0\) centered at time \(\hat{t}=0\), and arrives at some plane \(\hat{z}\) centered about time \(\hat{t}=\hat{z}/c\), then the pulse written in the delayed time coordinate \(t\) is centered on \(t=0\) at every plane along the amplifier.
We also rewrite the pulse intensity and the population difference in the new coordinate system in the modified forms
\[\tag{5}I(z,t)\equiv\hat{I}(\hat{z},\hat{t})\qquad\text{and}\qquad{N}(z,t)\equiv\Delta\hat{N}(\hat{z},\hat{t})\]
where we use \(N(z,t)\) instead of \(\Delta{N(z,t)}\) from here on merely to simplify the formulas. The basic equations for the pulse intensity and the population inversion are then transformed into the significantly simpler forms
\[\tag{6}\frac{\partial{I}(z,t)}{\partial{z}}=\sigma{N}(z,t)I(z,t)\]
and
\[\tag{7}\frac{\partial{N}(z,t)}{\partial{t}}=-\left(\frac{2^*\sigma}{\hbar\omega}\right)N(z,t)I(z,t)\]
where these equations are now expressed in the transformed or moving coordinate system.
Solution of the Pulse Equations
The first of these equations can then be rearranged and integrated over the length of the amplifier in the form
\[\tag{8}\int_{I=I_\text{in}(t)}^{I=I_\text{out}(t)}\frac{dI}{I}=\sigma\int_{z=0}^{z=L}N(z,t)dz\]
where \(I_\text{in}(t)\) is the input pulse intensity at the input plane to the amplifier, and \(I_\text{out}(t)\) is correspondingly the signal intensity at the output plane, both measured in the delayed time coordinate \(t\).
It will then be convenient to define the integral on the right-hand side of this equation as a kind of "total number of atoms" \(N_\text{tot}(t)\) in the amplifier, in the form
\[\tag{9}N_\text{tot}(t)\equiv\int_{z=0}^{z=L}N(z,t)dz\]
The first of the basic equations can then be expressed in the simple form
\[\tag{10}I_\text{out}(t)=I_\text{in}(t)e^{\sigma{N_\text{tot}(t)}}=G(t)I_\text{in}(t)\]
where \(G(t)\equiv\exp[\sigma{N}_\text{tot}(t)]\) is the time-varying or partially saturated gain at any instant within the pulse.
The second of the basic pulse equations can also be integrated over the amplifier length and then rewritten, using the first equation, in the form
\[\tag{11}\frac{\partial}{\partial{t}}\int_{z=0}^{z=L}N(z,t)dz\equiv\frac{dN_\text{tot}(t)}{dt}=-\left(\frac{2^*}{\hbar\omega}\right)\int_{z=0}^{z=L}\frac{\partial{I}(z,t)}{\partial{z}}dz\]
which simplifies to
\[\tag{12}\frac{dN_\text{tot}(t)}{dt}=-\frac{2^*}{\hbar\omega}[I_\text{out}(t)-I_\text{in}(t)]\]
The two basic equations thus reduce to two coupled equations in the delayed time coordinate \(t\) only. These will be the fundamental working equations from this point on.
Physical Interpretation
With this change of coordinates the equations take on the same form as if the total number of atoms \(N_\text{tot}(t)\) were condensed into an arbitrarily thin slab, as in Figure 10.2. Equation 10.10 is then a standard exponential gain equation, showing that the input signal \(I_\text{in}(t)\) is exponentially amplified by \(N_\text{tot}(t)\) to become the output signal \(I_\text{out}(t)\), and Equation 10.12 is essentially a conservation of energy equation, showing how \(N_\text{tot}(t)\) is burned up by the amplification of \(I_\text{in}(t)\) to produce \(I_\text{out}(t)\).
Note that the output pulse \(I_\text{out}(t)\) is actually delayed in real time \(\hat{t}\) with respect to the input pulse \(I_\text{in}(t)\) by the transit time through the amplifier, since a particular point \(t\) in the output pulse occurs \(L/c\) later than the same point \(t\) in the input pulse.
Also, the "number of atoms" \(N_\text{tot}(t)\) is actually not a physical number of atoms that exists at any instant of time \(\hat{t}\), or that could be seen by some snapshot of the amplifier at any single time \(\hat{t}\). It is rather a measure of the total space-integrated (or time-integrated) population difference \(\int{N}(z,t)dz\) seen by any one small segment of the pulse centered at time \(t\), as that segment passes through each successive plane \(z\equiv\hat{z}\) of the amplifier at the local time \(t\). The net result, however, still reduces to an equivalent thin slab in which everything seems to happen simultaneously in the delayed time coordinate \(t\).
Analytic Solutions
Several useful explicit solutions to these equations can be obtained as follows. We start by substituting Equation 10.10 into Equation 10.12 to obtain either of the alternative forms
\[\tag{13}\begin{align}\frac{dN_\text{tot}(t)}{dt}&=-\frac{2^*}{\hbar\omega}\{\exp[\sigma{N}_\text{tot}(t)]-1\}\times{I}_\text{in}(t)\\&=-\frac{2^*}{\hbar\omega}\{1-\exp[-\sigma{N}_\text{tot}(t)]\}\times{I}_\text{out}(t)\end{align}\]
Suppose the total initial inversion \(N_0\) in the laser medium, at a time \(\hat{t}_0\) prior to the arrival of any input pulse energy, is given by
\[\tag{14}N_0\equiv\int_{z=0}^{z=L}\hat{N}(\hat{z},\hat{t}_0)d\hat{z}\]
The initial single-pass power gain of the amplifier is then \(G_0=\exp[N_0\sigma]\).
We will also write the accumulated signal energies per unit area \(U_\text{in}(t)\) and \(U_\text{out}(t)\) in the input and output pulses, from starting time \(t_0\) up to normalized time \(t\) as
\[\tag{15}U_\text{in}(t)\equiv\int_{t_0}^tI_\text{in}(t)dt\qquad\text{and}\qquad{U}_\text{out}(t)\equiv\int_{t_0}^tI_\text{out}(t)dt\]
These pulse energies per unit area are sometimes referred to as energy fluences. It is also convenient to define a saturation energy per unit area \(U_\text{sat}\) sometimes called a saturation fluence) for the atomic medium by
\[\tag{16}U_\text{sat}\equiv\frac{\hbar\omega}{2^*\sigma}\]
This quantity is clearly the pulsed analog to the saturation intensity \(I_\text{sat}\equiv\hbar\omega/\sigma\tau\) we saw for continuous amplification. If a signal energy fluence \(U_\text{sat}\) flows by an atom in a time much less than the atom's recovery time \(\tau\), the atom has essentially a 50% chance of making a stimulated transition from one level to the other during the pulse. (As an example, a Nd:YAG laser, with cross section \(\sigma\approx5\times10^{19}\) cm\(^2\), has a saturation fluence of \(\approx\) 0.4 Joules per cm\(^2\).)
