Introduction

Science and technology need continuous revolutions, renovations, and replacements; for example, the invention of light amplification by stimulated emission of radiation (LASER) in 1960 revolutionized the science and engineering sectors by enabling the generation of highly coherent, monochromatic optical pulses [1], which acted as a gateway to novel spectroscopic methods of investigation that deepened our understanding of atomic structure. The time resolution of the equipment at our disposal limits our capacity to detect natural dynamics that occurs in short time intervals. For example, mechanical shutters provide millisecond resolution, stroboscopic lighting allows us to explore the microsecond range, and modern electronic sampling oscilloscopes have reduced the limit to the picosecond range. Beyond this limit, the development of ultrafast lasers has advanced the temporal resolution of measurement by another three orders of magnitude into the sub-ten-femtosecond regime, allowing for the direct observation of vibrational molecular dynamics [2]. In general, electromagnetic pulses of optical energy that have pulse durations of a picosecond (10⁻¹² s) or less are called ultrashort pulses. The word 'ultra,' meaning 'beyond,' originates from the Latin word 'ulter.' The first ultrashort pulse laser was demonstrated by De Maria et al after six years of the invention of Maiman's laser, using a passively mode-locked Nd:glass laser with an approximately measured minimum pulse width of $\approx 3.7\times {10}^{-13}$ s [3]. The ultrashort pulses are highly collimated beams, hence they propagate in a well-defined direction. Because of the high spatial coherence of ultrashort pulses, they can focus on very small spots, as small as 1 μm². A small spot size combined with a short pulse duration results in extremely high optical intensity. The spatial extent of short temporal pulses helps to put them in perspective. For example, a one-second light pulse may transverse a distance of 300 000 km, which is equal to the speed of light multiplied by one second. Meanwhile, a picosecond pulse has a spatial extent of 0.3 mm and a femtosecond pulse has a spatial extent of 0.3 μm. As a result, ultrafast events can be determined by these short pulses. The temporal confinement of light to durations close to the optical period [4] and the conversion of pulses of few cycles to extreme ultraviolet and x-ray wavelengths [5, 6] have enabled the measurement and control of electron dynamics on the sub-femtosecond timescale [7, 8]. The ability to scale peak and average power is an essential characteristic of the development of femtosecond laser technology [9]. An astounding capability to enhance laser peak power has been achieved by various ultrashort pulse-amplification techniques, such as chirped-pulse amplification (CPA), optical parametric chirped-pulse amplification (OPCPA) [10], and backward Raman scattering (BRA) during the last 25 years, and it has revolutionized laser science. These techniques help to amplify ultrashort pulses up to the peak power of PW without damaging the optical medium.

Even though these ultrashort pulse-amplification techniques provide light sources capable of generating hundreds of watts of average power; they seldom generate pulses shorter than 100 fs, which are extremely useful for frequency conversion to the extreme ultraviolet or the mid-infrared region. As a result, the development of power-scalable pulse-compression techniques is currently a work in progress. For the high-quality generation of ultrashort pulses, existing laser sources such as mode-locked lasers and Mamyshev oscillators require external pulse-compression techniques. In this context, several pulse-compression approaches based on linear and nonlinear optical components have been developed in an attempt to generate optical pulses with durations ranging from tens to hundreds of femtoseconds throughout a wavelength band spanning from the ultraviolet to the far infrared [11]. The confinement of all optical energy to a short time interval provides access to unprecedented peak powers, resulting in predominantly nonlinear interactions between light and matter, which has led to tremendous evolution in the field of ultrafast optics and opened new directions in the frontiers of high-field science and ultrafast spectroscopy [12, 13]. This book deals with such nonlinear pulse-compression techniques used for the generation of high-quality ultrashort pulses.

Ultrashort pulses are electromagnetic wave packets characterized by a time- and space-dependent electric field, which are the measurable quantities that are directly connected with the electric field. A complex representation of the field amplitude is especially useful when dealing with pulse propagation. The propagation of such fields and their interaction with matter are regulated by Maxwell's equations in a semi-classical manner, and the material response is represented by a macroscopic polarization. In general, the complex electric field of an optical pulse $E(t)$ as it propagates down the z-axis is usually expressed in the time domain by an envelope which is the product of an amplitude function and a phase term, as follows:

$$\begin{eqnarray}&&E(z,t)=U(z,t){e}^{i({\omega }_{0}t-{k}_{0}z+\phi (t))},\end{eqnarray} \tag{ 1.1 }$$

where $U(z,t)$ is the slowly varying envelope of the wave packet, $\phi (t)$ is the temporal phase, ${\omega }_{0}$ is the carrier frequency, and ${k}_{0}=\tfrac{n({\omega }_{0}){\omega }_{0}}{c}$ is a wave number that determines the carrier wavelength $\left({\rm{as}}\,{\lambda }_{0}=\tfrac{2\pi }{{k}_{0}}\right)$ of the wave packet. The time-varying temporal phase function $\phi (t)$ establishes a time-dependent carrier frequency (instantaneous frequency) $\omega (t)={\omega }_{0}+d\phi (t)/{dt}$. The most convenient way to describe the wave number in a Taylor expansion around the carrier frequency is

