## Abstract

The dispersive properties of materials, i.e., their frequency-dependent response to the interaction with light, in most situations determines whether an optical process can be observed. Although one can always search for a specific material with the sought-after properties, this material might be far from optimum or might not even exist. Therefore, it is of great interest to develop methods that could tune the dispersive properties of a medium independently of the working frequency band. Pulses with angular dispersion, or pulse-front tilt, precisely allow us to achieve this goal. In this tutorial, we show the basics of how angular dispersion can manage to tune the dispersion parameters that characterize the propagation of light in a medium, thus permitting the observation and application of various optical processes in nonlinear and quantum optics that could not be realized otherwise. To keep the focus on first principles, the list of topics addressed is not exhaustive. More specifically, we consider the role of angular dispersion for pulse stretching and compression, broadband second-harmonic generation, the generation of temporal solitons in nonlinear ${\chi}^{\left(2\right)}$ media, the tunable generation of terahertz waves by means of optical rectification of femtosecond pulses, and the tuning of the frequency correlations and of the bandwidth of entangled paired photons.

© 2010 Optical Society of America

## 1. Introduction

When light interacts with a material, the atoms and molecules that constitute the medium
respond differently depending on the frequency of the light, *ω*.
For low intensities, the relationship between the induced polarization $P\left(\omega \right)$ and the electric field $E\left(\omega \right)$ is linear, $P\left(\omega \right)={\u03f5}_{0}{\chi}^{\left(1\right)}\left(\omega \right)E\left(\omega \right)$, where the constant of proportionality is the frequency-dependent
susceptibility ${\chi}^{\left(1\right)}\left(\omega \right)$ and ${\u03f5}_{0}$ is the permittivity of free space. The frequency dependence of the
susceptibility is the so-called dispersion of the medium. One easily observable
manifestation of the dispersive nature of materials was described in 1704 by Isaac
Newton in his book *Opticks: or a Treatise of the Reflections, Refractions,
Inflections and Colours of Light* [1].
Newton showed that when a prism deflects a white-light optical beam, each of the colors
that make up the white light leaves the prism in a different direction (see Fig. 1). The refractive index of the medium is wavelength
dependent, and therefore the angles of refraction governed by Snell’s law
correspondingly change at each interface in accordance with wavelength.

Dispersion and diffraction are the main linear effects that describe the propagation of
a light beam in a medium. In the linear regime, the propagation of the light beam for a
distance *z* can be easily described by multiplying the electric field
amplitude at $z=0$, $a(\mathbf{q},\omega ,z=0)$, by a phase factor $\mathrm{exp}\left\{ik(\mathbf{q},\omega )z\right\}$, where **q** is the transverse wavenumber,
*ω* is the angular frequency, and *k* is the
longitudinal wavenumber. Normally, the bandwidth of a light pulse is much smaller than
its central frequency, so the wavenumber *k* can be expanded in a Taylor
series about the central frequency. The number of terms of the expansion to be included
increases with the bandwidth, but in most situations one can safely keep the expansion
to second order. In this case, the group velocity and the group-velocity dispersion
(GVD) are the two parameters that characterize the linear dispersion of the medium.

With the invention of the laser, it was possible to observe optical phenomena that generate material polarizations that are no longer linearly proportional to the electric field amplitude, but that depend on its higher powers. In this case, the strength of the different nonlinear terms is determined, among other things, by the nonlinear susceptibilities of the medium, ${\chi}^{\left(n\right)}$, with $n=2,3,\dots $. Examples of these nonlinear effects are the generation of harmonics [2, 3] and self-phase modulation (SPM) [4].

In order to observe any of these processes, the dispersive properties of the material have to be taken into account. In certain situations, the presence of dispersion is harmful and one must look for materials that exhibit negligible dispersion over a sufficiently large bandwidth. This is the case, for instance, for the generation of ultrashort pulses by means of second-harmonic generation (SHG): owing to its intrinsic dispersion, the material can severely limit the bandwidth of the generated second-harmonic pulse, imposing an effective lower limit on the duration of the pulse that can be generated. In other situations, the dispersion is not harmful, but, to the contrary, a certain amount of dispersion is required, and in addition must be of the appropriate sign. This is the case for the observation of quadratic temporal solitons in ${\chi}^{\left(2\right)}$ media and for the stretching and compression of pulses.

Although one can always search for a specific material with the sought-after linear dispersion properties, this material might be far from optimum or might not even exist. Moreover, certain applications need to be implemented in specific frequency bands. Even if an appropriate material can exist in one frequency band, it might be useless in another band. Therefore, it is of great interest to develop methods that could tune the dispersive properties of a medium independently of the material and the working frequency band.

In this tutorial, we will review how the introduction of angular dispersion into pulses, or pulse-front tilt, can tailor the dispersive response of any material in any frequency band. By applying angular dispersion to pulses, one can effectively generate a new group velocity and group-velocity dispersion as required by the specific application in mind. This capability has been used in different areas of nonlinear and quantum optics, but the different approaches, the emphases on different aspects of the process, and the different language used make it difficult to see what is the role of pulses with angular dispersion with a single view that might encompass all the different applications. The goal in this tutorial is to give an overall view of the role of angular dispersion that could be applied in the various applications of nonlinear and quantum optics that use these techniques.

Although the dispersive properties of a light beam that propagates in any given material can be changed in many different ways—for instance, by changing the temperature or by considering a different direction of propagation inside the nonlinear medium—these changes are related to modifications of the material properties of the media. Our interest here are the modifications of the dispersive properties introduced by changing the spatiotemporal structure of the light beam itself with appropriately designed dispersive elements, such as prisms or gratings.

In Section 2, we analyze the role of angular dispersion, or pulse-front tilt, in the propagation of light beams in dispersive media and derive the main equations that will be used throughout the tutorial. Sections 3, 4 will describe five optical process and applications that use pulses with angular dispersion.