The first of the differential forms in Equation 10.13 can then be integrated in the form
\[\tag{17}\int_{N_0}^{N_\text{tot}(t)}\frac{dN_\text{tot}}{\exp(\sigma{N}_\text{tot})-1}=-\frac{2^*}{\hbar\omega}\int_{t_0}^tI_\text{in}(t)dt=-\frac{2^*}{\hbar\omega}U_\text{in}(t)\]
to give the useful relation
\[\tag{18}U_\text{in}(t)=U_\text{sat}\times\ln\left\{\frac{1-\exp[-\sigma{N_0}]}{1-\exp[-\sigma{N_\text{tot}(t)}]}\right\}=U_\text{sat}\times\ln\left[\frac{1-1/G_0}{1-1/G(t)}\right]\]
where again \(G(t)=\exp[\sigma{N}_\text{tot}(t)]=I_\text{out}(t)/I_\text{in}(t)\) is the time-varying partially saturated gain within the pulse interval. This expression thus connects the cumulative input energy \(U_\text{in}(t)\) to the net remaining inversion \(N_\text{tot}(t)\) or the time-varying power gain \(G(t)\) at any instant of (normalized) time \(t\) within the pulse.
The second differential relation in Equation 10.13 can be similarly integrated to give the complementary relation
\[\tag{19}U_\text{out}(t)=U_\text{sat}\times\ln\left\{\frac{\exp[\sigma{N_0}]-1}{\exp[\sigma{N_\text{tot}(t)}]-1}\right\}=U_\text{sat}\times\ln\left[\frac{G_0-1}{G(t)-1}\right]\]
where \(U_\text{out}(t)\) is similarly the cumulative energy in the output pulse up to time \(t\) (in delayed time coordinates).
Gain Saturation
Either one of these results can then be inverted to express the instantaneous inversion and gain within the pulse in terms of either of the input or output pulseshapes, \(U_\text{in}(t)\) or \(U_\text{out}(t)\). For example, Equation 10.18 can be rewritten in the form
\[\tag{20}\sigma{N_\text{tot}}(t)=\ln\left[\frac{G_0}{G_0-(G_0-1)\exp[-U_\text{in}(t)/U_\text{sat}]}\right]\]
which gives
\[\tag{21}G(t)=\exp[\sigma{N}_\text{tot}(t)]=\frac{G_0}{G_0-(G_0-1)\exp[-U_\text{in}(t)/U_\text{sat}]}\]
For a given input pulseshape \(I_\text{in}(t)\) and a given initial gain \(G_0\) we can use this to calculate the output pulseshape \(I_\text{out}(t)=G(t)\times{I}_\text{in}(t)\).
Alternatively, if we want to specify a desired output pulseshape \(I_\text{out}(t)\) in the presence of saturation, we can calculate the necessary gain versus time from the output pulseshape, using
\[\tag{22}G(t)=1+(G_0-1)\exp[-U_\text{out}(t)/U_\text{sat}]\]
and then find the required input pulseshape from \(I_\text{in}(t)=I_\text{out}(t)/G(t)\). How to synthesize the required input pulseshape—which will be different for different output pulse energy levels or degrees of saturation—is, of course, a separate problem.
Pulseshape Distortion
Figure 10.3 illustrates the kind of output pulse distortion that is produced by amplifier gain saturation assuming typical input pulseshapes. In Figure 10.3, where we assume a square input pulse with a perfectly sharp leading edge, the initial gain right at the leading edge of the pulse is the unsaturated value \(G_0\). This gain immediately begins to saturate, however, falling rather slowly for a weak input pulse and thus producing only a certain amount of "droop" in the output pulse, as in (a), but dropping much more strongly for a strong input pulse, as in (b). The result is then a rapid decrease in the output signal, leaving a large spike on the leading edge of the amplified pulse.
This short pulse formation might seem potentially useful as a means of pulse sharpening, in order to obtain a shortened output pulse from a much longer input pulse. Its practicality is limited, however, because in order to obtain strong pulse sharpening the leading edge of the input pulse must have a rise time substantially shorter than the desired output pulse length. If a practical modulator is unable to create the desired short pulse to begin with, it may be no more capable of generating an input pulse with the required rise time on the leading edge.
A gaussian pulseshape, or any other shape with rounded leading and trailing edges, is generally a more realistic model for saturable pulse amplification. Figure 10.4 illustrates how the gaussian pulseshape (plotted on a log scale) changes as we increase the input energy level to an amplifier with an initial gain \(G_0=10,000\) (\(\equiv40\) dB), keeping the input gaussian pulseshape constant. Significant saturation effects begin to occur when the amplified output energy disregarding saturation, or \(G_0U_\text{in}\), begins to approach the saturation energy \(U_\text{sat}\). Even for input energies well above this value, however, the output pulsewidth appears to be very little changed from the input pulsewidth, even with quite strong saturation, although the pulse does seem to move forward slightly in time.
The output pulse is substantially changed, nonetheless, since there clearly is substantial gain saturation during the pulse. This saturation shows up primarily as an apparent advance in time of the peak of the output pulse. The pulse is not really advanced in time, but merely appears so because the leading edge receives essentially full amplification, whereas the gain is substantially reduced during the peak and trailing-edge portions of the pulse.
Figure 10.5 also plots the total pulse output energy \(U_\text{out}\) integrated over the full pulsewidth versus total pulse input energy \(U_\text{in}\) for the particular case of a homogeneously saturable amplifier with \(G_0=1,000=30\) dB. This plot clearly shows how the pulse energy gain saturates down as the output pulse energy increases much above \(U_\text{sat}\). (Note that these results are independent of the actual shape of the pulses.)
Pulse Energy Extraction and Energy Gain
The efficiency with which a signal pulse extracts the available energy from a laser pulse amplifier can be calculated in a simple fashion as follows. Subtracting the two earlier expressions for \(U_\text{in}(t)\) and \(U_\text{out}(t)\) gives
\[\tag{23}\frac{U_\text{extr}(t)}{U_\text{sat}}\equiv\frac{U_\text{out}(t)-U_\text{in}(t)}{U_\text{sat}}=\ln\left[\frac{G_0}{G(t)}\right]\]
Let \(U_\text{in}\) and \(U_\text{out}\) without a specific time-dependence henceforth denote the total energies in the complete input and output pulses, i.e., the limits of \(U_\text{in}(t)\) and \(U_\text{out}(t)\) as \(t\rightarrow+\infty\); and similarly let \(G_f\) denote the final value of \(G(t)\) after the pulse has passed, i.e., the limit of \(G(t)\) as \(t\rightarrow\infty\). The total energy extracted from the gain medium by the complete pulse is then given by
\[\tag{24}U_\text{extr}\equiv{U}_\text{out}-U_\text{in}=U_\text{sat}\times\ln\left(\frac{G_0}{G_f}\right)\]
The maximum available energy from the amplifier is obviously obtained when the residual gain is saturated all the way down to \(G_f\rightarrow1\). The maximum available energy that can be extracted from the amplifier, assuming an input pulse strong enough to completely saturate the initial inversion, is thus given by
\[\tag{25}U_\text{avail}=U_\text{sat}\times\ln{G_0}=\frac{N_0\hbar\omega}{2^*}\]
The right-hand expression confirms the physically obvious result that the available energy from the amplifier is either the total initial inversion energy, \(N_0\hbar\omega\), or half that value, depending on whether the lower level does or does not empty out rapidly during the pulse.
We might also define an overall or pulse-averaged "pulse energy gain" \(G_\text{pe}\) as the ratio of the total pulse energy output \(U_\text{out}\) to the total pulse energy input \(U_\text{in}\), or
\[\tag{26}G_\text{pe}\equiv\frac{U_\text{out}}{U_\text{in}}=\frac{\ln[(G_0-1)/(G_f-1)]}{\ln[(G_0-1)/(G_f-1)]-\ln[G_0/G_f]}\]
Figure 10.6 shows theoretical and experimental examples for the reduction in laser pulse energy gain with increasing input pulse energy for one particular experiment using a picosecond-pulse dye laser amplifier.