$$\begin{eqnarray}&&k(\omega )=k({\omega }_{0})+{\left.\frac{\partial k}{\partial \omega }\right|}_{{\omega }_{0}}\left(\omega -{\omega }_{0}\right)+\frac{1}{2}{\left.\frac{{\partial }^{2}k}{\partial {\omega }^{2}}\right|}_{{\omega }_{0}}{\left(\omega -{\omega }_{0}\right)}^{2}+\frac{1}{6}{\left.\frac{{\partial }^{3}k}{\partial {\omega }^{3}}\right|}_{{\omega }_{0}}{\left(\omega -{\omega }_{0}\right)}^{3}+\cdots .\end{eqnarray} \tag{ 1.2 }$$

Figure 1.1 depicts an electric field that has a pulse duration of 10 fs and its corresponding intensity pulse envelope. One can also describe the optical pulse using the spectral domain obtained from the time domain, in which the pulse consists of different frequency components. In general, the optical pulse in the spectral domain is described by its amplitude $U(z,\omega )$ and spectral phase $\phi (z,\omega )$ as follows:

$$\begin{eqnarray}&&\tilde{E}(z,\omega )=U(z,\omega ){e}^{i\phi (z,\omega )}\end{eqnarray} \tag{ 1.3 }$$

Given the temporal dependence of the electric field $E(t)$, the complex spectrum of the field strength $\tilde{E}(\omega )$ can be derived mathematically through the complex Fourier transform (${ \mathcal F }$):

$$\begin{eqnarray}&&\tilde{E}(z,\omega )={ \mathcal F }\,[E(t)]={\int }_{-\infty }^{\infty }E(t){e}^{-i\omega t}{dt}.\end{eqnarray} \tag{ 1.4 }$$

Given $\tilde{E}(\omega )$, the time-dependent electric field can be obtained through the inverse Fourier transform (${{ \mathcal F }}^{-1}$):

$$\begin{eqnarray}&&E(t)={{ \mathcal F }}^{-1}[\tilde{E}(\omega )]=\frac{1}{2\pi }{\int }_{-\infty }^{\infty }\tilde{E}(\omega ){e}^{i\omega t}d\omega .\end{eqnarray} \tag{ 1.5 }$$

For an electric field $U(z,t)$ propagating in a dispersionless material of refractive index n, the optical pulse energy can be easily calculated if the pulse parameters such as the pulse width (τ), peak power (P₀) and repetition rate ${T}_{\mathrm{rep}}$ are known. Figure 1.2 provides a pictorial representation of the measurement of a laser pulse's peak power, pulse width, and repetition rate. The pulse width in the time domain that spans a pulse is commonly calculated using the pulse's full width at half maximum (FWHM). For example, the pulse width of a Gaussian pulse is related to FWHM as follows (figure 1.3):

$$\begin{eqnarray}&&{T}_{\mathrm{FWHM}}=2\sqrt{{ln}\,2}\times \tau \approx 1.665\tau .\end{eqnarray} \tag{ 1.6 }$$

Given this value, the pulse energy can be calculated as follows:

$$\begin{eqnarray}&&E={\int }_{-{T}_{\mathrm{rep}}}^{{T}_{\mathrm{rep}}}| U(z,t){| }^{2}{dt},\end{eqnarray} \tag{ 1.7 }$$

where ${P}_{0}=| U(z,t){| }^{2}$. The energy per unit area is the instantaneous intensity (Wm⁻²) of a field and can be calculated from

$$\begin{eqnarray}&&I(t)={E}^{* }(t)E(t)=| E(t){| }^{2}\end{eqnarray} \tag{ 1.8 }$$

and the energy density per unit area (J cm⁻²),

$$\begin{eqnarray}&&W={\int }_{-\infty }^{\infty }I(t^{\prime} ){dt}^{\prime} .\end{eqnarray} \tag{ 1.9 }$$

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Laser light pulses with time durations on the order of a picosecond or less have a broad spectrum, which can be observed in packets of waves that are extremely localized in time. In this section, we will study about how the wave packets of ultrashort pulses can create a broad spectrum. An ideal light wave has a single frequency and it is spatially unlocalized. Consider the propagation of one light wave along the x-axis, as shown in figure 1.4, in the presence of the simultaneous propagation of light waves of equal amplitude but slightly different frequency. When these pulses, which have alternate out-of-phase and in-phase relationships, are added together, the single light wave is divided into beats via superposition. Figure 1.4 depicts the results of superimposing three waves, five waves, and seven waves to generate a longer, single light pulse.

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As in the generation of ultrashort pulses by combining a large number of slightly varying frequency components, as depicted in figure 1.4, by adding a larger number of slightly different frequency components, one can generate ultrashort pulses as shown in figure 1.5, which proves that an ultrashort pulse may be generated by broadening the spectrum of a longer pulse. However, due to the broad optical bandwidth of ultrashort pulses, chromatic aberrations that occur in the focusing optics can lead to complicated spatio-temporal effects, which may cause the focused pulse to have a larger duration than it had before focusing. To overcome this issue, refractive and diffractive optics with suitable lens combinations are required.

