Nonlinear Optics

Intense light rips apart ordinary materials \(-\) that is, strong light fields surrender so much energy to matter that electrons are ejected from their mother ions. But everyone knows that electrons can be freed (ionized) with ordinary light, as long as light's frequency is sufficiently high. This is why we wear sun screen during the summer, when high-frequency ultraviolet (UV) radiation is abundant. So what's unique about an intense light-matter interaction? And how is it that the applications of intense light range from lightning control to the visualization of electron and molecular dynamics?

Before diving into the exotic physics of intense laser-matter interactions, we should appreciate the technology that makes it all happen. Ultrafast laser systems are capable of generating ultrashort pulses with durations on the order of femtoseconds (\(1\) fs \(= 10\)^\(-15\) s) owing to mode-locked laser cavities, which consist of gain media with broad bandwidths (reliably amplifying many colors) and a mechanism for stabilizing the phase relationship between the colors (e.g., through electro-optic modulators or materials exhibiting saturable absorption) [1].

An important consequence of the short pulse duration is the resulting high peak powers, which occur even at low pulse energies! For example, a commercially-available Titanium–sapphire laser system with a repetition rate of \(80\) MHz (pulses are emitted every \(12.5\) ns) and an average power of \(100\) mW (measured using a typical power meter) will produce pulses containing about 1 measly nanojoule of energy. However, if the pulse duration is \(10\) fs, then the instantaneous (peak) power of the pulse is \(125\) kW! That's a result of simple physics: power is energy over time, so confining even a small amount of energy into a short time interval can have profound effects on peak power.

But that's not the end of the story. Modern ultrafast lasers are made to be even more powerful through chirped-pulse amplification (CPA), a Nobel Prize winning technology [2]. CPA enables the amplification of ultrashort pulses by preventing damage to the optics of the system (a typical complication at very high powers), and the strategy is surprisingly simple. The technique is illustrated in Figure 1. A pair of gratings is used separate the spectral content (the colors) of the pulse in time. For example, long wavelengths might be sent to the front of the pulse, and short wavelengths to the rear. This temporal redistribution of colors can broaden the pulse duration by a factor of \(\sim10^5\), which in turn reduces the peak power by the same factor. The reduced power allows us to amplify the pulse energy using the gain medium (Titanium–sapphire in our example) without damaging the crystal. Following the amplification, a second pair of gratings is used to recompress the pulse back to femtosecond durations. In many commercially available amplified systems, a final pulse energy of a few mJ can yield powers on the order of \(100\)s of GW \(-\) that's enough for a myriad of exotic laser-matter interactions. And so we can study nonlinear optics.

Now, what do we mean by nonlinear?

Figure 1: The important steps of chirped pulse amplification are shown. An ultrashort laser pulse is temporally broadened using a grating pair (stretcher), which separates the colors of the pulse in time. With a long pulse duration, the power is low enough to safely amplify the pulse energy in the gain medium (crystal amplifier). A second grating pair (compressor) temporally recompresses the beam, producing an ultrashort and ultrapowerful laser pulse.

When we say that a system is linear, we mean that the system's response to some input varies in direct proportion to that input's strength. In linear optics, the material response to an electric field \(E\) (of an electromagnetic wave) is described by the resulting material polarization: \begin{equation} P = \epsilon_0\chi E \tag{1} \end{equation}

Here \(\epsilon_0\) is the permittivity of free space (a fundamental constant) and \(\chi\) is the material's susceptibility (which quantifies its tendency to polarize). And what's polarizing? The electron clouds that surround the nuclei. That's intuitive: electrons are redistributed in the presence of a field, and the magnitude of the effect scales linearly with the field strength (e.g., twice the field strength will induce twice the polarization). Coupled with Maxwell's equations, this polarization response is the fundamental starting point of conventional optics, describing important physical processes like diffraction (spreading of light in space) and dispersion (spreading of pulses in time). We'd simply solve a linear wave equation to predict the evolution of a light field in a linear medium.