In Section 3 we consider applications in the field of nonlinear optics. We begin with techniques used to obtain pulse stretching and pulse compression to show that, even in free space, with no dispersive medium present, one can induce an effective anomalous group-velocity dispersion that can be used to broaden or compress optical pulses. Next we analyze how pulses with pulse-front tilt can enlarge the bandwidth of the nonlinear process of second-harmonic generation, which allows frequency upconversion of ultrashort pulses and the generation of ultrashort pulses in new frequency bands. We also analyze why the use of angular dispersion allows the observation of quadratic temporal solitons. Without angular dispersion, the currently used materials exhibit neither the appropriate amount of dispersion nor the appropriate sign. Furthermore, the group-velocity mismatch (GVM) between all the interacting waves is to large to let us observe temporal solitons with reasonable intensities. Finally, we consider the generation of tunable terahertz (THz) waves by means of changing the amount of angular dispersion present.

Section 4 reviews how it is possible to tune the frequency correlations and the bandwidth of entangled photon pairs generated during the process of spontaneous parametric frequency downconversion, which is achieved by tuning the group velocity of the interacting waves by means of angular dispersion.

## 2. Angular Dispersion: How Does it Work?

The aim in this tutorial is to demonstrate that angular dispersion can be used as a tool in nonlinear and quantum optics. The key point of this demonstration is to show that, in an appropriate configuration, dispersive elements allow us to modify the dispersive response of materials; in particular, they allow us to control the group velocities and group-velocity dispersion of light beams traveling through them.

In this section, the effects of angular dispersion in a medium flanked by two dispersive elements will be discussed. We will derive analytical expressions for the effective group velocities and effective group-velocity dispersion of materials placed between two gratings. These expressions will be the key points used throughout the rest of the tutorial to exemplify how angular dispersion can be considered a tool in nonlinear and quantum optics.

The tutorial will be centered mainly on angular dispersion introduced by diffraction gratings. At the beginning of this section, the effects of a grating on a pulse beam impinging on it will be considered. In particular, it will be shown that angular dispersion produces pulse-front tilt; i.e., it tilts the front of a pulse by a certain angle. The value of the angle that describes this tilt will be used to quantify the effect of dispersive elements. Moreover, it will be shown that the tilt angle is indeed the parameter that will allow us to tune the dispersive properties of different media. For completeness, the derivation of the pulse-front tilt angle produced by a prism is also included.

Finally, a brief discussion about the terms *angular dispersion* and
*pulse-front tilt* is presented. Although in most parts of the
tutorial both terms will be used without distinction, we will see that indeed they
represent different physical effects. In most situations they appear together, but one
can find scenarios where there are pulses with pulse-front tilt but without angular
dispersion and vice versa.

#### 2.1. Effect of a Diffraction Grating on an Optical Beam

A reflection diffraction grating (see Fig. 2) is formed by a periodically corrugated reflecting surface, where the reflecting elements, the grooves, are separated by a distance comparable with the wavelength of light. An incident beam of light is separated, on reflection, into different waves; i.e., several reflected waves come off at different angles that mark the individual diffraction orders. The amount of energy that is actually reflected in each diffraction order depends on the specific shape of the grooves on the grating surface and can be appropriately designed [5]. On the other hand, the directions of propagation of the reflected waves are independent of the specific shape of the grating surface and are given by the grating equation [6]

*λ*is the wavelength of the radiation,

*d*is the groove spacing, and

*m*is the diffraction order.

Let us consider an incident wave with a spectral distribution centered at angular
frequency ${\omega}_{0}$ (that corresponds to a wavelength ${\lambda}_{0}$ in the medium) impinging on a grating. In this case, the expression
for the angular dispersion, *γ*, produced by a grating can be
found by expanding Eq. (1) up to first
order. Let us assume that the grating is oriented in such a way that the angular
dispersion occurs in the *x* direction. The angles of incidence and
diffraction at the grating can be expressed as $\overline{\theta}={\theta}_{0}+\theta $ and $\overline{\u03f5}={\u03f5}_{0}+\u03f5$, where ${\theta}_{0}$ and ${\u03f5}_{0}$ are the central angles of incidence and diffraction, and
*θ* and *ϵ* are small deviations
from the corresponding central values. Analogously, $\lambda ={\lambda}_{0}+\Delta \lambda $. If we take into account that $\theta \ll {\theta}_{0}$, $\u03f5\ll {\u03f5}_{0}$, and $\Delta \lambda \ll {\lambda}_{0}$, Eq. (1) can be
written up to first order as [7, 8]

In the slowly varying envelope approximation, the amplitude of an electric field at a
spatial position $(x,y,z)$ and at time *t* can be written as

*c*is the velocity of light in vacuum. In general, $A(x,y,z,t)$ is conveniently written as the transverse wavenumber variable $\mathbf{q}=({q}_{x},{q}_{y})$ and the deviation from the central frequency

*Ω*:

Equation (2) allows us to find a
relationship between the wave vectors of the incident and the diffracted waves. For
clarity, we will use the subindex 1 to refer to the variables of the beam before the
grating and the subindex 2 for the variables after the diffraction at the grating.
When impinging on a grating, the incident field is diffracted, and each frequency
component is dispersed in a different direction. Consider a grating that introduces
angular dispersion in the *x* coordinate, as depicted in Fig. 3. In this case, the variables of interest become
the *x* component of **q** and the *x*
component of the transverse wavenumber after diffraction at the grating, $\mathbf{p}=({p}_{x},{p}_{y})$. By using geometrical considerations and the fact that
*θ* and *ϵ* are small, it is
possible to show that ${p}_{x}={k}_{0}\u03f5$ and ${q}_{x}={k}_{0}\theta $. For a grating immersed in a vacuum, the index of refraction is
equal to one.