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It is important to acknowledge the relationship between spectral width and pulse duration when considering the generation of ultrashort pulses. By measuring the pulse width in both the frequency and time domains, we can compare the results to the Heisenberg uncertainty principle, which limits the minimum uncertainty in these variables. The energy–time uncertainty principle is given by

$$\begin{eqnarray}&&{\rm{\Delta }}E{\rm{\Delta }}T\geqslant \frac{\hslash }{2},\end{eqnarray} \tag{ 1.10 }$$

where ${\rm{\Delta }}E$ is the standard deviation in the energy and ${\rm{\Delta }}T$ represents the deviation in time. Since, for photons, $E=\hslash \omega$, equation (1.10) can be written as

$$\begin{eqnarray}&&{\rm{\Delta }}T{\rm{\Delta }}\omega \geqslant \frac{1}{2}.\end{eqnarray} \tag{ 1.11 }$$

It can be seen that a smaller ${\rm{\Delta }}T$ demands a larger ${\rm{\Delta }}\omega$ or frequency range. The product of pulse duration and spectral bandwidth is known as the time–bandwidth product (TBP).

In theory, this indicates that a broad spectral bandwidth (${\rm{\Delta }}\omega$) is required to generate a short pulse of light with a specific duration (ΔT). According to figure 1.6, if one chooses a large bandwidth, the pulse width is reduced. The reverse is also true: a pulse's TBP is always greater than the theoretical minimum given by the uncertainty principle (for the appropriate width definition). When equality to 1/2 is reached in the context of equation (1.11), the pulse involved is called a Fourier-transform-limited pulse. The variation in the phase of such a pulse is beautifully uniform; thus, it has a linear time dependence and its instantaneous frequency becomes time independent. In real-time measurements, the TBP is a measure of the complexity of a wave or pulse. The TBPs of actual signals vary, but there is always a minimum TBP for a certain desired effect. In communications over a channel, transmitting a certain amount of data over a given bandwidth requires a certain time. The TBP measures how well one can use the available bandwidth for a given channel. Even though every pulse's time-domain and frequency-domain functions are related by the Fourier transform, a wave with the minimum TBP is called Fourier transform limited. For ideal pulses, the product of the pulse width multiplied by the bandwidth has a minimum constant value. More commonly, the pulse duration is defined according to the principle of the FWHM of the optical power versus time. Equation (1.11) then becomes

$$\begin{eqnarray}&&{\rm{\Delta }}\nu {\rm{\Delta }}t\geqslant K,\end{eqnarray} \tag{ 1.12 }$$

where ${\rm{\Delta }}\nu$ is the frequency at the FWHM and ${\rm{\Delta }}t$ is the duration at half maximum. The value of K depends upon the symmetrical shape of the pulse. For a Gaussian function, K = 0.441, and for a hyperbolic secant function, K = 0.315.

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The applications of ultrashort pulses stem from direct laser beam interactions with matter and from the interactions between matter and the secondary particle and photon sources they drive. Ultrashort pulses are helpful in studies of the first principles of fundamental processes and to monitor extremely fast events in biology (photosynthesis, vision, protein folding), chemistry (molecular vibrations, re-orientations, and liquid-phase collisions), and electronic processes (high-lying excited-state lifetimes, photo-ionization, and electron–hole relaxation times that determine the response times of light detectors and electronics). Because of the broad spectrum of ultrashort pulses, they are used in medical diagnostics, such as optical coherence tomography (OCT), hard tissue ablation, and brain surgery, frequency metrology [14, 15], standoff trace gas detection (particularly in the oil and gas industry), and the detection of chemical components in artifacts [16–19]. The extreme concentration of energy in femtosecond pulses is useful for the materials processing of semiconductors, composite materials, glasses, and plastics (because much finer structures can be created in the absence of thermal interaction caused by longer pulses), and in attosecond pulse generation, laser-driven particle acceleration, and defense applications [20–22]. The latest advances in femtosecond technology have strongly emphasized the control of ultrashort pulses in many applications in which the preservation of the pulse duration is most important. Some of the applications of ultrashort sources are elaborated below.

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1.5.1. Frequency metrology

1.5.2. Optical coherence tomography

OCT is a micrometer-scale high-resolution cross-sectional imaging technology used to acquire pictures of strongly scattering media. Due to the interferometric basis of this technique, the resolution depth $\bigtriangleup z$ of the cross-sectional image is related to the central wavelength ${\lambda }_{c}$ and the FWHM bandwidth $\bigtriangleup \lambda$ of the light source as follows (figure 1.8):

$$\begin{eqnarray}&&\bigtriangleup z=\frac{2\,\mathrm{ln}\,2}{\pi }\frac{{\lambda }_{c}^{2}}{\bigtriangleup \lambda }\approx 0.44\frac{{\lambda }_{c}^{2}}{\bigtriangleup \lambda }.\end{eqnarray} \tag{ 1.13 }$$

If the source spectrum is approximately Gaussian in form, the optimal value of ${\lambda }_{c}$ is determined by the medium being studied. To achieve a good penetration depth, the 800 nm wavelength region is optimal for OCT measurements of the eye, due to its lower absorption, whereas the 1300nm wavelength region is best for observations of highly scattering tissue, such as skin. Based on the foregoing, it is desirable to have an OCT light source with an exceptionally broad (hundreds of nanometers), relatively smooth, and flat spectrum, with a central wavelength tailored to the specific OCT application.