Now crank up the power. Once the beam is intense enough, interesting things start happening. The material's polarization \(P\) no longer scales in direct proportion to \(E\). Instead, a nonlinear dependence on the field strength arises: \begin{equation} P = \epsilon_0(\chi E + \chi^{(2)} E^2 + ... + \chi^{(m)} E^m) \tag{2} \end{equation}

Now the polarization scales as the field raised to varying powers \(m\), with each term weighted by a respective \(m^{th}\) order susceptibility \(\chi^{(m)}\). One might envision this as the electron cloud exhibiting a more complicated redistribution in response to the electric field of the intense pulse. Why? In part, this can happen because a strong electric field will drive electrons very far from their mother ions \(-\) so far that they might start interacting with neighboring atoms/ions, which alter the local electromagnetic environment of the cloud.

More technically, a linear polarization response to an electric field can be described by treating an electron as a linear harmonic oscillator. Just like a mass on a spring obeying Hooke's law, in the linear regime the nucleus is assumed to exert a force on the electron that is proportional to the electron's displacement from the nucleus. In contrast, an intense field warrants an anharmonic oscillator approach that describes the restoring force using terms beyond the linear [3]. This is the basis of nonlinear optics (to be technical once more, perturbative nonlinear optics).

So fundamentally, in nonlinear optics the electron cloud responds to an intense electromagnetic field in a nonlinear way. This simple fact manifests as a multitude of physically novel and technologically appealing effects. In particular, it is useful to consider the unique features of intense ultrashort pulse propagation in the spatial, spectral and temporal domains.

The most well-known spatial nonlinear effect is self-focusing, which, as implied by the term, describes the focusing of an intense beam on its own. No lens is involved! This is very strange indeed, since all light (including laser light) naturally diverges as a consequence of diffraction (a linear effect). Let's try to make this more natural with some simple physical intuition.

Fundamentally, self-focusing occurs because an intense beam modulates a material's refractive index during propagation. That modulation results in a refractive index of a higher value, which scales directly with the intensity \(I\) (and in turn scales quadratically with the field strength \(I \sim E^2\)). We call this the optical Kerr effect. To understand how this influences the focusing of the beam, we can write the phase acquired by the beam as: \begin{equation} \Phi = kz - \omega_0 t \tag{3} \end{equation}

Here \(k\) is the wavenumber, \(z\) is the propagation distance, \(\omega_0\) is the central angular frequency, and \(t\) is time. We can express the wavenumber in terms of the linear refractive index \(n_0\) and the change in the index due to the nonlinear (intensity-dependent) refractive index \(\Delta n = n_2I\), where \(n_2\) is the nonlinear index coefficient (a material-dependent factor): \begin{equation} k = \frac{n\omega_0}{c} = \frac{(n_0 + n_2I)\omega_0}{c} \tag{4} \end{equation}

Here \(c\) is the speed of light. Now the intensity of most laser beams is highest near the beam's cross-sectional center, with a peak value \(I_0\), and the intensity falloff is often described as a Gaussian across the profile \(I(r) = I_0e^{-2r^2/w_0^2}\). Here \(r\) is the radial coordinate along the direction of the beam's cross-sectional profile and \(w_0\) is a characteristic radius (the value in space, relative to \(r = 0\) μm, at which the intensity of the pulse falls to a value of \(I_0/e^2\), where \(e\) is Euler's number). This radial variation naturally extends to the spatial component of the phase: \(\Phi(r) \propto k \propto I \propto e^{-r^2}\). With some familiarity of geometrical optics, one might recognize that such a phase front resembles that induced by a focusing lens! Recall that a focusing (convex) lens is thickest near its center, so rays propagating near the center acquire more optical phase than those at the outskirts, resulting in a converging wavefront. Figure 2 illustrates the idea: nonlinear self-focusing overcomes the beam's natural tendency to diverge!

Figure 2: Self-focusing occurs due to the optical Kerr effect, which produces a phase front resembling that induced by a converging lens.

