Linear Pulse Propagation
This is a continuation from the previous tutorial - what are hollow-core fibers?
Extraordinarily short optical pulses can be generated in mode-locked lasers, and these pulses can then be amplified to very large energies in subsequent laser amplifiers. Such pulses can be used for laser ranging (laser radar, or lidar); for pulse-modulated optical communications, both in free space and especially along optical fibers; and as measurement probes for studying a very wide variety of ultrafast physical, chemical, and biological processes, in what has come to be known as picosecond spectroscopy.
Pulse propagation both in passive optical propagation systems and in laser amplifiers is therefore a subject of considerable practical interest. Understanding the propagation of optical pulses through both linear and nonlinear systems is also important in mode-locked lasers, in optical fibers and other propagation systems, and in picosecond spectroscopic applications.
In this tutorial we first introduce some of the fundamental ideas of pulse propagation in linear systems, including the concepts of group and phase velocities, and pulse compression and broadening in linear dispersive systems and laser amplifiers.
In the following tutorial we will discuss some elementary concepts in the amplification and distortion of optical pulses caused by saturation in laser amplifiers, and also some of the interesting and useful effects, such as nonlinear pulse compression and soliton propagation, that occur in nonlinear dispersive fibers and other propagation systems.
1. Phase and Group Velocities
In this section we will analyze some of the fundamental effects that can arise in pulse propagation through linear systems, especially systems with either group velocity or gain dispersion. Fundamental concepts we will examine include pulse delay, group velocity, and pulse compression or group velocity dispersion effects.
Gaussian Pulses
The concepts we will introduce in this section occur for pulses of any shape. The analysis of these effects becomes particularly simple, however, with little if any of the physics being lost, if we analyze these effects using primarily gaussian pulses. Such pulses are simple, mathematically tractable, and clearly exhibit all the essential physical features. Many real systems such as actively mode-locked lasers generate pulses that are very close to complex gaussian pulses.
We consider as our basic model, therefore, an optical pulse with a carrier frequency \(\omega_0\) and a complex gaussian envelope written in the form
\[\tag{1}\mathcal{E}(t)=\exp(-at^2)\exp{j}(\omega_0t+bt^2)=\exp(-\Gamma{t^2})\exp{j\omega_0t}\]
The complex gaussian parameter describing this pulse is thus
\[\tag{2}\Gamma\equiv{a-jb}\]
(This T has nothing to do with the axial wave propagation constant F we used in an earlier tutorials.) The instantaneous intensity \(I(t)\) associated with this pulse can be written as
\[\tag{3}I(t)=|\mathcal{E}(t)|^2=\exp(-2at^2)=\exp[-(4\ln2)(t/\tau_p)^2]\]
so that the pulsewidth \(\tau_p\), defined in the usual FWHM fashion, is related to the parameter \(a\) by
\[\tag{4}\tau_p=\sqrt{\frac{2\ln2}{a}}\]
Note that this is the FWHM pulsewidth for the intensity \(I(t)\), and not for the signal amplitude \(\mathcal{E}(t)\).
Instantaneous Frequency
The time-varying phase shift or phase rotation of the sinusoidal signal within this gaussian pulse is given by
\[\tag{5}\mathcal{E}(t)\propto\exp{j}(\omega_0t+bt^2)=\exp[j\phi_\text{tot}(t)]\]
so that the total instantaneous phase of the signal is
\[\tag{6}\phi_\text{tot}(t)=\omega_0t+bt^2\]
What then is the "instantaneous frequency" \(\omega_i(t)\) of this sinusoidal signal?
The total phase variation in this case can obviously be written in the form \(\phi_\text{tot}=\omega_0t+bt^2=(\omega_0+bt)t\). We might therefore be led to say that the instantaneous frequency of the pulse at time \(t\) should be written as \(\omega_i(t)=\omega_0+bt\). This is not, however, a correct interpretation.
The instantaneous radian frequency of an oscillating signal, in the way this term is usually interpreted, should instead be the rate at which the total phase of the sinusoidal signal rotates forward, or alternatively \(2\pi\) times the number of cycles completed per unit time, as measured in any small time interval.
In other words, the instantaneous frequency in radians per second is properly defined as
\[\tag{7}\omega_i(t)\equiv\frac{d\phi_\text{tot}(t)}{dt}\]
In the complex gaussian case this gives
\[\tag{8}\omega_i(t)\equiv\frac{d}{dt}(\omega_0t+bt^2)=\omega_0+2bt\]
The factor of 2 in the time-varying part of this expression is important.
A gaussian pulse with a nonzero imaginary part \(b\) thus has a linearly time-varying instantaneous frequency. Such a signal is often said to be chirped, with the parameter \(b\) being a measure of this chirp. Figure 9.1 shows a rather strongly chirped gaussian pulse with a sizable variation of the instantaneous frequency within the pulse.
Gaussian Pulse Spectrum
One of the major virtues of the gaussian pulse approach is that a gaussian pulse in time immediately Fourier-transforms into a gaussian spectrum in frequency, in the form
\[\tag{9}\mathcal{E}(t)=\exp(-\Gamma{t^2}+j\omega_0t)\quad\Rightarrow\quad\tilde{E}(\omega)=\exp\left[-\frac{(\omega-\omega_0)^2}{4\Gamma}\right]\]
The exponent in the Fourier transform thus has a complex-quadratic dependence on frequency of the form
\[\tag{10}\tilde{E}(\omega)=\exp\left[-\frac{1}{4}\left(\frac{a}{a^2+b^2}\right)(\omega-\omega_0)^2-j\frac{1}{4}\left(\frac{b}{a^2+b^2}\right)(\omega-\omega_0)^2\right]\]
A signal with a linear frequency chirp (or quadratic phase chirp) in time automatically also has a quadratic imaginary component or quadratic phase shift of its Fourier spectrum in frequency, as given by the \(b/(a^2+b^2)\) factor.
The power spectrum, or power spectral density, of this pulse is then given by
\[\tag{11}\begin{align}|\tilde{E}(\omega)|^2&=\exp\left[-\frac{1}{2}\left(\frac{a}{a^2+b^2}\right)(\omega-\omega_0)^2\right]\\&=\exp\left[-(4\ln2)\left(\frac{\omega-\omega_0}{\Delta\omega_p}\right)^2\right]\end{align}\]
where \(\Delta\omega_p\) is the FWHM spectral width (in radians/second) of the pulse. We can convert this into a pulse bandwidth measured in Hz by writing
\[\tag{12}\Delta{f}_p\equiv\frac{\Delta\omega_p}{2\pi}=\frac{\sqrt{2\ln2}}{\pi}\sqrt{a[1+(b/a)^2]}\]
For a given pulsewidth in time as determined by the real parameter \(a\), the presence of a frequency chirp as determined by the imaginary parameter \(jb\) increases the spectral bandwidth \(\Delta\omega_p\) by a ratio \(\sqrt{1+(b/a)^2}\), as compared to an unchirped pulse with the same pulsewidth in time.
Time-Bandwidth Products, and Transform-Limited Pulses
Combining the preceding equations shows in fact that the gaussian pulse has a time-bandwidth product given by
\[\tag{13}\Delta{f_p}\tau_p=\left(\frac{2\ln2}{\pi}\right)\times\sqrt{1+(b/a)^2}\approx0.44\times\sqrt{1+(b/a)^2}\]
The minimum or unchirped value of time-bandwidth product for a gaussian pulse is thus \(\Delta{f}_p\tau_p\approx0.44\). The presence of chirp increases this time-bandwidth product to a value given, in the limit of large chirp, by \(\approx{b/a}\) times the minimum value.
This particular time-bandwidth product is the gaussian-pulse, FWHM version of a general Fourier theorem which says the time-bandwidth product of any pulsed signal is constrained by the uncertainty principle \(\Delta{f}_\text{rms}\Delta{t}_\text{rms}\ge1/2\), where \(\Delta{f}_\text{rms}\) and \(\Delta{t}_\text{rms}\) are the root-mean-square widths of the signal in frequency and in time.