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1.5.3. Wavelength-division multiplexing

1.5.4. Materials processing

Laser materials processing is a major component of manufacturing and is used to accomplish tasks ranging from heating for hardening, melting for welding and cladding, and the removal of material for drilling and cutting [25]. Material removal is based on the fact that all materials have an ablation threshold, i.e. a point at which they are directly vaporized when hit with a laser beam of sufficient peak optical intensity [26]. The threshold fluence (energy per unit area) for ablation reduces as a function of pulse width [27]. Hence, when a pulse is short enough, most of the optical pulse excites electrons, which then quickly cause a small section of the material to ablate without heating the substrate during the interaction. As a result, laser materials processing via plasma formation is considered to be a 'cold process' [28]. Because of the minimum thermal energy deposition in materials, ultrashort pulses allow for highly precise cutting, resulting in high-aspect-ratio holes and finely imprinted patterns with no collateral damage outside the desired interaction volume [29]. The resulting ejected material is mostly gaseous or very fine particles, and leaves behind a very limited heat-affected zone (HAZ), typically much less than a micron. Short pulses are also used in surface processing in order to clean or texture surfaces, resulting in hydrophobic surfaces or chemically reactive surfaces [30]. The typical intensities required for such tasks include heat treatment at 10³–10⁴ W cm⁻², welding and cladding at 10⁵–10⁶ W cm⁻², and material removal at 10⁷–10⁹ Wm⁻² for drilling, cutting, and milling [31]. The typical operating parameters of the commercial lasers used for manufacturing include pulse widths of 100–200 fs, peak energies of 50–150 μJ, average powers of 100–150 W, and pulse repetition rates of up to 1 MHz. Due to the ability of femtosecond lasers to efficiently fabricate complex structures, state-of-art laser processing techniques with ultrashort pulses are used to structure materials with sub-micrometer resolution, such as intricate three-dimensional photonic crystals, micro-optical components, gratings, and optical waveguides. Such structures rely on the creation of increasingly sophisticated miniature parts. As a result of the precision, fabrication speed, and versatility of ultrafast laser processing, it is it well placed to become a vital industrial tool for advanced material 3D micro/nano processing (figure 1.9).

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1.5.5. Medicine

In the field of medicine, lasers have reduced the need for sterilization or anesthetics. Ultrashort pulsed laser technology is now commonly used in the medical industry to fabricate high-quality surgical stents that have micron-scale features such as 1 μm-diameter holes with a large length-to-diameter ratio. Single-mode femtosecond laser technology is proving the best tool for these needs. The femtosecond laser ablation depths achieved using a single laser pulse can be more precise than those of material removed by conventional laser melting. Cracks due to thermal damage appear as a result of picosecond to femtosecond pulses but nearly disappear when the pulse duration is reduced to 5 fs. Because of the reduced collateral damage, high-intensity ultrashort pulses are used in various kinds of surgery based on laser processing of tissues in which intense laser pulses are delivered to internal tissues via optical fibers (figure 1.10).

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Particularly well known is laser-assisted in-situ keratomileusis (LASIK), which uses ultrafast laser scalpels to make incisions in the eyeball as part of a laser sculpting protocol to improve eyesight [32]. In these surgeries, the peak power is limited by the microscopic nature of the instrumentation. In addition, the promised ability of ultrahigh-intensity laser pulses to create different kinds of high-energy particles and radiation through interaction with a variety of sources leads to applications such as hard x-ray and γ-ray imaging which offer imaging features that cannot be achieved using ordinary x-rays [33], therapies using high-energy x-rays and γ-rays, therapies using laser-accelerated electron beams, therapies using laser-accelerated ion beams (mostly protons), and transmutation to create radioactive positron sources for positron emission tomography (PET) [34, 35]. Likewise, radiotherapy requires high-energy electrons to selectively kill cancer cells. In this regard, high-intensity ultrafast lasers based on wake field acceleration are commonly used instead of cyclotrons and linacs for the generation of high-energy electrons [36, 37]. Since thermal damage and stress-induced cracking depend on the average power of the femtosecond laser source, absorbed laser power leads to melting or thermal shock, even with picosecond or femtosecond pulse durations. Hence, low-average-power or ultrashort low-energy pulses are needed in surgical applications.