Pulse Energy Extraction Efficiency
Equations 10.23 through 10.26 can also be manipulated in various other ways that may be useful. For example, we might define an energy extraction efficiency \(\eta\) as the ratio of the energy actually extracted from the laser medium to the maximum energy available in the medium, or
\[\tag{27}\eta\equiv\frac{U_\text{out}-U_\text{in}}{U_\text{avail}}=\frac{\ln{G_0}-\ln{G_f}}{\ln{G_0}}\]
Inverting this tells us that the final gain \(G_f\) can be related to the initial gain \(G_0\) and the energy extraction efficiency \(\eta\) in a simple fashion by
\[\tag{28}G_f=G_0^{1-\eta}\]
Obviously, if the energy extraction efficiency approaches anywhere near 100%, the final saturated gain \(G_f\) at the end of the pulse will be much less than the unsaturated gain \(G_0\) at the beginning of the pulse.
Putting this result into Equation 10.26 then gives a general relationship between the initial or unsaturated power gain \(G_0\), the time-averaged pulse energy gain \(G_\text{pe}\), and the energy extraction efficiency \(\eta\), entirely independent of input or output pulseshapes.
Figure 10.7 illustrates how the pulse energy gain is rapidly saturated downward as we attempt to obtain increased energy extraction from an amplifier (i.e., by putting in stronger and stronger input pulses).
Summary
This section has presented the simplest possible rate-equation analysis of pulsed amplifier saturation (often referred to as the Frantz-Nodvik analysis). More complicated analyses can, of course, be developed to take into account transverse intensity variations, finite pumping and relaxation times, large-signal and Rabi-flopping effects in the atoms, dispersive wave propagation effects, and other complications.
All of these results for pulse amplification are obviously similar in character to the power extraction and efficiency results we obtained for continuous amplification. As usual, efficient energy extraction is obtained only at the cost of substantial reduction in effective energy gain integrated over the full pulse. Except for relatively smooth pulses, like gaussian pulses, efficient energy extraction can also mean severe distortion of the output pulseshape.
All the results given in this section also apply directly to pulse propagation through a saturable absorber, if we simply invert the signs of \(N_0\) and \(N_\text{tot}(t)\) in all the expressions. We will apply these results to passive saturable absorber mode locking in a later tutorial.
2. Pulse Propagation in Nonlinear Dispersive Systems
When an optical pulse propagates through any kind of nonlinear system, we can expect to see at least some nonlinear distortion of the pulseshape with propagation distance, with stronger effects for larger amplitude pulses.
When such nonlinear distortion is combined, however, with linear dispersive pulse distortion effects such as those discussed in the linear pulse propagation tutorial, even more complex and interesting effects can be expected to occur—especially since the nonlinear distortion effects may in general tend either to combine with or to cancel out the linear dispersive effects.
In this section we will introduce several such nonlinear and dispersive effects that are of particular importance in real laser systems.
Nonlinear Pulse Propagation in Atomic Systems
Let us first make some general observations about the large-signal propagation of an optical pulse through an atomic medium which contains a resonant atomic transition, such as a typical laser medium.
When a pulsed signal with an electric field variation \(\mathcal{E}(z,t)\) propagates through any kind of atomic medium, the electromagnetic aspects of the pulse behavior are governed by the electromagnetic wave equation. This equation is a fundamentally linear equation for the electric field \(\mathcal{E}(z,t)\) in terms of the polarization \(p(z, t)\) in the atomic medium. If the polarization term on the right-hand side of this wave equation in turn represents an induced polarization which is also linear in the applied signal field, then the overall response of this system will be entirely linear; and the pulse propagation and distortion behavior in the system can be completely described by the dispersion curve for the atomic medium, or the \(\omega-\beta\) curve for the wave-propagating system.
Suppose however that the polarization \(p(z, t)\) arises from a resonant atomic transition, and also that the applied signal fields are strong enough to produce significant nonlinear or "Rabi flopping" behavior. The polarization response \(p(z, t)\) is then more complicated and no longer linear in the applied signal. The polarization response must be described instead by (at least in simple cases) a resonant dipole equation for the polarization response, together with an additional equation for the population difference \(\Delta{N}(z, t)\) on the relevant transition as a function of space and time. Both of these equations are basically nonlinear equations, at least at larger signal levels.
To find the complete pulse propagation behavior for a large-amplitude wave passing through a resonant atomic system, all three of these equations must then be solved simultaneously and in a self-consistent fashion, taking full account of both the effects of the pulse fields on the atoms and the effects of the atomic polarization back on the pulses. For larger signals where these equations become more strongly nonlinear, the resulting solutions will generally be quite complicated, in themselves and in how they depend on the pulse intensity. As a result there are many complicated analyses of such phenomena in the scientific literature. (Only the atomic equations are nonlinear; Maxwell's equations and hence the electromagnetic wave equation are entirely linear in \(\mathcal{E}\) and \(p\).)
The results that come out of these analyses (and experiments) include purely linear behavior, such as free induction decay, and other types of pulse propagation and distortion behavior, such as Rabi flopping behavior, "self-induced transparency," and "\(\pi\) and \(2\pi\) pulse propagation." We will not attempt to review any of these resonant-atom pulse phenomena in detail here, since such discussions are unavoidably lengthy and complex, and since the resulting large-signal phenomena, though sometimes experimentally interesting, do not seem to have major practical applications.
Nonlinear Optical Polarization: The Optical Kerr Effect
There is, however, another fundamental type of nonlinear polarization response that occurs in essentially all transparent optical materials, not just on resonant atomic transitions, and that can be of considerable practical importance in many laser situations.
This is a nonlinear change in the dielectric constant or index of refraction of almost any optical material with increasing optical intensity, often referred to as an optical Kerr effect. In the rest of this section we will examine several of the important propagation effects produced by this optical Kerr effect.
When an electric field \(\mathcal{E}\) is applied to a transparent dielectric medium, the force associated with this field produces a distortion of the electron-charge clouds in that medium, and also a possible reorientation of the molecular axes of molecules in a liquid medium, since such molecules generally like to line up with an applied field.
Both effects in turn lead to a macroscopic polarization p in the medium that in first order will be linear in the applied \(\mathcal{E}\) field. This linear response in a low-loss or transparent dielectric is, of course, just the linear dielectric constant or index of refraction of the medium.
If the applied field is strong enough, however, the polarization response of the medium may become nonlinear in the applied field. (The distortion of the electron-charge clouds, or the realignment of the molecular axes, becomes nonlinear—usually weakly nonlinear—in the applied field strength.)
This is very often expressed in a somewhat simplified but still fairly general fashion by writing the polarization as a series expansion in the applied field in the form
\[\tag{29}p=\chi_{(1)}\epsilon_0\mathcal{E}+\chi_{(2)}\mathcal{E}^2+\chi_{(3)}\mathcal{E}^3+\ldots\]
where \(\chi_{(1)}\) is the linear susceptibility, and \(\chi_{(2)}\) and \(\chi_{(3)}\) (which have quite different dimensions) represent weak higher-order nonlinearities in the dielectric response of the medium.
(In a more accurate picture, all three of the \(\chi\) quantities should be tensor quantities, and all three should have frequency dependences that become increasingly complex for the higher-order terms.)
Second-Order Susceptibility: Harmonic Generation and Modulation
The \(\chi_{(2)}\mathcal{E}^2\) term—sometimes written in an alternative notation as \(d_2\mathcal{E}^2\)— represents a second-order nonlinearity, which can be responsible for second-harmonic generation, optical rectification, optical parametric amplification, and other useful nonlinear effects.