If one uses the rms definitions of \(\Delta{f}\) and \(\Delta{t}\), the time-bandwidth product for a chirped gaussian pulse is in fact the same as Equation 9.13, except that the 0.44 factor is replaced by exactly 0.5. More generally, the exact value of time-bandwidth product \(\Delta{f}\Delta{t}\) for an arbitrary pulse shape depends on:
- The exact shape of the pulse (gaussian, square, exponential, etc.);
- How \(\Delta{f}\) and \(\Delta{t}\) are defined (rms, FWHM, etc.); and
- Especially on the amount of chirp or other amplitude or phase substructure within the pulse.
Pulses with little chirp or other internal substructure will have a time-bandwidth product close to the value of \(\approx0.5\). Such pulses are often referred to as transform-limited pulses. If separate measurements of pulsewidth and spectral width on a pulsed signal give a time-bandwidth product close to this limit, the pulsed signal must have little or no amplitude or phase substructure within the pulse duration.
Dispersive Systems and "Omega-Beta Curves"
Consider now a dispersive atomic medium, or any other kind of dispersive wave-propagating system, such as a transmission line, waveguide, or optical fiber. By "dispersive" in this context we mean any linear system in which the propagation constant \(\beta(\omega)\) as a function of frequency has any form other than a straight line through the origin, i.e., \(\beta=\omega/c\).
We have shown plots in earlier tutorials of the propagation constant \(\beta(\omega)\), or the total phase shift \(\phi(\omega)=\beta(\omega)L\) plotted versus frequency \(\omega\) for various atomic systems.
In discussing dispersive systems, however, it is convenient to plot \(\omega\) versus \(\beta\), rather than \(\beta\) versus \(\omega\), as shown in Figure 9.2.
Such an "omega-beta plot" may represent the dispersive effect of an atomic transition or of the background index in a host medium, in which case it is called material dispersion. Alternatively, it may represent the propagation characteristics of a guided mode in some waveguiding system such as a microwave waveguide, an optical fiber, or a general filter network, in which case the dispersion is commonly referred to as waveguide dispersion or modal dispersion.
Suppose we are concerned with narrowband signals having frequency components primarily near some center frequency \(\omega_0\). Then the propagation constant of a dispersive system can be conveniently expanded about its value at \(\omega_0\) in the form
\[\tag{14}\beta(\omega)=\beta(\omega_0)+\beta'\times(\omega-\omega_0)+\frac{1}{2}\beta^"\times(\omega-\omega_0)^2\]
where the derivatives \(\beta'\equiv{d\beta}/d\omega\) and \(\beta^"\equiv{d^2\beta}/d\omega^2\) are both evaluated at \(\omega=\omega_0\).
Besides the frequency-dependent propagation constant \(\beta(\omega\), we might at the same time consider the effects of a frequency-dependent gain or loss coefficient \(\alpha=\alpha(\omega)=\alpha(\omega_0)+\alpha'\times(\omega-\omega_0)+\frac{1}{2}\alpha^"\times(\omega-\omega_0)^2\) in the same system.
Both of these frequency variations will distort or modify a pulse propagating through the system. We wish to focus at this point, however, on pulse propagation and pulse-compression phenomena due only to velocity dispersion. We will therefore ignore for now any frequency-dependent gain coefficient, and assume that any gains or losses are either zero or at least independent of frequency.
Gaussian Pulse Propagation Through a Dispersive System
Suppose then that we put a gaussian pulse of the form
\[\tag{15}\mathcal{E}_0(t)=\exp(-\Gamma_0t^2+j\omega_0t),\qquad\tilde{E}_0(\omega)=\exp\left[-\frac{(\omega-\omega_0)^2}{4\Gamma_0}\right]\]
into such a dispersive system, where \(\Gamma_0\equiv{a_0}-jb_0\) is the initial pulse parameter at the input to the system. The output pulse spectrum \(\tilde{E}(z,\omega)\) after propagating a distance \(z\) through such a system will be the input spectrum \(\tilde{E}_0(\omega)\) of Equation 9.9, multiplied by the frequency-dependent propagation through the system, or
\[\tag{16}\begin{align}\tilde{E}(z,\omega)&=\tilde{E}_0(\omega)\times\exp[-j\beta(\omega)z]\\&=\exp\left[-j\beta(\omega_0)z-j\beta'z\times(\omega-\omega_0)-\left(\frac{1}{4\Gamma_0}+\frac{j\beta^"z}{2}\right)\times(\omega-\omega_0)^2\right]\end{align}\]
The output pulse in time from this system will be the inverse Fourier transform of the output spectrum, or
\[\tag{17}\mathcal{E}(z,t)\equiv\int_{-\infty}^{\infty}\tilde{E}(z,\omega)e^{j\omega{t}}d\omega\]
With some minor manipulations, this integral can be put into the form
\[\tag{18}\mathcal{E}(z,t)=\frac{e^{j[\omega_0t-\beta(\omega_0)z]}}{2\pi}\int_{-\infty}^{\infty}\left[-\frac{(\omega-\omega_0)^2}{4\Gamma(z)}+j(\omega-\omega_0)(t-\beta'z)\right]d(\omega-\omega_0)\]
where \(1/\Gamma(z)\equiv1/\Gamma_0+2j\beta^"\).
In this form, the carrier-frequency time and space dependence have been moved out in front, so that the integral part of this expression gives the time and space dependence of the output pulse envelope. The output pulse is still a gaussian pulse, but with an altered gaussian pulse parameter \(\Gamma(z)\) at the output of the system.
To interpret this mathematical result, we can carry out the integration explicitly using "Siegman's lemma," namely,
\[\tag{19}\int_{-\infty}^{\infty}e^{-Ay^2-2By}dy\equiv\sqrt{\frac{\pi}{A}}e^{B^2/A},\qquad\text{Re}[A]\gt0\]
or we can simply note that the integral in Equation 9.18 is obviously the Fourier transform of a gaussian pulse of the form \(\exp(-\Gamma{t}^2)\), with a shift in time by \(t-\beta'z\) included.
From either approach, the output pulse after traveling any arbitrary distance \(z\) through the system is given by
\[\tag{20}\begin{align}\mathcal{E}(z,t)&=\exp[j(\omega_0t-\beta(\omega_0)z)]\times\exp[-\Gamma(z)\times(t-\beta'z)^2]\\&=\exp\left[j\omega_0\left(t-\frac{z}{v_\phi(\omega_0)}\right)\right]\times\exp\left[-\Gamma(z)\times\left(t-\frac{z}{v_g(\omega_0)}\right)^2\right]\end{align}\]
where \(\Gamma(z)\) is the modified gaussian pulse parameter after traveling a distance \(z\), and where \(v_\phi(\omega_0)\equiv\omega_0/\beta(\omega_0)\) and \(v_g(\omega_0)\equiv1/\beta'(\omega_0)\).
Phase Velocity
The first exponent in each line of Equation 9.20 says that in propagating through the distance \(z\), the phase of the sinusoidal carrier frequency \(\omega_0\) is delayed by a midband phase shift \(\beta(\omega_0)z\), or by a midband phase delay \(t_\phi\) (in time) given by
\[\tag{21}\text{phase delay},\;t_\phi=\frac{z}{v_\phi(\omega_0)}=\frac{\beta(\omega_0)}{\omega_0}z\]
This says that the carrier-frequency cycles, or the sinusoidal waves within the pulse envelope, will appear to move forward with a midband phase velocity \(v_\phi(\omega_0\) given by
\[\tag{22}\text{phase velocty},\;v_\phi(\omega_0)=\frac{z}{t_\phi}=\frac{\omega_0}{\beta(\omega_0)}\]
The midband phase velocity is thus determined by the propagation constant \(\beta(\omega_0)\) at the carrier frequency \(\omega_0\).