Considering that ${(\partial \u03f5\u2215\partial \lambda )}_{{\lambda}_{0}}\Delta \lambda ={(\partial \u03f5\u2215\partial \omega )}_{{\omega}_{0}}\Omega $, ${q}_{x}$ can be rewritten as

Equation (7) relates the transverse wavenumber of the diffracted and incident beams. The transformation of the optical beam due to the presence of the grating placed at ${z}_{1}={z}_{2}=0$ can then be written as

To further illustrate the effects of a grating on the optical beam, let us consider the amplitude of the electric field at a distance ${z}_{2}$ from the grating. Neglecting the temporal and spatial broadening of the optical pulse due to diffraction and dispersion effects, propagation over a distance ${z}_{2}$ corresponds to the transformation

Equation (12) describes the
transformation of the shape of the electric field amplitude by the grating. This
provides us with a further physical insight into the grating effect. For $A({x}_{1},{y}_{1},{z}_{1},t)$ the transverse and longitudinal components are independent;
therefore, at any time, we observe that the front of the pulse is perpendicular to
the propagation direction ${z}_{1}$. The left-hand part of Fig. 3
depicts, for a fixed time, an input beam whose intensity is spatially described by a
Gaussian function with a beam waist in the *x* direction ${w}_{0x}$. On the other hand, after the beam has been reflected by the
grating, Eq. (12) tells us that the
transverse and longitudinal variables are no longer independent. As a consequence,
the pulse-front is no longer perpendicular to the propagation direction, as shown in
the right-hand part of Fig. 3.

To clarify this idea, consider the loci of the peak intensities

It is clear from this expression that the temporal and spatial variables are not independent after the pulses passes through the grating. For a fixed time $(t=0)$, the front of the pulse is given by the line ${z}_{2}=-\mathrm{tan}\phantom{\rule{0.2em}{0ex}}\Phi {x}_{2}\u2215\left(c{k}_{0}^{\prime}\right)$ in the plane $({x}_{2},{z}_{2})$. This corresponds to a straight line with a slope given by [9, 10]*ν*, and consequently the front of the pulse is no longer perpendicular to the propagation direction ${z}_{2}$. By using Eqs. (2, 8) we obtain that the tilt angle for a grating is given by

From Eq. (13), we also see that for a
fixed distance ${z}_{2}$
$({z}_{2}=0)$ the maximum of the field arrives at different times, $t={x}_{2}\phantom{\rule{0.2em}{0ex}}\mathrm{tan}\phantom{\rule{0.2em}{0ex}}\Phi \u2215c$, for each transverse coordinate ${x}_{2}$. Therefore, it becomes evident that the angle
*Φ* is the angle of the loci of the peak intensities in the $({x}_{2},ct)$ plane, as shown in Fig. 4.

In what follows, we will see that the tilt angle can be used as a control parameter to modify the dispersive response of a material at will, and in that way angular dispersion can become a tool in nonlinear and quantum optics.

#### 2.2. How to Use Angular Dispersion to Control Material Dispersive Properties

Angular dispersion can be effectively and expediently used to control the dispersive properties of materials. Let us consider a medium flanked by two gratings or prisms as shown in Fig. 5. As we will see below, this configuration allows us to tune at will the group velocities, group-velocity dispersion (GVD), and higher dispersive terms of the fields propagating in the medium. It is precisely this point that makes angular dispersion an enabling tool in many areas of optics. We will see particular examples in Sections 3 and 4.

The effect of a diffraction grating on the propagation of a pulse is to tilt the
front of the pulse by an angle *ν*, as depicted in Fig.
3. Alternatively, we can also see that the
line of the loci of peak intensities is tilted by an angle
*Φ*, as shown in Fig. 4.

To see how the particular configuration depicted in Fig. 5 can help tune the dispersive properties of the material, let us start by considering what happens when a pulse with pulse-front tilt enters a medium with a different refractive index. Let us assume that the beam enters the material normally. Making use of Snell’s law to first order in the angles and wavelength, we obtain

where ${\u03f5}_{1}$ $\left({\u03f5}_{2}\right)$ is the angle of refraction inside (outside) the material, and ${n}_{1}$ $\left({n}_{2}\right)$ is the refractive index. The wavenumbers inside $\left({k}_{1}\right)$ and outside $\left({k}_{2}\right)$ the material are different and are related by ${k}_{2}={n}_{2}{k}_{1}\u2215{n}_{1}$. With these expressions in hand, we can see that neither the transverse wavenumber nor the tilt angle*Φ*, given by Eq. (8), changes when the beam with pulse-front tilt enters the medium. For normal incidence, the amplitude of the field at the boundary after entering the material is still $a({q}_{x}\u2215\alpha -\Omega \phantom{\rule{0.2em}{0ex}}\mathrm{tan}\phantom{\rule{0.2em}{0ex}}\Phi \u2215\left(\alpha c\right),{q}_{y},\Omega ,{z}_{2}=0)$.

Let us now consider the medium to be dispersive; i.e., let us let its refractive
index vary with frequency. Further, we assume that the medium fills all the space
between the two gratings, as shown in Fig. 5.
From now on, we will work mostly with the diffracted beam, and, in order to simplify
the notation, we will drop the use of the subindex 2 used to refer to the diffracted
beams, so that $({x}_{2},{y}_{2},{z}_{2}\to x,y,z)$. The electric field amplitude at any propagation distance
*z* inside the medium can be written as

*z*in the absence of any other effects.

*ρ*is the Poynting vector walk-off angle, or spatial walk-off angle. In frequently used nonlinear materials, such as $\beta -\mathrm{Ba}{\mathrm{B}}_{2}{\mathrm{O}}_{4}$ (BBO), $\mathrm{Li}\mathrm{Nb}{\mathrm{O}}_{3}$, or $\mathrm{K}\mathrm{Ti}\mathrm{O}{\mathrm{PO}}_{4}$ (KTP), its typical values are $\rho \sim 0\xb0\u20135\xb0$ [11].