1.5.6. Fusion energy

1.5.7. High-harmonic generation

Ultrashort optical pulses are produced by a variety of methods. Although they differ in their technical details, each method relies on the same three key components: spectral broadening due to the nonlinear optical Kerr effect, dispersion control, and ultrabroadband amplification [10, 45–48]. In this chapter, we review state-of-the-art ultrashort pulse generation with a focus on pulse-compression schemes. In general, ultrafast light sources are either solid-state or fiber lasers. Ultrafast pulse generation is now a rapidly emerging field, because the development of optical communication system needs exceptionally high-speed transmission rates of 160 Gbps and higher. The development of ultrafast laser generation techniques in the visible and infrared regions of the electromagnetic spectrum has accelerated immense progress in numerous fields of fundamental science. In particular, high-repetition-rate ultrafast pulses have broad application prospects in laser micromachining, biomedical imaging, and photonic switching. However, it is difficult to directly generate laser pulses with so few cycles at visible and infrared wavelengths, even using the best available laser sources, because short pulses suffer from chromatic dispersion when they pass through optical elements, which changes their temporal shape. The wavelength dependence of the refractive index of the medium stretches the pulse temporally, thus lowering its peak power [49]. The evolution of ultrashort pulse-generation techniques is discussed in detail in this section.

Initially, femtosecond pulses were produced using dye lasers. In the late 1980s, the pulse duration of dye lasers was as low as 27 fs [50], which was later compressed to 6 fs [51]. At a wavelength of 600 nm, only three optical cycles fit under the FWHM of the intensity envelope of such a pulse. It took almost a decade to surpass these results with solid-state lasers. Later, solid-state quantum cascade and mode-locked lasers were utilized to generate ultrashort pulses because of their simplicity and low cost. However, as compared to mode-locked lasers, solid-state quantum cascade lasers have a number of fundamental limitations: limited average power at good beam quality and most crucially, poor energy storage capacity, which is critical for high peak power operation.

1.6.1. Mode-locking techniques

Until now, mode-locking techniques have been the technology most used for the generation of ultrashort pulses, particularly in the picosecond range [52]. In a mode-locked laser, pulse formation should start from normal noise fluctuations in the laser that initiate mode locking. In mode locking, the amplitude modulator opens and closes synchronously with the light propagating through the cavity, which causes the eigenfrequencies (or longitudinal modes) of the cavity to be phase locked when the modulation frequency is equal to the frequency spacing of the modes, generating a short pulse. In mode locking, the goal is to phase lock as many longitudinal modes as possible, because the broader the phase-locked spectrum, the shorter the pulse that can be generated. There are a number of potential operating characteristics that make mode-locking lasers particularly attractive. They have been shown to have: quantum limited noise, pulses shorter than 100 fs, more than 80 nm of bandwidth, repetition rates of up to 20 GHz, fundamental repetition rates from 100 kHz to 100 MHz, and transform-limited pulses. The first mode-locked laser was demonstrated in 1964, which suggested that as long as the unsaturated gain remains greater than the cavity losses, the modes of an incoming electromagnetic wave oscillate concurrently throughout all resonant frequencies of the cavity. In order to sustain a larger proportion of such longitudinal modes in a laser cavity, a broadband gain medium is needed. The competition within a wave packet traveling back and forth in the cavity causes the maxima to grow significantly stronger in the time domain. If the parameters are chosen correctly, a single focused pulse is created that oscillates with all of the energy of the cavity. Each time the pulse hits the output coupler mirror, a usable pulse is emitted, so that a regular pulse train leaves the laser. This is a mode-locked situation. In terms of spectral components, a short pulse is formed in the laser resonator when a fixed phase relationship is achieved between its longitudinal modes, or more precisely, between the lines in the spectrum of the laser output. The larger the number of frequency components involved, the shorter the duration of the generated pulses can be. In the steady-state femtosecond regime, dispersion and bandwidth limitations of the gain medium, mirrors, and so forth are mainly responsible for the temporal stretching of the pulse. Therefore, the pulse-shortening effect must be dominant for pulse durations ranging from nanoseconds at start-up to femtoseconds in steady-state operation. Mode-locking techniques are divided into two types based on the modulator utilized in the laser cavity, namely, active and passive mode-locking techniques (figure 1.11).

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1.6.1.1. Active mode locking

In active mode-locking operation, a modulator is inserted into the cavity or an external optical pulse is injected to actively modulate the light wave in the fiber cavity with a period equal to the round-trip time, which controls the cavity losses or the round-trip phase change, thus amplifying a specific portion of the radiation. Active mode locking can be implemented using gain modulation (switching the pump on and off), loss modulation (the periodic decrease of loss in some element), or cavity dumping (accumulating energy in the resonator cavity and releasing it in a short burst by misaligning one mirror). This can be accomplished using either an acousto-optic or an electro-optic modulator. If the external modulation frequency is equivalent to the intermode frequency, the sidebands and longitudinal modes compete with each other in the gain medium as long as the longitudinal modes lock their phases onto the sidebands, resulting in global phase locking and amplification of the whole spectral distribution. If the modulation is synchronized with the resonator round trips, this can lead to the generation of ultrashort pulses, usually with picosecond pulse durations. This technique provides high-order harmonic mode-locking operation.