Group Velocity
The second exponent in each line of Equation 9.20 says, however, that the pulse envelope, which remains gaussian but with a modified pulse parameter \(\Gamma(z)\), is delayed by the group delay time \(t_g\) given by
\[\tag{23}\text{group delay},\;t_g=\frac{z}{v_g(\omega_0)}=\beta'z\]
That is, the pulse envelope appears to move forward with a midband group velocity \(v_g(\omega_0)\) given by
\[\tag{24}\text{group velocity},\;v_g(\omega_0)=\left.\frac{1}{(d\beta/d\omega)}\right|_{\omega=\omega_0}=\left(\frac{d\omega}{d\beta}\right)_{\omega=\omega_0}\]
If we could take instantaneous "snapshots" of the pulse fields \(\mathcal{E}(z,t)\) from Figure 9.1 as the pulse propagated through the system, we would see the (invisible) pulse envelope moving forward at the group velocity \(v_g(\omega_0)\), while the individual cycles within the pulse envelope moved forward at the phase velocity \(v_\phi(\omega_0)\equiv\omega_0/\beta(\omega_0)\).
For \(v_g(\omega_0)\lt{v}_\phi(\omega_0)\), for example, we would appear to see cycles of the carrier frequency walking into the pulse envelope from the back edge and disappearing out through the front edge of the pulse envelope, while the envelope itself moved forward at a slower velocity.
Pulse Compression
Finally, the gaussian pulse parameter \(\Gamma(z)\) at distance \(z\), relative to the value \(\Gamma_0\) at the input, is given from Equations 9.16 and 9.18 by
\[\tag{25}\frac{1}{\Gamma(z)}=\frac{1}{\Gamma_0}+2j\beta^"z\]
The change in \(\Gamma(z)\) is determined by \(\beta^"\), the second derivative of the propagation constant at line center. We will discuss the meaning of this formula, which includes pulse compression in particular, in much more detail in the following section.
Summary
The successive coefficients in the power series expansion of \(\beta(\omega)\) thus have the meanings
\[\tag{26}\begin{align}\beta&\equiv\beta(\omega)|_{\omega=\omega_0}=\frac{\omega_0}{v_\phi(\omega_0)}\equiv\frac{\omega_0}{\text{phase velocity}}\\\beta'&\equiv\left.\frac{d\beta}{d\omega}\right|_{\omega=\omega_0}=\frac{1}{v_g(\omega_0)}\equiv\frac{1}{\text{group velocity}}\\\beta^"&\equiv\left.\frac{d^2\beta}{d\omega^2}\right|_{\omega=\omega_0}=\frac{d}{d\omega}\left(\frac{1}{v_g(\omega)}\right)\equiv\begin{split}\text{"group velocity}\\\text{dispersion"}\end{split}\end{align}\]
The physical interpretation of these coefficients in terms of group and phase velocities, although derived here using gaussian pulses, is very general, and applies to any sort of pulse signal.
If a pulsed signal has a carrier frequency \(\omega_0\) within a pulse envelope of any shape, and this pulse propagates through any lossless linear system that has a midband propagation constant \(\beta(\omega_0)\) and a first-order linear variation \(\beta'\times(\omega-\omega_0)\) across the pulse spectrum, then the carrier-frequency cycles within the pulse will move forward at the phase velocity \(v_\phi\), while the pulse envelope itself will move forward at the group velocity \(v_g\) evaluated at the center of the pulse spectrum. The pulse envelope itself may also change in shape with distance because of the \(\beta^"\) term, as we will discuss in the following section.
2. The Parabolic Equation
There is an alternative and somewhat more general way to derive the linear pulse propagation results we are presenting in this chapter, by using the so-called "parabolic wave equation."
This parabolic equation is widely used in the professional literature, and it also brings out an interesting analogy between dispersive pulse spreading and diffractive optical beam spreading.
We will therefore give a brief derivation of the parabolic equation in this section, although we will not make any further direct use of it here.
Derivation of the Parabolic Equation
The basic wave equation for a one-dimensional signal in a dispersive medium, or on a dispersive transmission line, may be written as
\[\tag{27}\frac{\partial^2\mathcal{E}(z,t)}{\partial{z^2}}-\mu_0\epsilon_0\frac{\partial^2\mathcal{E}(z,t)}{\partial{t^2}}=\mu\frac{\partial^2p(z,t)}{\partial{t^2}}\]
where \(p(z,t)\) is the potentially dispersive but linear polarization of the medium or transmission line. (In more sophisticated problems, a nonlinear polarization may be included here also.) Suppose we write this field \(\mathcal{E}(z,t)\) in the form
\[\tag{28}\mathcal{E}(z,t)\equiv\text{Re}\tilde{E}(z,t)e^{j[\omega_0t-\beta(\omega_0)z]}\]
where \(\omega_0\) is again a carrier or midband frequency for the signal, with propagation constant \(\beta(\omega_0)\) at this midband frequency, and \(\tilde{E}(z,t)\) is taken to be the complex envelope of the pulsed signal.
We can write the polarization \(p(z, t)\) in terms of its Fourier transform \(\tilde{P}(z,\omega)\) in the form
\[\tag{29}p(z,t)=\frac{1}{2\pi}\int_{-\infty}^{\infty}\tilde{P}(z,\omega)e^{j\omega{t}}d\omega\]
Assume this polarization arises from a linear but possibly dispersive response in the medium or transmission line. Then, we may write it in terms of the electric field in the form
\[\tag{30}\tilde{P}(z,\omega)=\tilde{\chi}(\omega)\epsilon_0\tilde{E}(z,\omega)\]
where \(\tilde{E}(z,\omega)\) is the Fourier transform of \(\mathcal{E}(z,t)\) given by
\[\tag{31}\begin{align}\tilde{E}(z,\omega)&=\int_{-\infty}^{\infty}\mathcal{E}(z,t)e^{-j\omega{t}}dt\\&=\int_{-\infty}^{\infty}\tilde{E}(z,t)e^{j(\omega_0-\omega)t}dt\end{align}\]
and where \(\tilde{\chi}(\omega)\) is the dispersive susceptibility of the propagation system.
By using these definitions, plus standard Fourier transform theorems, we can write the polarization term on the right-hand side of Equation 9.27 as
\[\tag{32}\begin{align}\frac{\partial^2p(z,t)}{\partial{t^2}}&=-\frac{1}{2\pi}\int_{-\infty}^{\infty}\omega^2\tilde{P}(z,\omega)e^{j\omega{t}}d\omega\\&=-\frac{\epsilon_0}{2\pi}\int_{-\infty}^{\infty}\omega^2\tilde{\chi}(\omega)e^{j\omega{t}}d\omega\int_{-\infty}^{\infty}\tilde{E}(z,t')e^{j[\omega_0t'-\beta(\omega_0)z]}dt'\end{align}\]
The derivation of the parabolic equation then proceeds by expanding the quantity \(\omega^2\tilde{\chi}(\omega)\) in Equation 9.32 about its midband value in the form
\[\tag{33}\begin{align}\omega^2\tilde{\chi}(\omega)\approx\omega_0^2\tilde{\chi}(\omega_0)&+\frac{d}{d\omega}[\omega^2\tilde{\chi}(\omega)]\times(\omega-\omega_0)\\&+\frac{1}{2}\frac{d^2}{d\omega^2}[\omega^2\tilde{\chi}(\omega)]\times(\omega-\omega_0)^2+\ldots\end{align}\]
with all derivatives evaluated at \(\omega=\omega_0\).