To understand the consequences of light with pulse-front tilt traveling through a dispersive media, let us expand $k\left(\omega \right)$ up to second order about ${\omega}_{0}$. Defining the inverse group velocity ${k}_{0}^{\prime}={(\partial k\u2215\partial \omega )}_{{\omega}_{0}}$ and the inverse GVD ${k}_{0}^{\u2033}={({\partial}^{2}k\u2215\partial {\omega}^{2})}_{{\omega}_{0}}$, it follows that

By substituting Eq. (18) into Eq. (17), we obtain the field amplitude at $z=L$:To see the effect of the second grating, let us consider a grating characterized by
the parameters ${\alpha}^{\prime}$ and ${\Phi}^{\prime}$. This second grating introduces angular dispersion in the
*x* direction as well, and it is oriented in such a way that it
satisfies $\alpha {\alpha}^{\prime}=1$ and $\mathrm{tan}\phantom{\rule{0.2em}{0ex}}{\Phi}^{\prime}=-\mathrm{tan}\phantom{\rule{0.2em}{0ex}}\Phi \u2215\alpha $. With these parameters, the transformation of Eq. (9) for the second grating
becomes

To simplify this expression and see more clearly the physics behind it, let us assume
that the input beam has an elliptical spatial shape with a large beam width in the
*x* direction, ${w}_{0x}\gg {w}_{0y}$. Then $a({q}_{x},{q}_{y},\Omega ,z)\simeq a({q}_{y},\Omega ,z)\delta \left({q}_{x}\right)$, and the field of Eq. (21) becomes

*x*transverse dimension, ${w}_{0x}$, should be larger than the lateral displacement of the beam due to spatial walk-off, $L\phantom{\rule{0.2em}{0ex}}\mathrm{tan}\phantom{\rule{0.2em}{0ex}}\rho $. This implies that the validity of Eq. (22) is restricted to propagation distances $z<{w}_{0x}\u2215\mathrm{tan}\phantom{\rule{0.2em}{0ex}}\rho $. For example, for ${w}_{0x}=1\text{\hspace{0.17em}}\mathrm{mm}$ and a walk-off angle $\rho =5\xb0$, the propagation length should be $L\le 1\text{\hspace{0.17em}}\mathrm{cm}$.

From Eq. (22), we can see that the
evolution of the spatial shape in the *y* transverse coordinate is the
well-known expression for the diffraction of a beam. In the time domain, the effect
of the presence of pulse-front tilt can be described by the introduction of two
effective dispersive parameters [12, 13, 14]:
an effective inverse group velocity,

It is important to remark that Eq. (23) refers to the effective group velocity along the propagation direction
*z*, ${k}_{0,\mathrm{eff}}^{\prime}$. In this case, the change of group velocity requires the existence
of spatial walk-off $(\rho \ne 0)$. For this reason, optical beams with pulse-front tilt propagating
in vacuum or in noncritical directions (with $\rho =0$) in birefringent crystals do not modify their group velocity when
placed between two gratings. On the other hand, the situation is different for the
effective GVD: even when no dispersive material is present, it is possible to observe
an effective anomalous dispersion. This can be clearly seen by setting ${k}_{0}^{\u2033}=0$ in Eq. (24) so that
the effective anomalous dispersion is ${k}_{0,\mathrm{eff}}^{\u2033}=-{\mathrm{tan}}^{2}\phantom{\rule{0.2em}{0ex}}\Phi \u2215\left({k}_{0}{c}^{2}\right)$.

The fact that an effective anomalous dispersion always accompanies angular dispersion in a vacuum was shown in [15]. It was demonstrated that the optical transfer function for a beam traversing a pair of gratings, arranged in tandem, contains a quadratic frequency term that is responsible of the appearance of anomalous dispersion. This fact allows GVD-free propagation to be achieved in dispersive media as demonstrated in [16].

Equations (23, 24) are the key results of this section.
They demonstrate that by means of the tilt angle *Φ*, i.e., by
the amount of pulse-front tilt introduced by diffraction gratings, it is possible to
modify the inverse group velocity and the GVD parameters of a material placed between
the two grating as depicted in Fig. 5. It is
precisely this capability that will become the crucial point to make angular
dispersion a tool in nonlinear and quantum optics.

The propagation of pulses with pulse-front tilt *Φ*
produced by angular dispersion in a dispersive medium can be described by an
effective inverse group velocity and an effective inverse group-velocity
dispersion (GVD) given by

#### 2.3. Angular Dispersion Produced by a Prism

Diffraction gratings are not the only optical devices that can introduce angular dispersion [17, 18]. One outstanding example is the prism [19]. Recalling Eq. (8), the tilt angle is determined solely by the amount of angular dispersion present. For completeness, here we derive the pulse-front tilt introduced by prisms following the general procedure used previously for the grating.

Let us consider a prism with an apex angle *C* and refractive index $n\left(\lambda \right)$. Light impinges on the prism at an angle $\overline{\theta}$ and exits at an angle $\overline{\u03f5}$ after suffering refraction at the two interfaces of the prism, as
depicted in Fig. 6. Using Snell’s law
to describe refraction at both surfaces, one has

The light impinging on the prism has a spatial and spectral distribution centered at ${\theta}_{0}$ and ${\lambda}_{0}$, respectively. The spatial and spectral variables can be conveniently written as $\overline{\theta}={\theta}_{0}+\theta $, $\overline{\u03f5}={\u03f5}_{0}+\u03f5$, and $\lambda ={\lambda}_{0}+\Delta \lambda $.

Expanding Eqs. (25) and (26) up to first order about ${\theta}_{0}$, ${\u03f5}_{0}$, and ${\lambda}_{0}$, one obtains

whereNotice that Eq. (27) is equal to Eq. (2), which was obtained for a diffraction grating, although each equation was derived in a different way. For the grating, Eq. (2) comes from the grating equation that is a consequence of interference at a periodic structure, while Eq. (27) comes from Snell’s law, which is a consequence of material dispersion. When expanded to first order, both devices lead to the same effect: the linear dependence of the angle of diffraction (refraction) on the angle of incidence and the wavelength. We mention that it is possible to derive a general framework that features the common working principle of different spectroscopic devices [20]. With this approach, the dispersion produced by a prism can be treated by the same formalism as the dispersion introduced by a grating.