Figure 3: Plasma-induced defocusing produces a phase front resembling that induced by a diverging lens.

Figure 4: Laser filamentation occurs when nonlinear self-focusing is balanced by diffraction and plasma-induced defocusing. In this way, an ultrashort pulse can sustain a confined spatial profile over an extended range, which is impossible in the linear optical regime.

The spectral dynamics of a typical beam of light propagating in the linear regime are quite bland: generally the colors that comprise the beam will either remain unaltered during propagation, or certain colors will be extinguished due to absorptive media. Usually new colors will not be added to the beam during linear propagation. On the other hand, nonlinear propagation often manifests a wildly broader spectrum compared to the initial spectral content. This spectral broadening is often called supercontinuum generation [3, 10, 11] due to the broadband continuum of light generated about the central wavelength of the ultrashort pulse. Although the mechanism of supercontinuum generation is not fully understood, at its most basic level the process can be explained in terms of the optical Kerr effect, leading to self-phase modulation.

In an analogous way to spatial phase modulation, which leads to a converging wavefront (see spatial effects above), an ultrashort pulse can modulate its own phase in the time domain! Consider the temporal intensity profile of a typical ultrashort pulse, which can be described over time using a Gaussian function: \(I(t) = I_0e^{-2t^2/\tau^2}\). Here \(I_0\) is the peak intensity and \(\tau\) is the pulse duration (the value in time, relative to \(t = 0\) s, at which the intensity of the pulse falls to a value of \(I_0/e^2\)). So the beam has a phase that is time-dependent, expressed as: \begin{equation} \Phi(t) = \frac{[n_0 + n_2I(t)]}{c}\omega_0z - \omega_0 t \tag{5} \end{equation}

To describe the variation of the beam's spectral content over time, we can calculate the instantaneous angular frequency \(\omega_i\) of the pulse, simply by taking the (negated) time derivative of the phase \(-\frac{\partial \Phi(t)}{\partial t}\): \begin{equation} \omega_i(t) = -n_2\frac{\partial I(t)}{\partial t}\frac{\omega_0z}{c} + \omega_0 \tag{6} \end{equation}

The negative sign in the definition above is used simply for notational convenience, which allows us to describe our central frequency \(\omega_0\) as a positive quantity. Here's how to interpret the form of \(\omega_i\): without a nonlinear (\(n_2I\)) contribution, the spectrum of the pulse is unaltered, always taking a central value of \(\omega_0\). However, the derivative of \(I(t)\) is nonzero. Therefore, the spectral content of the pulse will be modified based on the temporal profile of the pulse! For some intuition, consider Figure 5.

Figure 5: A pulse with a Gaussian temporal profile has an increasing intensity (positive slope) near its leading edge, and a decreasing intensity (negative slope) near the trailing edge.

Figure 6: Supercontinuum generation results in the spectral broadening of an ultrashort pulse. Lower frequencies are generated near the leading edge of the pulse, and higher frequencies are generated near the trailing edge.

Figure 7: Second-harmonic generation occurs when two photons of frequency \(\omega\) are annihilated, exciting an electron in the process, followed by relaxation of the electron and production of a new photon at frequency \(2\omega\).

The temporal structure of any waveform is inherently linked to its spectral content. At the most fundamental level, we can say that pulses with broader spectra have shorter time durations. Mathematically, we can think of the spectral bandwidth \(\Delta\omega\) and pulse duration \(\Delta t\) as conjugate Fourier variables, which physically manifests as a variant of the all-important Heisenberg uncertainty principle: \begin{equation} \Delta\omega\Delta t \geq 1/2 \tag{7} \end{equation}

The implication? Broadband spectral expansion during supercontinuum generation is often accompanied by a shortening of the pulse duration, providing a useful mechanism for temporal compression of ultrashort pulses.