This is of course exactly the same approximation as in the expansion of \(\beta(\omega)\) in the previous section. It is then possible to evaluate the polarization integral of Equation 9.32 by making use of the convenient identities
\[\tag{34}\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{j(\omega-\omega_0)(t-t')}d\omega\equiv\delta(t-t')=\delta(t'-t)\]
as well as
\[\tag{35}\frac{1}{2\pi}\int_{-\infty}^{\infty}(\omega-\omega_0)e^{j(\omega-\omega_0)(t-t')}d\omega\equiv{j}\delta^{(1)}(t-t')=-j\delta^{(1)}(t'-t)\]
and
\[\tag{36}\frac{1}{2\pi}\int_{-\infty}^{\infty}(\omega-\omega_0)^2e^{j(\omega-\omega_0)(t-t')}d\omega\equiv-\delta^{(2)}(t-t')=-\delta^{(2)}(t'-t)\]
In these relations, \(\delta^{(n)}(t)\) indicates an \(n\)-th order derivative of the Dirac delta function, with the useful property that
\[\tag{37}\int_{-\infty}^{\infty}\delta^{(n)}(t-t_0)f(t)dt\equiv\left.\frac{d^nf(t)}{dt^n}\right|_{t=t_0}\]
when applied to any reasonable function \(f(t)\). We will also use the various identities that
\[\tag{38}\beta^2(\omega)\equiv\omega^2\mu_0\epsilon_0+\mu_0\epsilon_0\omega^2\tilde{\chi}(\omega)\]
as well as
\[\tag{39}2\beta\frac{d\beta}{d\omega}\equiv2\omega\mu_0\epsilon_0+\mu_0\epsilon_0\frac{d}{d\omega}[\omega^2\tilde{\chi}(\omega)]\]
and
\[\tag{40}\left(\frac{d\beta}{d\omega}\right)^2+\beta\frac{d^2\beta}{d\omega^2}\equiv\mu_0\epsilon_0+\frac{\mu_0\epsilon_0}{2}\frac{d^2}{d\omega^2}[\omega^2\tilde{\chi}(\omega)]\]
where \([d\beta(\omega)/d\omega]_{\omega=\omega_0}\equiv{v_g}(\omega_0)\) is the midband group velocity in the system; and we will use the slowly varying envelope approximation to drop second derivatives of the pulse envelope \(\tilde{E}(z, t)\) with respect to distance \(z\).
When all these approximations are inserted and all the algebra is cleared away, the basic wave equation for \(\mathcal{E}(z,t)\) given in Equation 9.27 reduces to the parabolic equation for the pulse envelope \(\tilde{E}(z,t)\) given by
\[\tag{41}\frac{\partial\tilde{E}(z,t)}{\partial{z}}+\frac{1}{v_g}\frac{\partial\tilde{E}(z,t)}{\partial{t}}-j\frac{\beta^"}{2}\frac{\partial^2\tilde{E}(z,t)}{\partial{t^2}}=0\]
where \(v_g\) and \(\beta^"\) are both evaluated at midband, \(\omega=\omega_0\). This equation is called the parabolic equation both because of the parabolic expansion of \(\omega^2\tilde{\chi}(\omega)\) used in deriving it, and because of the second-derivative term in \(t\) which appears as a consequence of this approximation.
Group Velocity and Group Velocity Dispersion
We can note first that if the second-derivative term \(\beta^"=0\), then this equation is obviously satisfied by any solution of the form \(\tilde{E}(z,t)\equiv\tilde{E}(z-v_gt)\), where \(v_g\) is the midband value at \(\omega=\omega_0\) as defined in Equations 9.23. and 9.24.
This shows that the group velocity concept for propagation of the pulse envelope \(\tilde{E}(z,t)\) applies to much more than just the gaussian pulses discussed in the preceding section. For any reasonably narrowband pulsed or modulated signal, the pulse or modulation envelope moves forward at velocity \(v_g\), whereas the individual optical cycles move forward at velocity \(v_\phi\).
If the \(d^2\beta/d\omega^2\) term is nonzero, however, the propagation system will have a "group velocity dispersion," or a variation of group velocity with frequency, as we will discuss in the following sections.
The \(j(\beta^"/2)\partial^2\tilde{E}/\partial{t^2}\) term in Equation 9.41 then acts like a kind of generalized complex diffusion term for the pulse envelope \(\tilde{E}(z,t)\) in the time coordinate. This "complex-valued diffusion" leads to pulse broadening, pulse compression, and pulse reshaping effects that we will discuss in more detail in the following sections.
Alternative Form
There is also an alternative form for the parabolic equation, in which we begin with a pulseshape defined as
\[\tag{42}\begin{align}\mathcal{E}(z,t)&\equiv\text{Re}\tilde{E}(z,\eta)e^{j(\omega_0t-\beta(\omega_0)z)}\\&=\text{Re}\tilde{E}(z,t-z/v_g)e^{j(\omega_0t-\beta(\omega_0)z)}\end{align}\]
so that \(\eta\equiv{t}-z/v_g\) is a displaced time coordinate whose origin \(\eta=0\) is centered at the time of arrival of the pulse at each plane \(z\). The parabolic equation (9.41) then simplifies to the form
\[\tag{43}\frac{\partial\tilde{E}(z,\eta)}{\partial{z}}-\frac{j}{2}\left(\frac{d^2\beta}{d\omega^2}\right)\frac{\partial^2\tilde{E}(z,\eta)}{\partial\eta^2}=0\]
Obviously if the dispersion term \(d^2/d\omega^2\equiv0\), then the pulse shape \(\tilde{E}(z, \eta)\) becomes independent of \(z\), or \(\tilde{E}(z, \eta)\equiv\tilde{E}_0(\eta)\equiv\tilde{E}_0(t-z/v_g)\), as we have discussed. This form offers a slightly simpler way to express the same ideas.
Space-Time Analogy
The parabolic equation derived in this section has exactly the same mathematical form as the paraxial wave equation used in optical beam propagation analyses, if we identify the delayed time coordinate \(t-z/v_g\) (or \(\eta\)) in the parabolic equation with either of the transverse spatial coordinates \(x\) or \(y\) in the paraxial equation.
The dispersive or second-derivative term that leads to broadening (or compression) of a pulse's time envelope with distance \(z\) in the parabolic equation then plays exactly the same role as the diffractive term that leads to transverse spreading of a laser beam's transverse profile with distance \(z\) in the paraxial equation.
There is thus a very close analogy between signal pulse distortion with propagation distance in the dispersive equation, and changes in transverse beam profile with propagation distance due to diffraction effects in the paraxial situation.
An optical wavefront with positive or negative wavefront curvature (imaginary quadratic dependence on \(x\) or \(y\) in the exponent) is directly analogous to an optical signal with positive or negative chirp (imaginary quadratic dependence on \(t\) in the exponent); and this wavefront curvature may lead to a converging or diverging optical-beam profile, just as chirp may lead to pulse compression or expansion with distance.
As another example, an initially square signal pulse propagating on a dispersive transmission line will broaden into a \((\sin{t})/t\) pulseshape after a long enough distance, exactly as a uniform plane wave coming through a rectangular slit in the near field will broaden into a \((\sin{x})/x\) beam pattern in the far field. This general approach can give useful insights into the relationship between pulse-distortion effects on dispersive lines and beam-spreading effects in diffractive propagation.
3. Group Velocity Dispersion and Pulse Compression
If a pulse propagates through a system in which the group-velocity dispersion term \(\beta^"\times(\omega-\omega_0)^2\) has a significant amplitude, then we must consider not only the phase and group velocities as discussed in the preceding sections, but also the fact that the pulseshape itself will be significantly changed in propagating through the system.
Interesting and useful effects, such as pulse compression, pulse spreading, and pulse reshaping, can result from such second-order dispersion effects. Once again it is very convenient to derive and illustrate such effects using a chirped gaussian pulse model, as we will show in this section.
Gaussian Pulse Propagation
From the analysis of the preceding section, if we put a pulse with initial pulse parameter \(\Gamma_0=a_0-jb_0\) through a dispersive propagation system whose propagation constant has nonzero second derivative \(\beta^"\) at the carrier frequency of the pulse, then the change in the complex pulseshape parameter \(\Gamma(z)\) with propagation distance \(z\) through the system will be given by
\[\tag{44}\begin{align}\frac{1}{\Gamma(z)}&=\frac{1}{\Gamma_0}+2j\beta^"z=\frac{a_0}{a_0^2+b_0^2}+j\left(\frac{b_0}{a_0^2+b_0^2}+2\beta^"z\right)\\&=\frac{1}{a(z)-jb(z)}=\frac{a(z)}{a^2(z)+b^2(z)}+j\frac{b(z)}{a^2(z)+b^2(z)}\end{align}\]
This result can be interpreted graphically by noting that the quantity \(1/\Gamma(z)\) moves along a vertical straight-line trajectory in the complex \(1/\Gamma\) plane with increasing propagation distance \(2\beta^"z\), starting from an initial point \(1/\Gamma_0\), as indicated in Figure 9.3.