For minimum deviation configurations in which ${\theta}_{0}={\u03f5}_{0}$, and for incidence at the Brewster angle, $\mathrm{tan}\phantom{\rule{0.2em}{0ex}}{\u03f5}_{0}={n}_{0}$, Eq. (27) becomes [18]

Let us have a first glimpse at the amount of tilt induced by a prism and a grating. Let light at $800\text{\hspace{0.17em}}\mathrm{nm}$ illuminate a commercially available grating with $G=1\u2215d=1400\text{\hspace{0.17em}}\mathrm{lines}/\mathrm{mm}$. Let the input angle be ${\theta}_{0}=20\xb0$, which results in an output angle of the first diffraction order $m=1$ of ${\u03f5}_{0}=51.1\xb0$. In this case, the angular dispersion is ${(\partial \u03f5\u2215\partial \lambda )}_{{\lambda}_{0}}=0.128\xb0\u2215\mathrm{nm}$, and the tilt angle, given by Eq. (15), is $\Phi =-60.7\xb0$. On the other hand, for a prism [21] with a refractive index ${n}_{0}=1.457$ and ${(\partial n\u2215\partial \lambda )}_{{\lambda}_{0}}=-0.002\xb0\u2215\mathrm{nm}$ at a wavelength of $600\text{\hspace{0.17em}}\mathrm{nm}$, the tilt angle is $\Phi =2.4\xb0$. The resulting values of the tilt angle give a hint that one can obtain less angular dispersion with currently used prisms than with conventional gratings. Notwithstanding, losses can be much higher in setups with gratings than in those with prisms, unless an optimized grating is designed. We should notice that a larger amount of pulse-front tilt can be obtained by combining sequences of prisms [18] or by using combinations of gratings and prisms [22].

#### 2.4. Pulse-Front Tilt versus Angular Dispersion

Until now, we have seen that angular dispersion generates pulse-front tilt. In the cases considered, we have used both concepts indiscriminately to describe the effect of a prism and of a grating. But is the introduction of angular dispersion the only option to generate pulse-front tilt? This question was addressed in [23], where the authors showed that spatial chirp may also lead to the generation of pulses with pulse-front tilt. Although there may be alternative procedures, for the sake of simplicity we will restrict ourselves to this case.

Previously we analyzed the concept of the tilt angle in the spatial domain $(x,z)$ and in the spatiotemporal domain $(x,ct)$. Since the spatial chirp can be conveniently described [24] in the spatial and frequency degrees of freedom $(x,\Omega )$, in what follows we will focus our analysis on these variables. First, let us consider the amplitude of a pulse on diffraction off a grating in the $(x,\Omega )$ variables. This is obtained by Fourier transforming Eq. (12) from the time to the frequency domain and for ${z}_{2}=0$. To obtain analytical results, we assume the input beam to have a Gaussian shape in time and space, characterized by the widths ${T}_{0}$ and ${w}_{0}={w}_{0x}={w}_{0y}$, respectively. With these assumptions, the complex envelope of the beam on diffraction is written as

Let us consider light beams that apart from angular dispersion display linear spatial
chirp and GVD characterized by the parameters *ξ* and
*δ*, respectively. The amplitude of the beam is now written
as

Pulses diffracted by two gratings, as described in the previous sections, are a good example of pulses that show pulse-front tilt without angular dispersion. Between the two gratings, the beam propagates in free space. Performing the Fourier transform of $A(x,y,z,t)$ of Eq. (21) from time to frequency, we obtain

*z*is the distance between the gratings.

Comparing with Eq. (33), we can see that the diffraction effects arising from the propagation from one grating to the other introduce spatial chirp $\xi =\alpha z\mu \u2215k$ and an effective second-order dispersion parameter $\delta =-{\mu}^{2}z\u2215k$.

One can have pulses that show pulse-front tilt and no angular dispersion. This can be achieved by introducing, for instance, spatial chirp and group-velocity disperion (GVD). The combined effect of two dispersive elements and diffraction produced by the propagation of the pulse from one element to the other can effectively generate the spatial chirp and GVD required.

## 3. Angular Dispersion in Nonlinear Optics

Since the invention of the laser, nonlinear optics has become one of the most fruitful areas of optics. Nonlinear effects are employed in many important applications of optical technologies, such as optical fiber communications or high-resolution imaging and detection of biological tissue. Two important concepts that are at the core of nonlinear optics are considered here: the generation of optical waves at new wavelengths, analyzed in Subsections 3.2, 3.4, and the existence of an optical entity that can exist only in the realm of nonlinear optics: the soliton. Solitons are analyzed in Subsection 3.3.

Paradoxically, the observation of most effects in nonlinear optics depends on the specific linear properties of the materials used. It is here where angular dispersion, a linear effect, plays an important role, making it possible to modify the dispersive properties of materials to allow the observation of certain nonlinear effects [25].

#### 3.1. Pulse Compression and Pulse Stretching

The first two important applications that we analyze in this tutorial that use the angular dispersion introduced by a series of gratings or prisms are pulse compression and chirped pulse amplification (CPA). In both cases, the combined use of nonlinear effects and dispersive effects aims at shortening (pulse compression) or broadening (pulse stretching) pulses in the temporal domain in order to amplify ultrashort pulses.

### 3.1a. Pulse Compression Techniques

The goal of pulse compression techniques considered in this subsection is to modify the properties in frequency and time of a transform-limited pulse of time duration ${T}_{1}$ in order to generate a new pulse with a shorter duration ${T}_{2}<{T}_{1}$. This can be achieved with the scheme shown in Fig. 7. The goal of the first stage is to broaden the spectrum of the pulse. For this purpose, we can use, for instance, the nonlinear effect of self-phase modulation (SPM) in an optical fiber [4]. In SPM, the temporal width of the pulse is not changed, but because of the appearance of a quadratic temporal phase chirp, the bandwidth is enhanced. Afterward, one needs to translate the frequency broadening into pulse compression, erasing any frequency chirp introduced in the first stage. This will render the new pulse transform limited again, but this time with a shorter time duration thanks to the increased bandwidth.