1.6.1.2. Passive mode locking

A theory of passive mode locking was developed by Haus et al [53]. In this model, the intracavity elements are assumed to be continuous and change on a pulse per round trip, which are treated as perturbations. Passive mode locking causes lower losses in the more intense fractions of the radiation. Lower losses are caused by fractions such as saturable absorption mode locking and Kerr lens mode locking, in which more intense radiation obtains a desirable transverse profile by self-focussing. The passive amplitude modulator is a saturable absorber which has increased transmissivity or reflectivity for high peak powers and produces self-amplitude modulation (SAM). This SAM reduces the losses for short-pulse laser operation. An optical pulse traveling through a saturable absorber in a solid-state laser is shortened by the SAM, provided that the response time of the absorber is sufficiently fast. Traditionally, dyes have been used as saturable absorbers for passive mode locking. These dyes have been replaced by saturable absorbers obtained using Kerr or semiconductor nonlinearities. The precise control of optical nonlinearities, combined with the availability of a variety of bandgaps ranging from the visible to the infrared, makes semiconductor materials very attractive for use as saturable absorbers in solid-state lasers. Semiconductor materials typically provide an optical nonlinearity with two pronounced time constants. Intraband processes give rise to a very rapid relaxation in the 100 fs regime, while electron–hole recombination generates a slow response time in the picosecond regime. This slow response time can be reduced by several orders of magnitude using low-temperature epitaxy. For high pulse repetition rates with fundamental mode locking, very short laser resonators are required. Due to the high pulse repetition rate, the pulse energies obtained from mode-locked lasers are fairly limited—normally nanojoules or picojoules.

1.6.1.3. The influence of nonlinear effects on the mode-locking medium

Active and passive mode-locking techniques have been extensively researched in solid-state media in order to obtain high-repetition-rate pulses on the scale of a few GW. The most widely used solid-state medium is titanium-doped sapphire (Ti:Sa) crystal. The attractiveness of the Ti:Sa technology stems, in particular, from the ultrabroadband emission bandwidth of the gain material [54, 55]. With proper dispersion control, it readily enables the generation of pulses consisting of a few cycles. The scope of these techniques is limited by the stability of the pulse energy and low power scalability. This is due to the lack of available high-power pump diodes in the green wavelength region. This lack of availability of pump sources has been overcome by the development of diode lasers, which have covered an increasing number of wavelengths, allowing for a wide range of directly pumped diode lasers. This has motivated researchers to work toward the development of ultrashort pulses with millijoule pulse energies, fundamental mode beam quality, and average powers greater than one kilowatt. In order to stabilize the pulse energy, subcavities with a free spectral range have been used to match the modulation frequency. However, exact matching of the fundamental frequency, the free spectral range of the subcavity, and the modulation frequency is required. Later, additive pulse limiting (APL) and self-phase modulation (SPM) techniques were used to stabilize the pulse energy by properly adjusting the polarization bias to clamp the energy at a specific level, and by using the spectral filter inside the cavity to create more loss in the high-intensity pulse.

However, when generating femtosecond-duration pulses, the interaction of light with the cavity medium becomes nonlinear even at low pulse energies, which gives rise to phenomena such as chromatic dispersion, the Kerr effect, Raman scattering, self-phase modulation, and gain saturation. These phenomena lead to self-focussing and damage the laser medium by its interaction with the intense electric field of the light wave. The self-damage phenomena in an optical medium can be reduced by stretching out the optical pulse and thereby lowering the peak power. However, while achieving high energy density in a short pulse, the power density must not exceed the laser medium's damage threshold, which is achieved by compensating for the SPM effects. As the peak power triggers nonlinear effects, such as frequency conversion or multi-photon ionization, while the laser repetition rate determines the data acquisition rate, the combination of both is needed in various experiments, for example, in extreme ultraviolet and mid-infrared frequency comb spectroscopy [56, 57], time-resolved photo-emission electron microscopy [58], or coincidence spectroscopy [59]. Therefore, the simultaneous scaling of peak and average power is the key focus of current femtosecond technology developments.