Since the real part of \(1/\Gamma(z)\) determines the pulse bandwidth, it is evident from the left-hand part of Figure 9.3 that the pulse bandwidth stays constant, as it obviously should do in the absence of gain narrowing.
This trajectory in the \(1/\Gamma\) plane can then be converted to a trajectory in the \(a-jb\) plane (or more conveniently in the \(a+jb\equiv\Gamma^*\) plane) by simply inverting each complex point through the origin. An inversion of this form always converts a straight line in the \(1/\Gamma\) plane to a circle in the \(\Gamma\) or \(\Gamma^*\) plane. Propagation through a distance \(2\beta^"z\) now represents propagation about an arc of this circle, as illustrated on the right-hand side of Figure 9.3.
Differential Approach
These same results can also be derived using a differential approach. If we differentiate the complex beam parameter \(\Gamma(z)\) with respect to distance along the system, we obtain the differential equation
\[\tag{45}\frac{d\Gamma(z)}{dz}=-2j\beta^"\times\Gamma^2(z)\]
and this in turn separates into the two equations
\[\tag{46}\frac{da(z)}{dz}=-4\beta^"a(z)b(z),\qquad\frac{db(z)}{dz}=2\beta^"[a(z)^2-b(z)^2]\]
The solutions to these equations are a family of circular trajectories, such as the trajectory indicated in Figure 9.3.
Pulse Compression
The pulse parameters at the output of a length \(z\) of the dispersive line are
\[\tag{47}a(z)=\frac{a_0}{(1+2\beta^"zb_0)^2+(2\beta^"za_0)^2}\]
and
\[\tag{48}b(z)=\frac{b_0(1+2\beta^"zb_0)+2\beta^"za_0^2}{(1+2\beta^"zb_0)^2+(2\beta^"za_0)^2}\]
The point where the trajectory of \(\Gamma(z)\) crosses the positive real axis in either the \(\Gamma\) or \(1/\Gamma\) plots obviously corresponds to the maximum value of the output parameter \(a(z)\), and hence to the minimum value of the output pulsewidth \(\tau_p(z)\).
As the pulse propagates from the initial value \(\Gamma_0\) to this point, the pulse is being compressed in width, or shortened in time, at least with the choice of parameters we have used in Figure 9.3. Beyond this point, the pulse broadens again in time. This pulse compression for a chirped pulse passing through a dispersive propagation system can be very useful in a wide variety of not only optical but also microwave and radio frequency applications, as we will see later.
Optimum Compression Length
Suppose we can adjust the total dispersion \(2\beta^"z\), by changing either the group velocity dispersion \(\beta^"\) or the distance \(z\) that is traveled. The size of this parameter is related to the distance traveled along the straight-line trajectory or, in a more complicated way, to the arc length traveled along the circle in Figure 9.3.
What dispersion length \(2\beta^"z\) is then needed to reach the minimum pulsewidth point, starting with a given input pulse parameter \(\Gamma_0\)? Differentiating the quantity \(a(z)\) with respect to the parameter \(2\beta^"z\) shows that the maximum value of \(a(z)\) occurs for an optimum propagation distance related to the input pulse parameters by
\[\tag{49}(2\beta^"z)_\text{opt}=-\frac{b_0}{a_0^2+b_0^2}\approx-\frac{1}{b_0}\quad\text{if }b_0\gg{a_0}\]
The output pulse parameters at this optimum distance are given by
\[\tag{50}a_\text{opt}=a_0[1+(b_0/a_0)^2]\approx{b_0^2}/a_0\qquad\text{and}\qquad{b}_\text{opt}\equiv0\]
After propagating an optimum distance through the system, the output pulse is compressed in time down to a minimum pulsewidth \(\tau_{p,\text{min}}\) which is related to its input pulsewidth \(\tau_{p0}\) and to its initial pulse parameters \(a_0\) and \(b_0\) by
\[\tag{51}\frac{\tau_{p,\text{min}}}{\tau_{p0}}=\sqrt{\frac{1}{1+(b_0/a_0)^2}}\approx\left|\frac{a_0}{b_0}\right|\qquad\text{if }b_0\gg{a_0}\]
A large initial chirp compared to the pulsewidth, or \(b_0\gg{a_0}\)—which is the same thing as a large initial time-bandwidth product—leads to the possibility of large pulsewidth compression (Figure 9.4); whereas an initial condition such that \(b_0\le{a_0}\) permits only a negligible pulse compression.
It is also evident that at the optimum compression point \(b_\text{opt}=0\), meaning that all the chirp has been removed. A gaussian pulse will be compressed all the way down to its minimum time-bandwidth product \(\Delta{f}_p\tau_{p,\text{min}}=0.44\) at the optimum point.
Again these results, although derived for a gaussian pulse, are in fact quite general conclusions, which by no means apply only to gaussian pulses. All signals with large initial time-bandwidth products are potentially compressible; signals with near-transform-limited initial time-bandwidth products are not.
Physical Interpretation
We can give a physical interpretation of the preceding results as follows. A quadratic variation in \(\beta(\omega)\) means that the group velocity, which is related to the linear variation of \(\beta(\omega)\), must itself vary significantly across the pulse spectrum.
The dispersion parameter \(\beta^"\) is related in fact to the so-called group-velocity dispersion \(dv_g(\omega)/d\omega\) by
\[\tag{52}\beta^"=\frac{d}{d\omega}\left(\frac{1}{v_g(\omega)}\right)=-\frac{1}{v_g^2(\omega_0)}\frac{dv_g(\omega)}{d\omega}\]
Hence the group velocity \(v_g(\omega)\) as a function of the frequency deviation of a signal away from \(\omega_0\) will be given by
\[\tag{53}v_g(\omega)\approx{v_g}(\omega_0)-\beta^"v_g^2(\omega_0)\times(\omega-\omega_0)\]
i.e., the group velocity itself is frequency dependent.
Consider then a strongly chirped pulse whose instantaneous frequency varies with time in the form \(\omega_i(t)=\omega_0+2bt\) at the input plane to a linear system. Instead of thinking of this as a single pulse with carrier frequency \(\omega_0\), let us mentally break this pulse up into a number of segments or subpulses, each with a slightly different carrier frequency, and hence a slightly different group velocity.
Suppose in particular that the center portion of the pulse, which has the central frequency \(\omega_0\), leaves \(z=0\) at \(t_0=0\), and travels a distance \(z\) with a group delay given by \(t_{d0}=z/v_g(\omega_0)\). Any other part of the pulse starting at some slightly earlier or later time \(t_1\) has an instantaneous carrier frequency \(\omega_1\approx\omega_0+2b(t_1-t_0)\). Hence we can, in a crude way, say that this other portion of the pulse will travel with slightly different group velocity \(v_g(\omega_1)\).
Let us assume that the chirp \(b_0\) is > 0, so that the instantaneous frequency \(\omega_1\) is greater than \(\omega_0\) for \(t_1\gt{t_0}\) (i-e., the part of the pulse that starts late). Then we can say that this part of the pulse travels at a velocity
\[\tag{54}v_{g1}\approx{v_g}(\omega_0)-2\beta^"v_g^2(\omega_0)b_0(t_1-t_0)\]
Hence it travels the distance \(z\) in a time
\[\tag{55}t_{d1}\approx\frac{z}{v_g(\omega_0)-2\beta^"v_{g0}^2b_0(t_1-t_0)}\approx\frac{z}{v_g(\omega_0)}[1-2\beta^"v_g(\omega_0)b_0(t_1-t_0)]\]
In order for the reduction in travel time for this portion of the pulse to just match the amount \(t_1-t_0\) by which it started late, so that it will exactly catch up with the center of the pulse, we should have
\[\tag{56}t_{d0}-t_{d1}=t_1-t_0\]
Substituting the above equations into this leads to the condition
\[\tag{57}2\beta^"L\approx-1/b_0\]
which is the same as the optimum result for large chirp given above.
We can thus view the pulse-compression process as one in which different parts of a chirped pulse, which start out down the line at slightly earlier or later times, also have slightly different frequencies. They can then travel slightly more slowly or rapidly down the line because of group-velocity dispersion, in such a way that they just exactly catch up with the central portion of the pulse.