Let us assume that the input pulse is a transform-limited Gaussian pulse that can be written in the temporal and frequency domains as

*Ω*is the frequency deviation from the central frequency ${\omega}_{0}$ of the pulse. The full width at half-maximum (FWHM) bandwidth of the pulse is ${B}_{1}={\left(8\phantom{\rule{0.2em}{0ex}}\mathrm{ln}\phantom{\rule{0.2em}{0ex}}2\right)}^{1\u22152}\u2215{T}_{1}$. In an optical fiber, depending on the peak power and the time duration of the pulse, many dispersive and nonlinear effects might have to be considered. For the sake of argument, let us suppose that the main effect that affects the pulse propagation in the optical fiber is SPM. In SPM, there is a time-dependent phase change, proportional to the intensity of the pulse, that is added during propagation: where $P\left(t\right)={\left|A(t,z)\right|}^{2}$ is the power,

*z*is the length of the fiber, ${\gamma}_{f}={\omega}_{0}{n}_{2}\u2215\left(c{A}_{\mathrm{eff}}\right)$ describes the effective nonlinearity induced by the fiber, ${n}_{2}$ is the nonlinear index coefficient, and ${A}_{\mathrm{eff}}$ is the effective mode cross section. To get further physical insight and obtain some analytical results, we expand the expression of the power about $t=0$ so that $P\left(t\right)\sim {P}_{0}(1-2{t}^{2}\u2215{T}_{1}^{2})$. In this way, we get an approximate expression of the pulse in time at the end of the nonlinear fiber that is written as

After Fourier transforming this equation, we get

Equation (42) shows that SPM also introduces a positive quadratic frequency chirp of the form $\mathrm{exp}(i{\alpha}_{\mathrm{SPM}}{\Omega}^{2}\u22152)$ with

The angular dispersion introduced by gratings or prisms can be used to compensate
such a quadratic chirp. As was shown in Subsection 2.2, a pair of gratings separated by a distance *L* in a
vacuum introduces a quadratic negative frequency chirp ${k}_{\mathrm{eff}}^{\u2033}=-{\mathrm{tan}}^{2}\phantom{\rule{0.2em}{0ex}}\Phi \u2215\left(k{c}^{2}\right)$. Therefore, to achieve compensation it is required that ${\alpha}_{\mathrm{SPM}}={k}_{\mathrm{eff}}^{\u2033}$.

The combined effect of self-phase modulation, which broadens the bandwidth of a pulse while it propagates down an optical fiber, and of the angular dispersion introduced by dispersive elements (gratings and/or prisms), which renders the resulting pulse transform limited, allows the generation of a new, compressed pulse with a shorter time duration.

As an example, in [26] $90\text{\hspace{0.17em}}\mathrm{fs}$ input pulses at the wavelength of $619\text{\hspace{0.17em}}\mathrm{nm}$ were focused into a polarization-preserving fiber $15\text{\hspace{0.17em}}\mathrm{cm}$ long. The authors observed a factor of 3 increase in the frequency spectrum, from $6\text{\hspace{0.17em}}\mathrm{nm}$ to about $20\text{\hspace{0.17em}}\mathrm{nm}$. A pair of gratings with $600\text{\hspace{0.17em}lines}\u2215\mathrm{mm}$ and a slant angle of 30° set $6.4\text{\hspace{0.17em}}\mathrm{cm}$ apart were used to compress the pulse after propagation in the fiber, measuring the output pulses with a time duration of $30\text{\hspace{0.17em}}\mathrm{fs}$.

For the sake of simplicity, we have considered only the effects of the presence of quadratic phase terms in the frequency domain. Notwithstanding, apart from SPM, other dispersive and nonlinear effects might cause the appearance of nonquadratic frequency chirp terms, especially when we are dealing with ultrashort pulses. The effects to be included are higher-order dispersion, cross-phase modulation, self-steepening, and the self-induced Raman effect [4]. On the other hand, pairs of gratings also introduce nonquadratic phase terms that should also be taken into account. With this information in hand, one can use appropriately engineered combinations of prisms and gratings to erase the frequency chirp generated in the nonlinear fiber. In [27], the authors used combinations of prisms and gratings to compensate not only for the quadratic but also for the cubic phase of ultrashort optical pulses. They obtained compressed pulses as short as $6\text{\hspace{0.17em}}\mathrm{fs}$.

In general, as one moves to shorter pulses and higher peak powers, the phase terms, induced by nonlinear and dispersive effects during propagation in the fiber, present more complicated shapes in an increasingly larger bandwidth. In order to compensate for these phases for generating ultrashort pulses, one needs to consider appropriately tailored arrangements of prisms and gratings that are correspondingly more sophisticated [19].

### 3.1b. Chirped Pulse Amplification

Until now, we have analyzed a two-stage pulse compression scheme: the first stage broadens the spectrum, and the second stage renders the pulse transform limited. Certain applications require not only pulse compression at a certain stage, but also pulse stretching in another stage. This is the case for chirped pulse amplification (CPA), a scheme to amplify pulses that avoids the serious damage that high peak powers of several gigawatts per square centimeter can cause to gain media due to the effect of self-focusing [28]. To avoid this damage, one should reduce the peak power of the pulse before the pulse enters the amplifying stage.

The general scheme of CPA is shown in Fig. 8(a). The first stage is intended to reduce the peak power of the input pulse by introducing a quadratic phase term. Although this can be achieved by propagation in an optical fiber, the high powers used can nonetheless generate other undesirable effects that can be avoided by employing angular dispersion with pairs of gratings. If we again consider an input pulse given by Eq. (39) with peak power ${P}_{1}$, the pulse at the end of the pulse stretching stage is written as

*L*is the separation between them. The peak power ${P}_{2}$ of the pulse after the first stage isand its time duration ${T}_{2}$ isAs an example, let us consider a pair of gratings with $1400\text{\hspace{0.17em}lines}\u2215\mathrm{mm}$, separated by a distance $L=20\text{\hspace{0.17em}}\mathrm{cm}$. For an angle of incidence of 20°, the effective GVD at $800\text{\hspace{0.17em}}\mathrm{nm}$ is ${k}^{\u2033}=-4495\text{\hspace{0.17em}}{\mathrm{fs}}^{2}\u2215\mathrm{mm}$. A $100\text{\hspace{0.17em}}\mathrm{fs}$ pulse is stretched to $18\text{\hspace{0.17em}}\mathrm{ps}$, and the peak power after the pulse stretching stage is ${P}_{2}\sim 6\times {10}^{-3}{P}_{1}$.