By symmetry arguments, however, this effect must be identically zero in any material that has a centrosymmetric arrangement of atoms. Effects of this type are found therefore primarily in certain special crystals having a noncentrosymmetric crystal structure—in essence, only in those materials that are also piezoelectric.
This includes, for example, barium titanate or BaTiO3, crystal quartz, potassium dihydrogen phosphate (KDP), ammonium dihydrogen phosphate (ADP), cesium dihydrogen arsenate (CDA), and lithium niobate (LiNbO3); these are some of the nonlinear optical crystals most widely used in optical modulators and harmonic generators.
The Third-Order Susceptibility
The third-order susceptibility term \(\chi_{(3)}\mathcal{E}^3\) can be present, however, with varying strength, in essentially all optical materials of any crystal structure or class, including liquids and gases. If we include this term in the polarization \(p\), the total electric displacement \(d\) in the medium can then be related to the applied field \(\mathcal{E}\) in the form
\[\tag{30}d=\epsilon_0[1+\chi_{(1)}]\mathcal{E}+\chi_{(3)}\mathcal{E}^3=\epsilon_0[1+\chi_{(1)}+\epsilon_0^{-1}\chi_{(3)}\mathcal{E}^2]\mathcal{E}\]
Hence the dielectric constant \(\tilde{\epsilon}\) is now a nonlinearly varying quantity given by
\[\tag{31}\tilde{\epsilon}=\epsilon_1+\epsilon_2\mathcal{E}^2\]
in which \(\epsilon_1\equiv\epsilon_0(1+\chi_{(1)})\) is the linear or first-order dielectric constant, and \(\epsilon_2\mathcal{E}^2\equiv\chi_{(3)}\mathcal{E}^2\) is the nonlinear change in the dielectric constant, produced by the applied field.
Since the optical index of refraction \(n\) is related to the optical-frequency value of \(\epsilon\) by \(n=\sqrt{\tilde{\epsilon}/\epsilon_0}\), we can also view this as a nonlinear dependence of the index of refraction on applied signal strength, as given by
\[\tag{32}n=n_0+n_{2E}\mathcal{E}^2\]
where \(n_0=\sqrt{\epsilon_1/\epsilon_0}\) is the linear value and \(n_{2E}\mathcal{E}^2\) the nonlinear variation of the index of refraction.
Kerr Cell Light Modulators
Suppose we construct a liquid cell containing CS2 or nitrobenzene or some similar liquid, as in Figure 10.8, to which we apply both a strong low-frequency modulation field \(E_0\) (by using suitable electrodes) and a much weaker optical-frequency field \(\mathcal{E}\) (in the form of a traveling optical wave). A strong enough modulation field will then change the index of refraction seen by the optical wave according to the relationship
\[\tag{33}n=n_0+n_{2E}E_0^2\]
and this provides a way of phase modulating the light beam.
To be slightly more accurate, the modulation field \(E_0\) usually causes an increase in index of refraction for optical \(\mathcal{E}\) fields polarized parallel to the dc field, and a decrease in index of refraction for fields polarized perpendicular to \(E_0\). This then creates an induced birefringence in the modulation cell, with a magnitude proportional to the modulating voltage squared.
This birefringence can in turn be converted into amplitude modulation by placing the modulation cell between suitable crossed polarizers. This physical effect is known as the Kerr effect, and the resulting device is a Kerr cell modulator.
Pockels Cell Light Modulators
Practical Kerr cells, when they are used at all, usually employ one of the molecular liquids mentioned earlier, since the reorientation of the molecules in these liquids under the influence of the applied field \(E_0\) produces the strongest available Kerr coefficient, or change of index with voltage. Even with these liquids, however, voltages on the order of 25,000 volts are necessary to produce sizable amplitude-modulation effects.
Most of the electrooptic modulators used with lasers, therefore, are instead Pockels cell modulators, which use one of the noncentrosymmetric crystals mentioned earlier, and produce the index change or birefringence through the second-order nonlinearity term \(\chi_{(2)}E_0^2\).
The induced birefringence is then linear rather than quadratic in the modulation field \(E_0\), and adequate modulation can be obtained in practice with modulation voltages of a few thousand volts, or even less in some especially favorable cases. Figure 10.9 shows two examples of simple Pockels cell modulator designs.
Optical Kerr Effect
Suppose, however, we consider an optical signal with a sufficiently strong optical field strength that we can have a significant \(\chi_{(3)}\mathcal{E}^3\) or \(n_{2E}\mathcal{E}^2\) term produced by the optical beam itself. If \(\mathcal{E}(t)\) is in fact an optical-frequency signal, at frequency \(\omega\), then this term will produce two effects.
On the one hand the \(\chi_{(3)}\mathcal{E}^3\) term will produce a third-harmonic polarization \(p(t)\) at frequency \(3\omega\); this may radiate—typically very weakly—at the third harmonic of the applied optical frequency \(\omega\). We will neglect this third-harmonic generation process here, since it is usually very weak; is not of direct interest at this point; and will not be properly phase-matched in most situations.
On the other hand, this same \(n_{2E}\mathcal{E}^2\) effect will also produce a zero-frequency or time-averaged signal-induced change in the refractive index for the signal field, which can be written as
\[\tag{34}n=n_0+n_{2E}\langle{\mathcal{E}^2}\rangle=n_0+n_{2I}I\]
where \(\langle{\mathcal{E}^2}\rangle\) represents the time-averaged value of the field squared, and \(I\equiv\sqrt{\epsilon/\mu}\langle{\mathcal{E}^2}\rangle\) represents the optical intensity. This change in the average value of the optical refractive index produced by the optical signal itself is commonly referred to as the optical Kerr effect.
Such an optical Kerr effect will be present, with a positive sign (\(n_{2I}\gt0\)), in nearly all optical materials. A representative value for the optical Kerr coefficient in a typical glass (such as might be used in an optical fiber) might be \(n_{2E}\approx10^{-22}\) m\(^2\)/V\(^2\) or \(n_{2I}\approx10^{-16}\) cm\(^2\)/W.
Certain strongly polarizable molecular liquids, such as CS2 or certain long-chain organics, can have values 10 to 20 times larger than this. (Essentially all condensed materials have an electronic optical Kerr effect of roughly comparable magnitude; the strongly enhanced response in molecular liquids comes from an orientational Kerr effect similar to that produced by low-frequency electric fields).
We will review several nonlinear phenomena produced by this optical Kerr effect in subsequent paragraphs. We can estimate, however, that this optical Kerr effect might produce significant effects if it produces an additional path length \(\Delta{nL}\) of half a wavelength, or an additional half cycle of phase shift, in a path length of, say, \(L=1\) cm, or \(2\pi{n}_{2I}IL/\lambda=\pi\).
If we use \(L=1\) cm and \(\lambda\) = 0.6 μm, then a significant nonlinear effect will occur for an optical intensity of \(I\approx1\) to \(10\) GW/cm\(^2\), depending on the value of \(n_{2I}\). It is in fact in this range of intensities that significant optical Kerr effects do occur in optical systems like high-power laser rods, focusing lenses, and other optical elements.
Note, however, that intensities in this range will occur for a total input power of less than 1 watt in a 4 μm-diameter optical fiber; moreover, in an optical fiber it may take kilometers rather than centimeters for the resulting phase-modulation effects to accumulate. This can make possible very strong and useful nonlinear optical effects in optical fibers, as we will see in more detail in the following section.