Pulse Compression With Other Pulseshapes: Chirp Radars
We have discussed pulse compression for the analytically tractable case of a gaussian pulse. In other cases, however, it may be necessary to work with other pulseshapes.
In certain radar systems, for example, it is easy to generate rectangular pulses which have constant output amplitude during a long time duration, thus combining low peak power with large total energy per pulse. Suppose we then give these same pulses a linear frequency chirp within the pulse as shown in Figure 9.5. A pulse like this, after propagating through a properly designed dispersive system, can also be substantially compressed in time (by roughly its initial timebandwidth product). However it will also be distorted in shape, and generally will acquire side-lobes something like a sine function, as illustrated in Figure 9.5.
Pulses much like this are often used in microwave chirped radar systems, since they can combine a comparatively long low-intensity pulse, easily obtainable from a microwave transmitter, with the much sharper range resolution achieved by using substantial pulse compression in the microwave receiver. Such systems are commonly referred to as chirped radar systems. The name dates back to an early classified memo during World War II which described such systems under the title "Not With a Bang But a Chirp." The echo-locating properties of bats are also related to a form of sonic chirped radar.
Other Dispersive Optical Systems
In addition to dispersive atomic media, and to dispersive propagation effects in waveguiding systems such as optical fibers, various other dispersive optical systems have been invented and used, particularly to compress naturally or deliberately chirped laser pulses.
One such system is the Gires-Tournois interferometer (Figure 9.6). This device is simply a lossless etalon with a partially reflecting front surface and a 100% reflecting back surface, so that there is regenerative interference between the front and back surfaces. If the back surface is truly 100% reflecting, and the material between the surfaces is sufficiently lossless, then this device must have a reflectivity magnitude equal to unity at all frequencies.
The interference between the front and back surfaces leads, however, to a periodic phase-versus-frequency curve for the complex reflectivity, which varies periodically with the axial mode spacing of the etalon. Portions of this phase-versus-frequency curve can then exhibit the correct dispersion to be used as a pulse-compression method. It is physically obvious, however, that the interferometer itself must be physically very thin compared to the physical length of the laser pulse, or it will obviously break a single laser pulse into two or more multiply reflected pulses, rather than compressing it.
Another and much more useful dispersive system consists of a pair of gratings arranged as shown in Figure 9.7. Different frequencies or wavelengths have a different geometrical propagation distance through this system because they diffract from the gratings at slightly different angles. Such grating pairs have been used very successfully to compress short chirped optical pulses in several different experiments, as will be discussed later.
Still another kind of dispersive system is a sequence of prisms as shown in Figure 9.8. Since the angular dispersion from a prism is generally smaller than from a diffraction grating, prism systems generally produce considerably smaller dispersion effects than grating pairs. On the other hand the insertion losses are also much smaller with prisms. Systems such as Figure 9.8 can thus be placed inside laser cavities to provide small corrections to the round-trip group velocity dispersion which are important in controlling the mode-locking behavior in very short-pulse lasers.
Sign of the Group-Velocity Dispersion
The second derivative of the dispersion parameter \(\beta\) with respect to frequency can be related to other frequency or wavelength derivatives in the forms
\[\tag{58}\beta^"\equiv\frac{d^2\beta(\omega)}{d\omega^2}=\frac{4\pi^2c_0}{\omega^3}\frac{d^2n(\lambda_0)}{d\lambda_0^2}=-\frac{1}{v_g^2}\frac{dv_g(\omega)}{d\omega}\]
where \(c_0\) and \(\lambda_0\) are the velocity of light and the optical wavelength in free space, and \(n(\lambda_0)\) is the index versus wavelength in a dispersive medium.
The usual practice in optics texts has been to plot index of refraction \(n(\lambda_0)\) of a dispersive medium versus free-space wavelength \(\lambda_0\), and then to speak of regions where this plot is concave upward (positive \(d^2n/d\lambda_0^2\)) as regions of positive dispersion, and regions of opposite sign as regions of negative dispersion. Positive dispersion in this sense thus means positive values for both \(\beta^"\) and \(d^2n/d\lambda_0^2\), but a negative derivative for \(dv_g/d\omega\), i.e., for group velocity versus frequency. Most common optical materials exhibit positive dispersion in the visible region, turning to negative dispersion somewhere in the near infrared.
With the recent advent of extremely short (femtosecond-time-scale) pulses, as well as growing realization of the role that self-chirping and pulse compression play in the mode-locked lasers that generate these pulses, there is growing interest in low-loss dispersive systems which can generate dispersion of either sign. The prism configuration in Figure 9.8 can be designed to give either negative or positive dispersion, with the additional advantages that the beams enter and leave the prisms at Brewster's angle, and there is neither displacement nor deviation of the input and output beam paths.
4. Phase and Group Velocities in Resonant Atomic Media
Particularly strong and interesting dispersion effects can occur when signals are tuned close to the transition frequency of a narrow atomic resonance in an absorbing or amplifying atomic medium. In this section we give a brief description of these atomic dispersive effects, showing how they confirm both the absorption and especially the phase-shift properties of a resonant atomic transition.
Phase and Group Velocities Near Atomic Transitions
The total phase shift for a wave making a single pass through a laser amplifier (or an absorbing atomic medium) can be written as \(\exp[-j\phi_\text{tot}(\omega)]=\exp[-j(\beta+\Delta\beta_m)L]\), where the total phase shift consists of
\[\tag{59}\Delta\phi_\text{tot}(\omega)\equiv[\beta(\omega)+\Delta\beta_m(\omega)L]=\frac{\omega{L}}{c}+\frac{\beta{L}}{2}\chi'(\omega)\]
The first term gives the basic "free-space" phase shift \(\omega{L}/c\) through the laser medium, a phase shift which is large and linear in frequency; the second term is the small added shift \(\Delta\beta_m(\omega)L\) due to the atomic transition. The phase velocity \(v_\phi(\omega)\) of the wave in the medium is then given by
\[\tag{60}v_\phi(\omega)=\frac{\omega{L}}{\phi_\text{tot}(\omega)}\]
and the group velocity \(v_g(\omega)\) by
\[\tag{61}v_g(\omega)=\frac{L}{d\phi_\text{tot}(\omega)/d\omega}=\frac{L}{(d/d\omega)[\beta(\omega)+\Delta\beta_m(\omega)]}\]
The phase velocity in a medium with an atomic transition is thus given by
\[\tag{62}v_\phi(\omega)=\frac{\omega}{\beta(\omega)+\Delta\beta_m(\omega)}=\frac{c}{1+\chi'(\omega)/2}\]
and the group velocity is given by
\[\tag{63}v_g(\omega)=\frac{v_\phi(\omega)}{1-(\omega/v_\phi)(dv_\phi/d\omega)}\]
Figure 9.9 shows these quantities for a wave passing through a resonant amplifying laser medium, with the atomic or \(\chi'(\omega)\) contributions very much exaggerated.
The group velocity over the central portion of the amplifying bandwidth in a laser medium is slightly slower than the free-wave velocity in the medium. A physical explanation for this is that as a pulse travels through the medium, the leading edge of the pulse must first build up a coherent induced polarization in the inverted atomic transition, before this polarization can begin radiating back into the input pulse to amplify it.
This build-up, however, requires a short but finite build-up time, on the order of \(T_2\), as described in the preceding chapter. The leading edge of the pulse thus gets slightly "under-amplified" compared to the steady-state gain of the medium, and by similar arguments the continuing reradiation of the oscillating atoms slightly "over-amplifies" the trailing edge of the pulse. The net pulse envelope in an amplifying medium thus appears to travel slightly more slowly than the free-space wave velocity.
Group Velocities Faster Than the Velocity of Light?
The phase and group velocities in Figure 9.9 are associated with an amplifying atomic medium. Going from an amplifying to an absorbing medium will reverse the signs of both \(\chi^"\) and \(\chi'\), and thus reverse the sign of all the atomic phase-shift contributions in this figure.