The second stage of CPA is the amplification of the pulse. Assuming that the gain bandwidth is broad enough to amplify the whole frequency spectrum of the pulse and that it does not introduce any frequency chirp, the final stage should compensate for the angular dispersion introduced in the first stage. Therefore, it should introduce an effective quadratic chirp of value ${k}_{\mathrm{eff}}^{\u2033}L$.

The role of angular dispersion in chirped pulse amplification (CPA) is to stretch a pulse before it is amplified in order to reduce its peak power and to recompress it afterwards.

We have seen in Subsection 2.2 that a pair of
gratings separated by a distance *L* can introduce only an
effective anomalous dispersion (negative dispersion). To generate effective GVD
values with a positive sign, we have to introduce elements that modify the
propagation of the pulse between the two gratings. One example is shown in Fig.
8(b). Two lenses are separated a distance
*d* from each grating. The separation between the two lenses is $2f$, where *f* is the focal lens of each lens. After
some straightforward calculation [29],
which takes into account the paraxial propagation of the pulse from one grating to
the other, one finds that the transfer function of the system can be written
as

#### 3.2. Achromatic Phase Matching

### 3.2a. Broadband Second-Harmonic Generation

Second-harmonic generation (SHG) is a nonlinear process in which light at a certain frequency ${\omega}_{1}$ (fundamental frequency, FF) interacts with a nonlinear crystal and generates a nonlinear polarization in the medium at double frequency ${\omega}_{2}=2{\omega}_{1}$ that causes the generation of optical radiation at the same frequency ${\omega}_{2}$ (the second harmonic, SH). At a more fundamental level, SHG is a process in which the atoms or molecules that make up the nonlinear material absorb two photons at frequency ${\omega}_{1}$ and re-emit a single photon at frequency ${\omega}_{2}$. To do so, the interacting waves have to satisfy the energy and momentum conservation laws.

In the field of nonlinear optics, momentum conservation is usually called the phase-matching condition, which comes from the requirement to match the phases of the interacting waves [29, 31, 32]. The electric field of the fundamental and second-harmonic waves can be written as ${E}_{i}(x,y,t,z)=1\u22152{A}_{i}(x,y,t,z)\mathrm{exp}(i{k}_{i}z-i{\omega}_{i}t)+\mathrm{h.c.}$, $i=1,2$, and the consequent phase-matching condition is

Normally we try to work with optical materials that are transparent at the wavelengths of interest. Due to Kramers–Kronig relations that relate the real and imaginary parts of susceptibility (i.e., relate dispersion to absorption), the absence of losses results in normal dispersion. Since ${k}_{i}={n}_{i}\left({\omega}_{i}\right){\omega}_{i}\u2215c$, the condition imposed on refractive indices, i.e., ${n}_{2}\left({\omega}_{2}\right)={n}_{1}\left({\omega}_{1}\right)$ that results from Eq. (50), would be difficult to satisfy.On the other hand, for one wavelength it is relatively easy to satisfy the condition given by Eq. (50) in birefringent media where the polarization of the fundamental and second-harmonic waves are chosen to be mutually orthogonal and their refractive indices follow different dispersion curves. It is, however, more complicated to satisfy the phase-matching condition for a broader range of frequencies, which is necessary for the conversion of short light pulses that exhibit a correspondingly broader spectrum.

The **k** vectors of the interacting fields can be expanded in a Taylor
series about the central frequency, i.e., ${k}_{i}({\omega}_{i}+{\Omega}_{i})={k}_{i}+{k}_{i}^{\prime}{\Omega}_{i}+1\u22152{k}_{i}^{\u2033}{\Omega}_{i}^{2}$, where ${\omega}_{i}$ are the central frequencies and ${\Omega}_{i}$ are frequency deviations. The fulfilment of the phase-matching
condition given by Eq. (50) over a
broader range of frequencies can then be satisfied when the group velocities of
the interacting waves are equal,

The GVM could be neglected when the crystal length *L* is much
smaller than the temporal walk-off length ${L}_{\mathrm{gvm}}$, $L\ll {L}_{\mathrm{gvm}}$, which is defined as

*L*is proportional to [33, 34]

It is possible to increase the bandwidth of spectral acceptance by using a shorter nonlinear crystal, because the spectral acceptance is inversely proportional to $({k}_{2}^{\prime}-{k}_{1}^{\prime})L$. A major disadvantage is a serious reduction of the conversion efficiency, because the signal energy at the central frequency is proportional to the square of the crystal length. The decrease of the crystal length would require a corresponding increase of the input power to compensate for the reduction of conversion efficiency. By extension, the reduction of the spectral acceptance of the nonlinear crystal effectively sets a minimum duration of the SH pulse that can be achieved. For example, in [35], the spectral acceptance of a $1\text{\hspace{0.17em}}\mathrm{mm}$ long BBO crystal was measured at a FF of $496\text{\hspace{0.17em}}\mathrm{nm}$. The obtained acceptance of $0.52\text{\hspace{0.17em}}\mathrm{nm}$ imposes an approximate minimum pulse duration of $700\text{\hspace{0.17em}}\mathrm{fs}$, assuming a Gaussian pulse shape.

It is here where the angular dispersion comes to help. It was shown in Subsection 2.2 that with the help of angular dispersion it is possible to control the group velocities and higher-order dispersion terms. Simply changing the angle of incidence of light at a diffraction grating, we can tune the dispersive properties to the desired values.