1.6.1.4. Mode locking in fiber

The mode-locking technique has also been investigated in fiber media based on the incorporation of trivalent rare-earth ions, such as those of neodymium, erbium, and thulium, into glass hosts. The fiber itself provides the waveguide, and the availability of various fiber components minimizes the need for bulk optics and mechanical alignment. These fiber lasers can be actively or passively mode locked. In active mode locking, a modulator produces amplitude or phase modulation, while passive mode locking uses a nonlinear amplifying loop mirror, nonlinear polarization rotation, and semiconductor saturable absorbers. In passive mode locking, an intensity fluctuation acts in conjunction with fiber nonlinearity to modulate the cavity loss without external control [60]. The pulse duration of an actively mode-locked fiber laser is typically within the order of a picosecond, due to the limited response time of the modulator. In addition, the modulator can reduce the environmental stability of actively mode-locked fiber lasers. In rare-earth-doped fiber lasers, the upper-state lifetime is very long ($\approx$ ms), implying that the gain does not react significantly within the cavity round-trip time (<0.5 μs). A fast saturable absorber is therefore required to clean up both the leading and trailing edges of the pulse. Mode locking of fiber lasers generates subpicosecond pulses. However, in fiber lasers with discrete lumped elements, a pulse undergoes appreciable nonlinear and linear evolutions. One example of this effect is dispersion management in soliton transmission. Soliton-like mode-locked pulses require a saturable absorber for stability to preferentially support pulses as opposed to continuous-wave (CW) background or dispersive waves. Furthermore, saturable absorbers play a critical role in pulse shaping. Therefore, the optimization of mode-locked laser design inevitably involves consideration of the right rate and the depth of saturation, which are important parameters of a saturable absorber. Semiconductor structures have frequently been employed for this purpose. Various optical switches, such as nonlinear loop mirrors and nonlinear polarization evolution have been more effective in mode-locking fiber lasers, because of their instantaneous responses and the fact that the short photon lifetime of fiber lasers demands a large amount of saturation.

Because of the potential applications of ultrashort pulses in numerous fields of science and technology, ultrashort pulse-generation techniques have gained traction in a number of nonlinear optics domains. A well-known technique for generating ultrashort optical pulses is to employ a mode-locked fiber laser or semiconductor laser and parametric processes. Although a mode-locked laser can generate a high-quality ultrashort optical pulse, all of the parameters involved in mode-locked operation are interdependent. For example, a small change in cavity length might cause the pulse to function with a completely new set of parameters, such as pulse width, repetition rate, carrier frequency, etc. and therefore the operation of mode-locked oscillation can be unstable. In addition, mode-locking and parametric processes necessitate more sophisticated and expensive state-of-the-art setups than those required for the generation of longer pulses. As a result, pulse-compression techniques are required to provide few-cycle pulses at any specified wavelength with greater simplicity. Similarly, many applications, such as laser fusion, need a high energy density in a short time span (figure 1.12). Pulse-compression methods are linear or nonlinear processes used to obtain bandwidth-limited pulses of extremely short duration by balancing SPM effects. Post-pulse-compression processes have been identified as the most effective method for obtaining ultrashort pulses; these work on the idea of compressing the optical energy of the electromagnetic waves in time to attain the required pulse characteristics. In order to reduce alignment and loss issues and improve the ease of coupling with transmission systems, short-pulse sources must take the form of guided-wave photonic structures. In any pulse-compression technique, the bandwidth of the coherent optical pulse restricts the degree of compression. As a result, when working with ultrashort pulses it is important to thoroughly characterize their features in order to adjust their shape to meet the application requirements.

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1.7.1. Linear pulse compression

Linear pulse-compression techniques are based on the idea of chirp compensation. An instantaneous change in frequency over time of an optical pulse propagating through an optical medium is called chirp. In linear pulse-compression techniques, the temporal length of the pulse can be shortened by lowering the chirp, i.e. by flattening the spectral phase (figure 1.13).

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Dechirping can be achieved by passing pulses through an optical device with an appropriate degree of chromatic dispersion (dispersion compensation), such as a grating compressor, a prism pair, an optical fiber, a chirped mirror, a chirped fiber Bragg grating, or a volume Bragg grating. The shortest possible pulse duration is then determined by the optical bandwidth of the pulse, which is unaffected by dispersive (linear) compression. In the ideal case, bandwidth-limited pulses can be obtained.

1.7.2. Nonlinear pulse compression

In a nonlinear pulse-compression system, nonlinear properties, such as self-phase modulation (SPM) (intensity-dependent phase modulation of the pulse), arise as a result of the optical Kerr effect (intensity-dependent refractive-index variation), which helps to broaden the spectrum of the pulse along with subsequent compression of the broadened spectrum. Because of the conjugate relationship between the transform-limited duration and its spectral width, a pulse becomes shortened or compressed compared to its input duration (figure 1.14). The post nonlinear compression of short pulses involves techniques such as free propagation in bulk materials, multiple-plate continuum generation, multi-pass cells, filaments, photonic crystal fibers (PCFs), hollow-core fibers, and self-compression. In most of the nonlinear pulse-compression schemes, optical fibers in the normal dispersion regime are used for spectral broadening of the pulse. For high-intensity pulses, spectral broadening has been achieved using a gas-filled hollow fiber or capillary. For soliton compression, anomalous dispersive fibers are used, but this limits the energy due to the small soliton pulse energies of typical fibers. The compressed pulse energy can be increased by amplification in a doped fiber with constant dispersion properties. Pulse compression can also occur during nonlinear frequency conversion processes, for example, those occurring in optical parametric oscillators, and in HHG. Subsequent dispersive compression can be achieved using chirped mirrors, dispersive mirrors, or bandpass filters. The selection of a pulse-compression process depends on a number of factors, including the initial and required pulse durations, the pulse energy, and the required pulse quality.