Whole-Beam Self-Focusing Effects
As a first illustration of an important effect produced by the optical Kerr effect, we can consider so-called self-focusing of an optical beam. Suppose an optical beam with a moderate intensity \(I\) and a smooth transverse profile passes through a medium having a finite and positive optical Kerr coefficient \(n_{2I}\), as in Figure 10.10.
The higher intensity in the center of the beam will then cause an increase in the index of refraction seen by the center of the beam, as compared to the wings; in other words, the optical medium will be given a focusing power, or converted into a weak positive lens.
If this self-focusing effect in propagating through a given length of the medium exceeds the diffraction spreading of the optical beam in the same length, the optical beam profile will begin to be focused inward as the beam propagates.
But, inward focusing then increases the intensity in the center of the beam, and makes the sides of the beam profile steeper; and this in turn increases the strength of the self-induced lens. The beam will then continue to be focused ever more strongly inward, in an essentially runaway fashion.
This type of self-focusing of the entire beam profile is known as whole-beam self-focusing. A more detailed analysis shows that, when the effects of self-focusing and diffraction spreading are both taken into account, runaway whole-beam self-focusing will begin to occur when the total power in a beam with a smooth transverse profile exceeds a certain critical power \(P_\text{crit}\) independent of the diameter of the beam. Typical values of \(P_\text{crit}\) range from a few tens of kilowatts in strong Kerr liquids to a few megawatts in materials with typical weak Kerr coefficients.
Small-Scale Self-Focusing Effects
There is a similar phenomenon known as small-scale self-focusing, in which any small-amplitude variations or ripples on a transverse beam profile will begin to grow in amplitude exponentially with distance because of the optical Kerr effect. In essence the transverse spatial variation of the optical beam intensity produces a transverse spatial variation in refractive index, or a refractive index grating.
This grating diffracts some of the optical beam energy into small-angle scattering, and this diffracted light interferes with the original beam in exactly such a way as to make the initial intensity ripples on the beam profile grow in amplitude with distance.
Figure 10.11 shows the dramatic results of an experiment in which initially small-amplitude ripples were put on the transverse amplitude profile of an optical beam, and the beam then sent through a strong optical Kerr medium with an intensity sufficient to cause significant growth in these ripples after a few tens of cm. The runaway growth of the periodic amplitude variations is evident.
In general, if either whole-beam or small-scale self-focusing becomes significant, this self-focusing will continue in a runaway fashion. The optical beam may then rapidly collapse with distance into one or several very small filaments or selffocused focal spots.
Once this happens, not only does the beam become badly distorted in its transverse profile, but the power density also usually becomes large enough to cause optical damage, optical breakdown, or other undesirable nonlinear effects to ensue.
The type of nonlinear self-focusing produced by the optical Kerr effect can thus be a major problem in many high-power lasers, especially pulsed lasers, as well as in scientific experiments using focused high-power laser beams.
Self-Phase Modulation
The optical Kerr effect can also produce a very similar self-phase-modulation effect, which occurs for pulsed or modulated signals in the time rather than the spatial domain. To demonstrate this, we can next suppose that an optical pulse with some given intensity variation \(I(t)\) in time propagates through a certain length \(L\) of a medium with a finite optical Kerr coefficient \(n_{2I}\), and that the pulse amplitude is large enough to produce a significant index change \(\Delta{n}(t)\equiv{n}(t)-n_0=n_{2I}I(t)\) and a significant change in optical path length \(\Delta{n(t)L}\), at least near the peak of the pulse. (Assume for now that the transverse beam profile is uniform, or that we somehow otherwise avoid the self-focusing effects just discussed.)
The pulse fields will then experience a time-varying phase shift or phase modulation \(\exp[j\Delta\phi(t)]=\exp[-j2\pi\Delta{n}(t)L/\lambda]=\exp[-j2\pi{n}_{2I}I(t)L/\lambda]\) produced by the intensity variation of the pulse itself.
If the optical Kerr coefficient is positive (\(n_{2I}\gt0\)), as it usually is, this self-phase modulation will represent in effect a lowering of the optical frequency of the pulse during the rising or leading edge of the pulse, since \(dn/dt\gt0\) and hence \(\Delta\omega_i(t)=(d/dt)\Delta\phi(t)\lt0\). (In physical terms, the medium is getting optically longer; so the arrival of optical cycles is delayed or slowed down.)
Similarly there will be an increase in the instantaneous frequency of the pulse signal during the trailing or falling edge of the pulse. The maximum frequency shift will occur at the points of maximum slope or maximum \(dI(t)/dt\).
A pulse with a smooth time envelope, as in Figure 10.12, will thus acquire a more or less linear frequency chirp across the central region of the pulse, as shown in the lower plot. The magnitude of this chirp will increase more or less linearly with distance through the medium, at least as long as the pulse intensity profile remains unchanged.
In most practical situations, however, the optical medium will also have a certain value of group-velocity dispersion \(dv_g(\omega)/d\omega\). The different portions of the pulse, with their slightly different optical frequencies, will thus begin to travel at slightly different group velocities; and as a result the pulseshape will begin to change by an increasing amount with increasing distance. Depending on circumstances, this effect can lead to at least three different types of behavior: severe pulse distortion and breakup, soliton formation and propagation, or pulse broadening and enhanced frequency chirping. We can discuss each of these in turn.
Approximate Analysis of Self-Phase Modulation
We can develop an approximate analysis to indicate the magnitude of these self-phase modulation effects as follows. Suppose an initially unchirped gaussian input pulse has the time-variation
\[\tag{35}\mathcal{E}(t)=\mathcal{E}_0e^{-at^2}\quad\text{or}\quad{I}(t)=I_0e^{-2at^2}\]
The net phase shift for this pulse in passing through a length \(L\) of nonlinear medium will then be
\[\tag{36}\phi(t)=\frac{2\pi(n_0+n_{2I}I)L}{\lambda}\]
and hence the phase-shift derivative will be
\[\tag{37}\frac{d\phi}{dt}=\frac{2\pi{n_{2I}L}}{\lambda}\frac{dI(t)}{dt}\approx\frac{4\pi{a}n_{2I}LI_0}{\lambda}\times{t}e^{-2at^2}\]
where we assume for simplicity that to first order the pulseshape \(I(t)\) is not changed in passing through the length \(L\).
But this phase modulation corresponds to giving the pulse a frequency chirp at the center of the pulse which is given (in our earlier gaussian pulse notation) by
\[\tag{38}\frac{d\phi}{dt}=2bt\approx\frac{4\pi{a}n_{2I}LI_0}{\lambda}t\]
This means that the pulse will acquire a chirp parameter \(b=a\), and thus increase its time-bandwidth product by a factor of \(\sqrt{2}\), after passing through a length of nonlinear medium given by
\[\tag{39}\frac{2\pi{n_{2I}I_0L}}{\lambda}=1\]
That is, the pulse will acquire a significant amount of self-phase modulation in length \(L\) if its peak intensity exceeds a value given by
\[\tag{40}I_0=\frac{\lambda}{2\pi{n_{2I}}I}\]
If we take \(n_{2I}=3\times10^{-16}\) cm\(^2\)/W, \(L\) = 10 cm, and \(\lambda\) = 0.5 μm, this gives a threshold intensity for significant self-phase modulation of \(I_0\approx3\) GW/cm\(^2\), which is close to the damage threshold in most optical materials. Suppose, however, we consider a single-mode optical fiber with a diameter of 4 μm and a length of \(L\) = 10 m. Its threshold intensity of 30 MW/cm\(^2\) is reached with a total power in the fiber of \(\approx\) 3 W.