The careful reader may then note that for a strongly absorbing atomic transition the group velocity at the center of the transition can apparently become faster than the velocity of light \(c\) in the host medium, and also the group velocity off line center can become negative for strong enough absorbing transitions. In fact, for a strong enough transition in a gas, \(v_g\) might even become greater than the velocity of light \(c_0\) in vacuum. But this would seem to be in serious conflict with a fundamental axiom of relativity, that no signal or information can ever be transmitted at a velocity greater than \(c_0\).
The resolution to this apparent paradox lies in the fact that the group velocity \(v_g\), as defined here, can significantly exceed \(c_0\) only at the center of a very strongly absorbing transition. But this is also a situation in which any incident signal will be both strongly attenuated and distorted by this attenuation. Both detailed calculations and experimental measurements for a pulsed signal sent into a strongly absorbing atomic medium, with a carrier frequency anywhere within the absorption line, have been carried out in recent years.
These results have shown that when a smooth pulse is sent into a strongly absorbing medium, the observed signal-pulse envelope will indeed appear to travel at very close to the group velocity \(v_g\equiv{d}\omega/d\beta\) in almost all cases, with very little distortion of the pulse shape, even when \(v_g\) has values that are greater than \(c_0\), or even negative (i.e., the peak of the pulse appears to come out of the absorbing sample before it goes in.
This is not a violation of special relativity, however, nor does it mean that the signal is transmitted at greater than the (vacuum) speed of light. The pulses used for all such calculations (and measurements) necessarily always have long tails that are weak but finite. In passing through the lossy medium, different portions of the pulse spectrum are attenuated and phase-shifted very differently, in such a way that the peak of pulse envelope appears to move faster than \(c\). In reality, however, the pulse is being severely modified under the envelope; and no part of it is actually moving faster than light.
When such calculations are done for an input signal with a sharp discrete leading edge, it is always found that no output ever emerges at the output face at a time earlier than the transit time \(L/c_0\) through the system at the vacuum velocity of light.
Group Velocities Much Slower Than Light
The dispersion in the wings of a strongly absorbing transition can produce a group velocity much less than the phase velocity \(c\) in the medium (and also obviously less than \(c_0\) in free space); and if the entire signal spectrum of a pulse is sufficiently far out in the wings, the net absorption for the pulse may simultaneously be made very small. With a sufficiently strong and narrow transition, one can thus obtain a very large slowing of the group velocity of a pulse, even while there is very small net absorption of the pulse energy.
As an example of this, cells filled with alkali vapor (such as rubidium or sodium) have been used to produce some rather large (though comparatively narrowband) dispersive effects at wavelengths on the sides of the strong resonance absorption lines in these atoms. In experiments done at the IBM Laboratories, Yorktown Heights, N.Y., the pulse-envelope group delay was measured when short pulses from a tunable dye laser were sent through a gas absorption cell one meter long. By using pulses whose center frequencies were located far out on the wings of a very strongly absorbing transition (i.e., by operating at \(|\omega-\omega_a|\gg\Delta\omega_a\)), the experimenters were able to obtain a power absorption loss of only \(\approx\) 20% per pass, and yet have a group velocity as slow as 1/10 to 1/15 the velocity of light.
9.5. Pulse Broadening and Gain Dispersion
Pulses can be broadened as well as compressed in a propagation system having significant group-velocity dispersion, and this broadening can be very important both in measurement applications of picosecond optical pulses and in the transmission of such pulses through optical fibers. In this section, therefore, we extend the discussion of the previous sections to cover pulse broadening in such dispersive systems.
In addition, pulses will also be broadened—and more rarely narrowed—in passing through systems with gain dispersion, that is, in passing through amplifiers with a finite bandwidth. In this section therefore we also consider the complementary effects of gain or absorption dispersion in a propagating system.
Dispersive Pulse Broadening
Gaussian pulses starting out in the lower half-plane in Figure 9.3 will be initially compressed as they propagate through the dispersive system. On the other hand, pulses starting out—or moving into—the upper half-plane will be broadened (unless, of course, the sign of \(\beta^"\) is reversed, in which case the arrowheads on the trajectories must be reversed).
Group-velocity dispersion can thus either compress a pulse with the right initial chirp, or broaden a pulse with the wrong initial chirp. In particular, a pulse with no initial chirp will begin to acquire a growing amount of chirp and then be broadened in any such system.
Consider for example an initially unchirped pulse, with \(b_0=0\). Its gaussian pulse parameter after propagating a distance \(z\) through a dispersive medium becomes
\[\tag{64}a(z)=\frac{a_0}{1+(2\beta^"z)^2a_0^2}\]
Reduction in \(a(z)\) means that the pulsewidth \(\tau_p(z)\) has broadened, as given by the expression
\[\tag{65}\tau_p^2(z)\equiv\frac{2\ln2}{a(z)}=\tau_{p0}^2+\left(\frac{(4\ln2)\beta^"z}{\tau_{p0}}\right)^2=[1+(z/z_D)^2]\times\tau_{p0}^2\]
The initial unchirped pulsewidth \(\tau_{p0}\) will increase by a factor of \(\sqrt{2}\) after a propagation distance \(z_D\) given by
\[\tag{66}z=z_D\equiv\frac{\tau_{p0}^2}{(4\ln2)\beta^"}\]
This "dispersion length" \(z_D\) is a kind of Rayleigh length for pulse broadening in time, analogous to the Rayleigh range for transverse beam spreading we will meet in a later tutorial.
It is convenient to rewrite this dispersion length in the form
\[\tag{67}z_D=\frac{(\omega_0\tau_{p0})^2}{8\ln2}\times\frac{\lambda}{D}\]
where the quantity
\[\tag{68}D\equiv\frac{\omega_0^2\beta^"}{\beta(\omega_0)}\]
is a dimensionless group-velocity dispersion parameter for a propagating system or an atomic medium. The first factor in the \(z_D\) expression depends only on pulse parameters—it is essentially the number of optical cycles in the pulse squared—and the second factor is the wavelength in the medium, \(\lambda=\lambda_0/n\), divided by the dimensionless dispersion parameter \(D\).
Dispersive Broadening in Real Materials
We might want to evaluate this dispersion parameter, for example, for an ultrashort optical pulse propagating through a typical optical material with a frequency-dependent index of refraction, so that the propagation constant is given by
\[\tag{69}\beta(\omega)=\frac{n(\omega)\omega}{c_0}\]
A common form for the variation of the index of the refraction across the visible region in transparent optical materials is the Sellmeier equation
\[\tag{70}n^2(\omega)-1=\frac{A\omega_e^2}{\omega_e^2-\omega^2}\]
For typical optical glasses and crystals \(A\) has a value between 1 and 2, and \(\omega_e\) corresponds to an effective resonant frequency or absorption band edge for the material, often located in the ultraviolet at a wavelength \(\lambda_e\) somewhere between 1000 and 2000 Å. (For semiconductors this wavelength generally corresponds to the band-gap wavelength, and the Sellmeier equation then gives the index of refraction for the semiconductor in the transparent region at wavelengths longer than the band-gap wavelength.)
We can then find that for many typical transparent materials the magnitude of the dimensionless dispersion parameter is given by
\[\tag{71}D\equiv\frac{\omega_0^2\beta^"}{\beta(\omega_0)}\approx0.10\text{ to }0.20\]
If we assume a value of \(D\approx0.10\) and an index of refraction \(n=1.5\), the approximate \(\sqrt{2}\) broadening lengths for initially unchirped pulses of different initial pulsewidths \(\tau_{p0}\) are given by
This type of dispersive pulse broadening becomes very important in the use of picosecond and femtosecond optical pulses from mode-locked lasers. Pulses \(\le\) 20 femtoseconds long have already been generated in mode-locked dye lasers. Such a pulse obviously cannot propagate more than a few mm through a typically dispersive medium before it will be significantly broadened by the self-broadening effect.