Once a broad spectrum is generated, it's important to consider the effect of group velocity dispersion, which describes the speed at which the colors of the pulse travel. Group velocity dispersion is material-dependent, and it is a linear process. However, its interplay with nonlinear effects such as supercontinuum generation makes it a crucial component in the unique temporal dynamics of intense light pulses. The dispersive nature of a material is one of two types, either normally dispersive, where long wavelengths propagate faster than shorter ones, or anomalously dispersive, where short wavelengths outpace longer ones. Figure 8 demonstrates the effect in a normally dispersive material.

Now recall that during supercontinuum generation, longer wavelengths are created at the front of the pulse. Therefore, if supercontinuum generation occurs in an anomalously dispersive medium, pulse compression can be amplified: the long wavelengths generated near the front of the pulse are swept towards the back of the pulse, while the short wavelengths made in the rear are swept forward, leading to further compression [23]. A myriad of other physically exotic processes can lead to compression, including plasma defocusing of the trailing edge [24], a steepening of the trailing edge due to the Kerr effect [25], and high-harmonic generation, which might one day enable the generation of pulses with durations on the order of zeptoseconds [26](\(1\) zs \(= 10\)^\(-21\) s!).

Finally, how can we use these ultrashort pulses to study novel physical processes? To gain a sense of the possibilities, consider a classical problem in photography. In 1878, the English-American photographer Eadweard Muybridge was interested to know whether a galloping horse ever becomes completely airborne during its gait. Do all four feet ever completely lift off the ground at any one moment in time? The question was hotly debated, since the unaided human eye is not able to resolve events that evolve too quickly, including the trotting or galloping of a horse. Muybridge's solution? A series of cameras configured to capture the horse during its gait, providing an effective frame rate that is high enough to accurately capture the motion of a galloping horse. Through his efforts, Muybridge was able to show that the animal indeed becomes fully airborne. His work, called The Horse in Motion [27] (see Figure 9) is now considered to be a crucial step in the development of the motion picture [28].

Using ultrashort pulses, we can extend these ideas to the ultrafast visualization of exotic physical systems that evolve very rapidly. To completely capture the physics of a system that changes quickly in time, we require a visualization mechanism that (a) has fine enough time resolution compared to the pace of the system's evolution and (b) can record the system over a long enough time so that all important events are measured. This is analogous to speed-racing movie that captures the entire race without blurry frames. However, instead of a race, we are interested in measuring dynamically-evolving microscopic events such as plasma-induced particle acceleration [29] or laser-induced fusion [30].

A myriad of ultrafast visualization schemes have been demonstrated, including frequency-domain holography (FDH) [31], single-shot supercontinuum spectral interferometry (SSI) [32], and multi-object-plane phase-contrast imaging (MOP-PCI) [33]. Although most of these techniques rely on rather sophisticated forms of microscopy (see my primer on digital holography), at the heart of ultrafast visualization is a simple strategy called the pump-probe method, shown in Figure 10.

The pump-probe technique relies on a well-timed interaction between two pulses: the pump and the probe. The pump pulse is usually more energetic than the probe, and a lens is often used to focus the pump into a medium of interest. The pump modifies the medium in some way. For example, it can generate plasma. And that plasma isn't stagnant \(-\) it's a sea of ions and free electrons in a state of rapid evolution, including the abrupt generation of pressure (shock) waves, an acceleration of electrons in the presence of the pump electric field, and finally a recombination of electrons to their mother ions. That's alot of interesting physics to study, but to describe these processes we require a visualization tool with a "frame rate" on the time scale of these events: femtoseconds! Enter the probe: a second ultrashort pulse is directed through the plasma at a specific time delay relative to the pump, which is controlled by sending the probe over an optical delay line. For each delay, the plasma modulates the probe in a slightly different way (for example, absorbing or defocusing the probe). This modulation can be captured using a detector such as a charge-coupled device camera or a photodiode. Since the delay between the pump and probe can be controlled to femtosecond precision, an effective "movie" of the plasma dynamics can be generated. A myriad of applications follow naturally, including the characterization of nanostructures [34] and fundamental biochemical processes [35], to name a few.

It's a bit like filming molecular movies \(-\) a novel art enabled by ultrafast science.

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