Pulse Distortion and Breakup Effects
We can recall that the group velocity in a dispersive medium is given by \(1/v_g(\omega)=d\beta(\omega)/d\omega\equiv\beta'(\omega)\). Hence the variation of group velocity with frequency is given by
\[\tag{41}\frac{dv_g}{d\omega}=-v_g^2\frac{d^2\beta}{d\omega^2}=-v_g^2\beta^"\]
where \(\beta^"\equiv{d^2\beta}/d\omega^2\) is commonly referred to as the group-velocity dispersion of the medium. A negative value of \(\beta^"\), which corresponds to negative dispersion according to this conventional definition, thus means that the group velocity increases with increasing frequency.
Suppose that an optical Kerr medium in fact has such a negative dispersion, so that \(v_g\) increases with increasing \(\omega\). This means physically that the leading edge of the chirped pulse in Figure 10.12 will begin to travel more slowly, and to fall back against the main part of the pulse, while the trailing edge of the pulse will begin to travel faster and to catch up with the main part of the pulse. In other words, the pulse will generally become compressed as it propagates, as a consequence of the self-phase-modulation process.
As the pulse becomes more compressed, however, its peak intensity will increase, on the one hand, and its rise and fall times will become shorter, on the other hand. Both effects will then combine to greatly increase the self-chirping effects on the leading and trailing edges of the pulse, and this in turn will increase the pulse compression, in another runaway type of process.
If the dispersion in the medium is of opposite sign, an initially smooth pulse will become broadened in time rather than compressed (as we will discuss in more detail in a later section). Even in this situation, however, the pulse will also acquire a growing amount of chirp, and its spectrum will be continuously broadened by the combination of nonlinear effects plus dispersion.
(Note also that the analog to dispersion for pulse distortion in time is diffraction for pulse distortion in space—i.e., in the transverse coordinates—and this diffractive dispersion always has a sign corresponding to pulse compression in space for positive \(n_{2I}\). Self-focusing thus always leads to beam compression in the transverse direction.)
In either situation, if the time envelope of either a pulsed or a cw signal contains any significant amount of initial amplitude (or phase) modulation or pulse substructure—that is, if either the phasor amplitude or the phase angle of the signal field has significant time modulation within the overall pulse envelope— this will increase these nonlinear distortion effects.
The phase substructure will represent additional chirp, and the amplitude structure will reinforce the self-phase-modulation effect. The result will often be that the envelope of a high-power laser beam will not retain a smooth shape, if indeed it initially has one, but will begin to break up into increasingly complex subpulses within the main pulse envelope.
This increasingly strong phase and amplitude modulation will also broaden the frequency spectrum of the pulse (but not the overall time envelope) by an amount that can increase rapidly with increasing distance.
Pulse Breakup in Practical Laser Systems: The B Integral
These pulse-breakup and spectral-broadening effects, especially when accompanied and intensified by self-focusing effects, can be a source of considerable difficulty in many high-power lasers, and particularly in mode-locked lasers, where the peak power can be very high even though the total energy or the average power may be quite low.
As we have said, the effects of nonlinear modulation and dispersion will grow exponentially as a laser signal begins to break up, because the time-variation becomes faster across the substructure within the pulse, and because the pulse energy gets compressed into shorter subpulses with higher peak intensities.
Self-phase-modulation and self-focusing effects are thus especially strong "runaway" effects in such devices as multistage Nd:glass laser amplifier chains and mode-locked Nd:glass laser oscillators. In both the pulse is continually being further amplified, and the laser medium has a broad enough atomic linewidth to continue amplifying the pulse even after its spectrum has been substantially broadened by the nonlinear effects.
When a mode-locked Nd:glass laser is pumped too strongly, for example, the early pulses in the mode-locked and \(Q\)-switched burst may be reasonably clean and well-formed, but the pulses near and after the peak of the \(Q\)-switched burst often become severely distorted and spectrally broadened.
Self-phase modulation of this type is often accompanied by, and reinforced by, self-focusing effects in the same system. It is also a common characteristic of such self-phase modulation that the pulse spectrum gets greatly broadened, generally in a one-sided fashion, to the low-frequency side of the original carrier frequency; and catastrophic optical damage may occur, often in small self-focused spots, if the peak intensity is not limited.
As a generalization of the self-phase-modulation criterion we developed a few paragraphs back, it has become conventional to define the "\(B\) integral" for a multipass laser system as a cumulative measure of the nonlinear interaction, where this integral is given by
\[\tag{42}B\equiv\frac{2\pi}{\lambda}\int_0^Ln_{2I}(z)I(z)dz\]
taking into account the changes in diameter and power level of the laser beam through the complete system. A generally accepted criterion for high-power laser systems is that the cumulative \(B\) integral must be kept somewhere below the value \(B\le3\) to \(5\) to avoid serious nonlinear damage and distortion effects due to either self-phase modulation or self-focusing.
3. The Nonlinear Schrödinge Equation
The basic equation of motion for analyzing signal propagation through a weakly nonlinear optical medium, or along a nonlinear transmission line (such as an optical fiber with an optical Kerr coefficient), is a nonlinear extension of the parabolic equation we derived in the linear pulse propagation tutorial. We can derive this nonlinear form in a simplified manner as follows.
Derivation of the Nonlinear Schrödinge Equation
Consider an optical signal of the form \(\mathcal{E}(z,t)\equiv\tilde{E}(z,t)\exp{j}[\omega_0t-\beta(\omega_0)z]\) traveling in the \(+z\) direction, where \(\tilde{E}(z,t)\) is the slowly varying amplitude of this signal.
If we Fourier-transform this signal \(\tilde{E}(z,t)e^{j\omega_0t}\) into its frequency spectrum \(\tilde{E}(z,\omega)\) at any arbitrarily chosen plane \(z\), propagate each frequency component forward by a small distance \(dz\) using the frequency-dependent and intensity-dependent propagation constant \(\beta(\omega)\), and then Fourier-transform these components back into the time domain, we can find that the signal envelope \(\tilde{E}(z,t)\) at the plane \(z=z+dz\) is given by
\[\tag{43}\tilde{E}(z+dz,t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}d\Delta\omega\int_{-\infty}^{\infty}dt'\tilde{E}(z,t')e^{j\Delta\omega(t-t')}e^{-j\Delta\beta{dz}}\]
where \(\Delta\omega\equiv\omega-\omega_0\) and \(\Delta\beta\equiv\beta(\omega)-\beta(\omega_0)\).
The partial derivatives of the signal \(\tilde{E}(z,t)\) with respect to time can be calculated by multiplying the integrand in Equation 10.43 by \(\partial/\partial{t}\equiv{j}\Delta\omega\), and with respect to distance by either expanding in powers of \(dz\) or by multiplying the integrand by \(\partial/\partial{z}\equiv-j\Delta\beta\). We can also suppose that the propagation constant \(\beta(\omega)\) has the weakly nonlinear and dispersive form
\[\tag{44}\beta(\omega)=\beta(\omega_0)+\beta_2\langle\mathcal{E}^2\rangle+\beta'(\omega_0)\times(\omega-\omega_0)+\frac{1}{2}\beta^"(\omega_0)\times(\omega-\omega_0)^2\]
where \(\beta'\) and \(\beta^"\) are the first and second derivatives of \(\beta\) with respect to \(\omega\), and where
\[\tag{45}\beta_2\langle\mathcal{E}^2\rangle\equiv\frac{2\pi\omega_0n_{2E}\langle\mathcal{E}^2\rangle}{c}\]
gives the (small) change in midband propagation constant due to the optical Kerr effect. We can then show, after some algebra, that the integral form for \(\tilde{E}(z,t)\) in Equation 10.43 is equivalent to the differential equation
\[\tag{46}\left[\frac{\partial}{\partial{z}}+\beta'\frac{\partial}{\partial{t}}-j\frac{\beta^"}{2}\frac{\partial^2}{\partial{t^2}}+j\frac{\beta_2|\tilde{E}|^2}{2}\right]\tilde{E}(z,t)=0\]
where we have used \(\langle\mathcal{E}^2\rangle\equiv\frac{1}{2}|\tilde{E}|^2\) for a sinusoidal signal.