Dispersive Broadening in Optical Fibers
The pulse-broadening effects caused by group-velocity dispersion in optical fibers are also of great importance in determining the maximum distance that a pulse of given width can be propagated through an optical-fiber communications systems before being significantly broadened. A 100 ps visible-wavelength pulse, for example, can propagate only a few km in a typical single-mode fiber before being significantly broadened (the situation in multimode fibers is much worse, because of mode-mixing effects). Dispersive pulse broadening can become the primary factor limiting the potential data rate for long-distance communication in low-loss high-capacity optical fibers.
The dispersive behavior of an optical fiber is usually a combination of materials dispersion, associated with the index variation of the glass in the fiber, and waveguide or modal dispersion associated with the propagating normal mode patterns in the fiber. In typical fibers this net dispersion passes through zero at a wavelength around 1.3 μm, so that in principle very short pulses tuned to this wavelength could be propagated for very long distances without dispersive spreading.
It can be difficult to match the transmitted wavelength exactly to the zero dispersion point, however, and in addition the lowest-loss wavelength for optical fibers is typically closer to \(\lambda=1.5\) μm (at which wavelength the absorption and scattering losses in real fibers can have values as extraordinarily small as \(\approx\) 0.2 dB/km). To transmit a pulse with the minimum possible input and output pulsewidth through a given length of such a fiber, we should launch a pulse with an input pulsewidth \(\tau_p\equiv\sqrt{2}\tau_{p0}\) and just the right amount of initial compressive chirp into the fiber, where the fiber length is given by \(z=2z_D\), and \(\tau_{p0}\) and \(z_D\) are connected by the analytical relations given in Equations 9.65 and 9.66. This pulse will then compress down to \(\tau_{p0}\) at the middle of the fiber and broaden back to \(\tau_p\) at the output end. For a small enough value of normalized dispersion—perhaps \(D\approx0.005\), which might be typical of a single-mode quartz fiber at \(\lambda=1.5\) μm—the relationship between fiber length and minimum input-output pulsewidth will have typical values given by
Note that a data transmission rate of 10 Gbits per second, such as optical-fiber communications designers hope to achieve, requires a pulsewidth at least as short as 100 ps, and preferably somewhat less.
(One very interesting alternative approach for accomplishing the propagation of very much shorter optical pulses in fiber communications systems is the use of nonlinear solitons in optical fibers, as described briefly in the following tutorial.)
Pulse-Broadening Effects of Gain Dispersion
Velocity or phase-shift dispersion, which is produced by a frequency-dependent propagation constant \(\beta(\omega)\), causes one set of pulse distortion effects. Gain dispersion, by which we mean a frequency-dependent gain coefficient \(\alpha(\omega)\), produces a different set of effects. The primary effects that result when a pulse passes through a linear but frequency-dependent gain medium (now leaving out dispersion effects) are pulse-broadening in time, due to finite bandwidth of the gain medium, and possibly more complex frequency-shifting and time-shifting effects that can occur if the gain medium has a linear variation of gain with frequency across the pulse spectrum.
In this section we will consider only quadratic or pulse-broadening effects, since they are the most fundamentally important, leaving the more complex effects of linear frequency dependence to an exercise. We therefore consider again a gaussian pulse with carrier frequency \(\omega_0\) and initial pulse parameter \(\Gamma_0\), which now passes through a linear gain medium whose gain coefficient has the quadratic frequency dependence
\[\tag{74}\alpha_m(\omega)=\alpha_{m0}-\frac{1}{2}\alpha^"_m(\omega_0)\times(\omega-\omega_0)^2\]
where \(\alpha_m^"\equiv-d^2\alpha_m(\omega)/d\omega^2\) evaluated at midband. The first term in this expansion gives a uniform amplitude gain which applies equally to all frequency components and hence simply increases the pulse amplitude uniformly without changing its shape. The \(\alpha^"_m\) term, however, leads to a change in the gaussian pulse parameter given by
\[\tag{75}\frac{1}{\Gamma(z)}=\frac{1}{\Gamma_0}+2\alpha^"_m(\omega)z\]
The trajectory of \(\Gamma(z)\) now moves horizontally in the \(1/\Gamma\) plane, rather than vertically as in Figure 9.3. For \(\alpha_m^"\gt0\) (that is, a gain peak at line center) this increases the real part of \(1/\Gamma\) and hence, as is physically obvious, decreases the spectral bandwidth of the pulse. The result is commonly, though not universally, to broaden the pulse in time as it propagates through the amplifier.
Pulse Broadening in Amplifiers
Consider as a particular example a lorentzian atomic transition with linewidth \(\Delta\omega_a\) and a spectrum centered at \(\omega_0\equiv\omega_a\). The gain variation around line center may be written to a first approximation as
\[\tag{76}\alpha_m(\omega)=\frac{\alpha_{m0}}{1+[2(\omega-\omega_a)/\Delta\omega_a]^2}\approx\alpha_{m0}-\alpha_{m0}\times\left(\frac{2}{\Delta\omega_a}\right)^2\times(\omega-\omega_0)^2\]
A length \(z\) of such an amplifier then produces an output pulse parameter given by
\[\tag{77}\frac{1}{\Gamma(z)}=\frac{1}{\Gamma_0}+\frac{16\alpha_{m0}z}{\Delta\omega_a^2}\]
Laser amplifiers with finite bandwidths will then usually broaden pulses in time, just as occurs in any other finite bandpass system.
Consider, for example, an input pulse with no initial chirp, i.e., with \(\Gamma_0=a_0\) and \(b_0=0\). Equations 9.75-9.77 then convert into the simple pulsewidth broadening result
\[\tag{78}\tau_p^2(z)=\tau_{p0}^2+\frac{(16\ln2)\ln{G_0}}{\Delta\omega_a^2}\]
where \(G_0=\exp(2\alpha_{m0}z)\). In a Nd:YAG laser amplifier with \(\Delta\omega_a/2\pi\approx120\) GHz and the very high (possibly multipass) gain \(G_0= 10^5\), this gives approximately
\[\tag{79}\tau_p^2\approx\tau_{p0}^2+(\sim15\text{ ps})^2\]
Such an amplifier will convert an ideal delta-function input pulse into an ~15 ps output pulse, and will broaden a 50 ps input pulse to ~52 ps at its output.
Also, the initially unchirped pulse develops an added chirp in passing through the amplifier. These kinds of results are important in understanding the amplification of short pulses in a laser amplifier, and, as we will see in a later tutorial, in understanding mode-locking in laser oscillators.
Pulse Narrowing in a Chirped Laser Amplifier
Laser amplifiers with finite bandwidths can also, under certain special circumstances, shorten chirped pulses in time. Suppose the input pulse also has a significant initial chirp \(b_0\). The general result for the pulsewidth parameter \(a(z)\) after an amplification distance \(z\) is then
\[\tag{80}a(z)=\frac{a_0(1+Ka_0)+Kb_0^2}{(1+Ka_0)^2+(Kb_0)^2}\]
where \(K\equiv2\alpha_m^"z=8\ln{G_0}/\Delta\omega_a^2\). Suppose for simplicity that the bandwidth-broadening factor \(K\) is small compared to \(1/\Gamma_0\) or \(1/\Gamma\). Equation 9.80 expanded to first order in \(K\) then becomes
\[\tag{81}a(z)\approx{a_0}[1-Ka_0+Kb_0^2/a_0]\]
This shows the first-order pulsewidth-broadening effect due to the \(-Ka_0\) term, but also a pulsewidth-narrowing term in the \(Kb_0^2/a_0\) term.
The physical interpretation of this effect is the following. If the pulse has a sizable chirp during its time duration (i.e., \(b_0\gg{a_0}\)), we may think approximately of the pulse frequency sweeping across the gain profile of the amplifying transition. The center section of the pulse (in time) is at line center and hence gets maximum amplification, whereas the frequencies in both the leading and trailing edges of the pulse are somewhat off line center and get less amplification; hence the pulseshape gets somewhat narrowed in time.
This last explanation mingles time and frequency descriptions in a way that is not rigorously correct, but which still gives a reasonably correct physical picture of the result for \(b_0\gg{a_0}\). Note that the pulse narrowing or compression here is independent of the sign of the chirp, as is compatible with our physical reasoning.
The next tutorial introduces spectrally efficient multiplexing - OFDM.