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This section describes the experimental realization of the various pulse-compression techniques. Initially, Nakazawa et al experimentally demonstrated the generation of 170–190 fs, 10 GHz pulses using a dispersion-decreasing erbium-doped fiber amplifier based on adiabatic fundamental soliton compression and adiabatic fundamental soliton-narrowing processes in a fiber with a length of 1 km [61]. Clark et al used a pulse-replica compression technique in a single-mode fiber with prisms and obtained 82 fs pulses with an energy of 0.4 mJ in a fiber with a length of 0.75 m [62]. With the development of PCF, Druon et al reported the direct compression of 110 fs pulses down to 75 fs using a PCF at a central wavelength of 1 μm with a tunability of 1–1.3 μm [63]. Ouzounov et al demonstrated the generation of 140 fs and 3 nJ pulses using 1.3 m microstructured large-mode-area fiber using a grating compressor [64]. Phase-compensated supercontinuum generation using a chloroform-infiltrated PCF was experimentally investigated by Le et al. In the case of CCl₄-filled fiber, a microfluidic pump system maintaining a constant pressure of 10 kPa was used to minimize microbubble-related effects and suppress thermal effects in the liquid under high-repetition-rate pumping conditions. CCl₄ was infiltrated into all the air holes in the cladding region of the fiber by capillary action and the pressure of the liquid pump system [65] (figure 1.15). Yang et al demonstrated the amplification of 120 fs pulses using an optical parametric chirped-pulse amplification system (OPCPA) consisting of two lithium triborate optical parametric preamplifiers, a potassium dihydrogen phosphate (KDP) optical parametric amplifier and neodymium-doped yttrium aluminum garnet (Nd:YAG)–Nd:glass hybrid amplifiers for the generation of 570 mJ, 155 fs pulses with a peak power of 3.67 TW [66]. Turchinovich et al reported a stable laser compression scheme using a 9.5m hollow-core polarization-maintaining PCF, which provided output pulses around 370 fs in duration at an energy of 4 nJ with high mode quality [67]. Pulses lasting for 77 fs with energies as large as 2.7 nJ have been generated by stretched-pulse erbium fiber lasers, although 1 nJ was produced by diode-pumped versions [68]. Tamura et al experimentally demonstrated a fiber amplifier/grating pair compression system that achieved compression from 350 to 77 fs at a gain of 18 dB in an erbium-doped fiber amplifier [69]. Seidel et al described the compression of pulses that had an average power of 90 W and a duration of 190 fs to 70 W and 30 fs, respectively, with an increase in peak power from 18 MW to 60 MW. This compression scheme was based on the cascaded phase-mismatched quadratic nonlinearities in β-barium borate (BBO) crystals [70].

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Pulse compression is the process used to generate the short-duration optical pulses required for specific applications. To date, various linear and nonlinear pulse-compression processes have been proposed. In this book, an attempt is made to address the fundamental principles and limitations of the numerical modeling of the linear and nonlinear phenomena involved in the process of pulse compression in PCFs used for the generation of ultrashort pulses. The chapters are structured as follows:

Chapter 2 reviews the basic properties of optical fibers, including different types of fiber and fiber modes. It introduces PCFs, their types, and their characteristics. It explains how their optical properties are well suited for ultrafast nonlinear optics.

Chapter 3 gives a brief overview of the current numerical methods used to solve the nonlinear Schrödinger equation (NLSE) and the numerical models used to calculate PCF parameters.

Chapter 4 provides a brief description of linear and nonlinear polarization effects, followed by a short discussion of the propagation properties of light in a dielectric waveguide. It then describes the interesting coupling between linear and nonlinear effects in an optical fiber under different circumstances, and shows how they lead to the spectral broadening and soliton dynamics that facilitate the results presented in later chapters.

Chapter 5 explores existing conventional soliton-compression techniques, such as higher-order soliton compression and adiabatic pulse compression, which are used with PCFs for the generation of ultrashort pulses and discusses quality analysis of compressed pulses.

Chapter 6 discuss a pulse-compression scheme based on self-similar analysis that aims to obtain a large compression factor and minimal pedestal energy by the compression of fundamental solitons with low input pulse energies over small propagation distances. The numerical model of a PCF compressor based on self-similar properties is discussed in detail.

Chapter 7 briefly reviews the numerical simulation of simultaneous pulse compression and pedestal suppression in a tapered PCF-based nonlinear optical loop mirror (NOLM) used to model a highly efficient optical pulse compressor that generates femtosecond pulses.

Chapter 8 offers a detailed discussion of a unique cascaded soliton-compression approach using chloroform-filled cascaded PCF used for the generation of low-energy, few-cycle laser pulses. Higher-order soliton compression is introduced for the compression of femtosecond pulses.

Chapter 9 theoretically explains the generation of a tunable broadband light spectrum using a PCF by varying its temperature, which is found to have a significant impact on the process of supercontinuum generation (SCG)-induced pulse compression. Temperature-dependent fiber parameters are used to solve the NLSE, which determines the evolution of the optical pulse through the proposed fiber model.