Discussion
Equation 10.46 is a generalization of the parabolic equation 9.41 derived in the linear pulse propagation tutorial, with a nonlinear optical Kerr effect term added. (This approach to its derivation also illustrates another way of arriving at Equation 9.41.) Equation 10.46 has the form of a nonlinear Schrödinger equation, with a nonlinear potential function, except that \(z\) and \(t\) are interchanged from the roles they usually play in the conventional Schrödinger equation.
This same equation arises in other physical situations, including deep-water wave propagation, ion-acoustic waves in plasma physics, superconductivity, and vortex motions; so many techniques for its solution have been developed.
Note again that the group-velocity dispersion term \(\beta^"\) in the propagation constant translates into what is essentially a complex diffusion term \(j\beta^"\partial^2/\partial{t^2}\) in the differential equation. Since this diffusion coefficient can have either sign, it can correspond to either pulse spreading or pulse compression in different situations. Its effects must, however, be balanced against the nonlinear propagation effects, as we will see further in the following section.
4. Nonlinear Pulse Broadening in Optical Fibers
To illustrate the importance of self-phase-modulation effects in fiber optics, we can consider what happens when a low- to moderate-power optical pulse (e.g., a few hundred milliwatts to a few watts peak power) of very short time duration (picoseconds to femtoseconds) is injected into a very low-loss single-mode optical fiber, typically a few microns in diameter.
Self-focusing effects will then be effectively eliminated by the strong waveguiding properties of the optical fiber; at the same time, the low losses and small area of the fiber will permit strong self-phase-modulation and dispersion effects to accumulate over very long distances in the fiber, at energy and power levels well below those that will produce optical damage.
This permits the demonstration of some very interesting and useful nonlinear propagation effects, including in particular pulse self-chirping and subsequent compression, and optical soliton propagation.
Dispersive Effects in Optical Fibers
We need first to describe the dispersion effects versus wavelength in an optical fiber, especially a single-mode optical fiber. Figure 10.13 shows the typical variation of the index of refraction versus wavelength or frequency across the visible and near-infrared regions, and the resulting variations of \(\beta'\) and \(\beta^"\) across the same regions, for typical transparent optical materials such as quartz or glass.
The group-velocity dispersion parameter \(\beta^"\) changes from being positive at shorter wavelengths or higher frequencies to negative at longer wavelengths. There is some confusion in the literature over how dispersive properties should be labeled; but the situation where \(\beta^"\gt0\), which is equivalent to \(dv_g/d\omega\lt0\) or \(dv_g/d\lambda\gt0\), is usually referred to as positive or normal dispersion, and the opposite case is referred to as negative or anomalous dispersion.
The effective dispersion values for a propagating wave in a single-mode optical fiber will differ somewhat from the purely material dispersion properties, because of modal dispersion effects, or waveguide propagation effects, which depend in essence on the mode shape, and on how the fields of the propagating mode are distributed between the core and the cladding of the optical fiber.
The general rule for quartz optical fibers, however, is that the group-velocity dispersion is positive (according to the preceding definition) across the visible region, goes through zero in the vicinity of 1.3 μm, and becomes increasingly negative at longer wavelengths. Note that the lowest-loss region for such optical fibers occurs, however, in the vicinity of 1.5 μm, where the dispersion has become significantly negative.
Nonlinear Pulse Broadening and Self-Chirping
Let us first consider the propagation properties of an optical fiber in the visible region, where the group-velocity dispersion is positive. From the arguments given two sections back, this means that the frequency chirp produced by the optical Kerr effect across the center of an optical pulse with a smooth time envelope will lead to broadening of the pulse envelope in time. It also turns out that such a pulse, as it propagates, will not only broaden, but acquire a growing amount of frequency chirp.
Suppose a short optical pulse propagates through an optical fiber at a wavelength in the visible region where the group-velocity dispersion in glass fibers is positive. Propagation of this pulse can then be calculated by the nonlinear Schrödinger equation derived in the preceding section, with an appropriate sign for the group-velocity-dispersion parameter.
It is found that an initially smooth input pulse gradually broadens out to acquire an essentially rectangular shape, with increasing width and increasingly sharp rising and falling edges at increasing distances.
Figure 10.14 shows predicted pulseshapes if we transmit a 5.5-ps pulse with 10 W peak power at 590 nm through increasing lengths ranging from 0 to 70 meters of a typical 4 μm-diameter single-mode optical fiber. The self-broadening effect on the pulse profile is evident.
Figure 10.15 shows more details from a similarly calculated result for an initially 6-ps 100-W pulse propagated through 30 m of single-mode fiber. The pulse has broadened from a initial smooth hyperbolic secant pulse with initial pulsewidth of 6 ps to a rectangular pulse \(\approx\) 24 ps in duration as shown in (a).
This pulse has also acquired a nearly linear frequency chirp over the full pulse duration as shown in (b). In agreement with this, the pulse spectrum has broadened from the initial transform limit of \(\approx\) 2.5 cm\(^{-1}\) to a characteristic phase-modulation spectrum nearly 50 cm\(^{-1}\) wide, as shown in (c). Plot (d) shows the greatly shortened pulse that could result from taking the self-chirped pulse in this particular theoretical example and compressing it externally using an optimum linear dispersion element.
Chirped Pulse Recompression
It was in fact realized and demonstrated by Grischkowsky and co-workers at the IBM Research Laboratories that this kind of strongly self-chirped pulse is an essentially ideal input signal for subsequent pulse recompression using any type of auxiliary dispersive medium following the fiber, such as the diffraction grating pair shown earlier.
Figure 10.16 illustrates an experiment in which an initial pulse from a modelocked laser is first self-chirped and broadened using a length of optical fiber, and then compressed to less than a tenth of its initial pulsewidth by using a simple grating pair of the type described in an earlier section as the auxiliary linear dispersive element. (Note that Grischkowsky and colleagues in fact used a retroreflective prism to achieve the desired dispersion with only a single grating.)
By cascading two stages of this type of self-broadening and linear recompression, this group has in fact converted initial 5.9-ps pulses into 0.09-ps (or 90-fs) pulses, as illustrated in Figure 10.17. In other experiments, Shank and co-workers have used a single stage to convert initial 90-fs pulses from a colliding-pulse mode-locked dye laser into 30-fs optical pulses. These pulses are approximately 14 optical cycles long, and are the shortest pulses known to date.
5. Solitons in Optical Fibers
In 1834 John Scott Russell, then a young Scottish university scientist and later to become a famous Victorian engineer and shipbuilder, recorded the following observations from the banks of the Glasgow-Edinburgh canal, where he first developed many of the fundamental principles of hydrodynamics and of ship's hull design: "I was observing the motion of a boat which was rapidly drawn along a narrow channel by a pair of horses, when the boat suddenly stopped—not so the mass of water in the channel which it had put in motion; it accumulated round the prow of the vessel in a state of violent agitation, then suddenly leaving it behind, rolled forward with great velocity, assuming the form of a large solitary elevation, a rounded, smooth and well-defined heap of water, which continued its course along the channel apparently without change of form or diminution of speed. I followed it on horseback, and overtook it still rolling on at a rate of some eight or nine miles an hour, preserving its original figure some thirty feet long and a foot to a foot and a half in height. Its height gradually diminished, and after a chase of one or two miles I lost it in the windings of the channel. Such, in the month of August 1834, was my first chance interview with that singular and beautiful phenomenon ..."
This represented one of the first detailed observations of a solitary wave, or soliton, a special large-signal solution to some nonlinear dispersive propagation equation, which either propagates with a certain fixed and unchanging steady-state pulseshape over very long distances, or else displays a slow periodic oscillation with distance between a certain set of similar characteristic pulseshapes.
Solitons having these properties represent a very interesting and useful physical phenomenon, with applications in many fundamental areas of physics. It has very recently been realized that very short optical pulses in optical fibers can also propagate as solitons, at very modest power levels; and that these soliton pulses may be very important for carrying pulsed optical communications signals through such fibers over very long distances and at very high data rates. In this section therefore we will briefly review some of the basic concepts of solitons, and particularly of their properties in single-mode optical fibers.
Solitons in General
A soliton is in general any member of a class of solutions to some nonlinear equation or nonlinear propagation problem, in which each such solution is characterized by a certain amplitude or power level and a certain pulseshape, with these two usually being interrelated; and in which these solutions can either propagate with an unchanging pulseshape over an indefinite distance, or else display a slow periodic oscillation with distance through a set of recurring characteristic pulseshapes.
Depending on the particular nonlinear equation, the soliton pulses may have different shapes; and the velocities of propagation and the distances for periodic recurrence generally depend on both the nonlinear equation and the pulse amplitude.
(According to a more precise classification, any solution to a nonlinear equation which will propagate with unchanged shape, or will repeat its shape periodically in distance, is known as a solitary wave; but only those classes of solitary waves which can collide or pass through each other and then resume their solitary propagation without change of shape after such a collision are called solitons. Hence not all solitary-wave solutions are solitons.)
Three nonlinear equations which are known to have soliton solutions are the Korteweg-deVries equation, the Sine-Gordon equation, and the nonlinear Schrodinger equation already introduced in this chapter. Equations like these arise in many physical situations, including shallow- and deep-water wave propagation, waves in plasmas, lattice waves in solids, superconductivity, vortex motions in liquids, and propagation in optical fibers.
Solitons in Optical Fibers
At wavelengths longer than \(\lambda_0\approx\) 1.35 μm, the group-velocity dispersion in quartz single-mode optical fibers has the appropriate sign (namely, \(\beta^"\lt0\)) such that the chirp produced by self-phase modulation through the optical Kerr effect will lead, at least at first, to time-compression of the central part of the optical pulse, rather than to pulse broadening as described in a preceding section.
A smooth pulse of sufficient amplitude will thus be steadily compressed, and also progressively distorted in shape, at least until it becomes sufficiently short that higher-order nonlinear effects begin to compete with the dispersive pulse compression.
It is found, in fact, that such a pulse can approach a limiting pulseshape which does not change further with distance, and which represents in fact the lowest-order soliton solution to the nonlinear Schrödinger equation that governs the nonlinear wave propagation in the fiber.
This solution has a sech dependence of the pulse amplitude on time, in the form
\[\tag{47}\mathcal{E}(z,t)=\mathcal{E}_0\text{sech}\left(\frac{t-t_0-z/v_g'}{\tau_0}\right)\exp[j(\Omega{t}-\kappa{z})]\]
where the peak amplitude \(\mathcal{E}_0\), the pulsewidth \(\tau_0\), the modified group velocity \(v_g'\), the (small) frequency shift \(\Omega\), and the wavenumber shift \(\kappa\), are all interrelated.
Different values for these parameters thus describe a continuous family of solutions that have the same basic shape but different amplitudes and pulsewidths, that carry different amounts of energy per pulse, and that travel at very slightly different group velocities.
For example, such a pulse in a single-mode fiber might have a width of 3 ps and a peak power of 100 mW, and thus carry a total energy of \(\approx\) 0.3 pJ. Lower peak powers mean longer pulsewidths, and the converse.
Higher-Order Soliton Solutions
There also turn out to be higher-order soliton solutions, characterized by an index \(N\ge1\), which do not propagate with constant shape, but which instead, if launched with a proper initial shape and amplitude will return to that same initial shape at periodic distances along the fiber. Analytical solutions for these periodically recurring solitons are difficult to obtain, and they are often studied by means of large-scale numerical simulations.
These higher-order solutions generally require higher amplitudes and energies than the lowest-order soliton, and the soliton period generally decreases with increasing pulse amplitude. They have also now been seen experimentally.
Fermi-Pasta-Ulam Recurrence
One of the counterintuitive properties of these higher-order periodic soliton solutions in certain systems—including optical fibers—is that the frequency spectrum of the nonlinear pulse signal, starting from a narrow and even transform-limited initial pulse, can first broaden out substantially with distance (or with time), but then can later condense back again to the same narrow initial spectrum.
This seems quite counter to what might be an initial expectation, that nonlinear and intermodulation effects should generally always act to continually broaden a signal spectrum.
This spectral broadening and subsequent recondensation is sometimes referred to as "Fermi-Pasta-Ulam recurrence"; these three men carried out an early set of calculations on the first large-scale computers at Los Alamos in order to trace the long-term dynamics of a computer model of nonlinear springs and discrete masses having many nonlinearly coupled resonant modes.
Instead of displaying a long-term trend from initial order toward eventual quasi random thermalization, as had been expected, these computer simulations frequently revealed a mysterious periodic recurrence behavior, which was not understood, and which is now sometimes explained as the excitation of periodically recurring solitons in the nonlinear system.
The Soliton Laser
The circulating pulses in some of the narrowest-pulse mode-locked lasers are in fact probably solitons, in the sense that nonlinear chirping and dispersion in the various laser elements begins to play a significant role in the reshaping of the pulses. This reshaping can be beneficial, if it leads to narrower pulses, or deleterious, as in the pulse break-up effects discussed earlier. There is an even more interesting way of using solitons in a mode-locked laser, which is accomplished as follows.
Suppose a length of fiber is connected to one end of a laser cavity in such a fashion that a pulse can come out the end of the laser, propagate down the fiber and reflect at the far end, and come back and enter the laser cavity again; and suppose the fiber length corresponds to the periodic recurrence or reshaping distance for a certain optical soliton.
If the laser operates at a wavelength where the fiber supports soliton propagation, and if the laser energy is properly adjusted, it is possible for the pulse to travel through the laser medium as a comparatively wide pulse in time, with a correspondingly narrow bandwidth that remains within the amplification bandwidth of the laser medium; but then it enters the fiber to propagate as a higher-order soliton (\(N\gt1\)).
As the pulse travels down the fiber and back, it can thus narrow in time and broaden in bandwidth, and then reverse this process as it returns back to the laser. This makes it possible to generate mode-locked pulses which are much narrower than can normally be supported by the finite amplification bandwidth of the mode-locked laser medium. This important (and very recent) development is referred to as the "soliton laser."
The next tutorial introduces polarization and nonlinear impairments in fiber communication systems.