Let us try to get a further insight into the effects of the use of pulses with angular dispersion in the process of SHG [36, 37]. Let us consider the full frequency dependence of the FF and SH wave vectors ${k}_{1}({\omega}_{1}+{\Omega}_{1})$ and ${k}_{2}({\omega}_{2}+{\Omega}_{2})$, respectively. In order to enhance the efficiency of SHG over a larger frequency range, the phase-matching condition must be fulfilled over a broader range of frequencies. If the beam passes through a prism or a grating that introduces angular dispersion, each frequency of the outgoing diverging beam propagates in a different direction. One can choose such angular dispersion so that each frequency enters the nonlinear medium at such an angle that the phase-matching condition is satisfied for every frequency along its direction of propagation. The phase-matching condition, given by Eq. (50), requires now that

We let ${\lambda}_{10}$ and ${\lambda}_{20}$ denote the central wavelengths of the FF and SH waves in vacuum, respectively, and ${\u03f5}_{0}$ the angle of the direction of propagation of the central frequencies. The refractive indices of both waves are denoted ${n}_{1}$ and ${n}_{2}$. If we expand Eq. (54) to first order and use the expression for the total derivative ${(d{n}_{1}\u2215d\lambda )}_{{\lambda}_{10}}={(\partial {n}_{1}\u2215\partial \lambda )}_{{\lambda}_{10}}+{(\partial {n}_{1}\u2215\partial \u03f5)}_{{\u03f5}_{0}}{(\partial \u03f5\u2215\partial \lambda )}_{{\lambda}_{10}}$, and similarly for the SH wave, we obtain

The use of pulse-front tilt allows us to fulfill the condition $L\ll {L}_{\mathrm{gvm}}$ by effectively increasing the temporal walk-off length ${L}_{\mathrm{gvm}}$. Its maximum value is obtained when the effective group velocities are equal, ${k}_{1,\mathrm{eff}}^{\prime}={k}_{2,\mathrm{eff}}^{\prime}$. Without loss of generality, let us consider a type-I SHG process, where both the FF and the SH waves might be extraordinary waves. Let us designate the spatial walk-off angles as ${\rho}_{1}$ and ${\rho}_{2}$. The effective group velocities of the interacting waves are (see Subsection 2.2)

Let us consider SHG in a uniaxial birefringent crystal. The walk-off angle $(\rho \equiv {\rho}_{1},{\rho}_{2})$ can be written as [11]

*θ*is the angle of the direction of propagation of the wave with respect to the optic axis of the crystal and ${n}_{o}$ and ${n}_{e}\left(\theta \right)$ are the ordinary and extraordinary refractive indices, respectively. After some tedious but straightforward calculations, one finds that

We should notice that Eq. (55) does not explicitly show the need to use a configuration with a nonzero spatial walk-off for the FF and/or SH waves. However, using Eq. (59), we conclude that a configuration where the FF or SH waves experience spatial walk-off is required. It should be noted that this is true in collinear configurations only, where the FF and SH waves propagate in the same direction inside the nonlinear crystal. Group-velocity matching can also be obtained without spatial walk-off in noncollinear geometries, where the FF and SH waves propagate in different directions, as described in [25, 38]. In most of these configurations, even if the spatial walk-off is nonzero, it plays a minor role, because the main correction to the pulse group velocity results from the angle of the noncollinear interaction.

The previous analysis gives us a clear picture of how the use of beams with pulse-front tilt allows us to increase the spectral acceptance bandwidth of the SHG process. Through selection of the appropriate amount of angular dispersion, one generates new effective group velocities of the interacting waves, making the group velocities of the FF and SH waves equal. This is equivalent to selecting directions of propagations for each frequency such that each frequency fulfills the condition of phase-matching in its own direction.

Figure 9(a) shows the evolution of a SH pulse with GVM compensation. Thanks to the perfect phase matching over a broader range of frequencies, the SH quickly and efficiently builds up. By comparison, in Fig. 9(b) without GVM compensation, we can see the SH wave hardly appears, and after a few millimeters of propagation, it disappears completely owing to backconversion that transfers the energy back into the fundamental wave. The evolution of the slowly varying envelopes depicted in Fig. 9 is calculated by use of the evolution equations described in Subsection 3.2b.

Achromatic phase matching uses angular dispersion to modify the group velocity of the fundamental and the second-harmonic (SH) waves in the process of second-harmonic generation (SHG). When the effective group velocities are made equal, the spectral acceptance of the frequency-doubling crystal increases, which allows the generation of SH pulses with enhanced bandwidths and thus shorter time durations.

One of the advantages of the method described here is the possibility to control the group velocities of the interacting waves in any frequency region of interest and in any nonlinear crystal, which significantly extends the range of frequencies and materials that can be used for the generation of ultrashort pulses through the SHG of short input pulses. It is especially important when no materials are available that can directly be used at specific wavelengths of interest [39]. For instance, broadband SH pulses have been generated in type-I BBO at $258\text{\hspace{0.17em}}\mathrm{nm}$ in a $7\text{\hspace{0.17em}}\mathrm{mm}$ long crystal [40], at $330\text{\hspace{0.17em}}\mathrm{nm}$ in a $4\text{\hspace{0.17em}}\mathrm{mm}$ long crystal [41], or around $456\text{\hspace{0.17em}}\mathrm{nm}$ and at $527\text{\hspace{0.17em}}\mathrm{nm}$ in a type-II $3.77\text{\hspace{0.17em}}\mathrm{mm}$ long BBO crystal [39]. In all of these cases, if the nonlinear crystals were used without pulse-front tilt, the bandwidth and the efficiency of the SH wave would have been severely reduced.

Another example of the capability of the pulse-front tilt technique to enhance the
bandwidth of SHG was demonstrated in [42].
The authors used $25\text{\hspace{0.17em}}\mathrm{nm}$ wide FF pulses at $1550\text{\hspace{0.17em}}\mathrm{nm}$ that were to be upconverted in a $1\text{\hspace{0.17em}}\mathrm{cm}$ long periodically poled $\mathrm{Li}\mathrm{Nb}{\mathrm{O}}_{3}$ crystal (PPLN). A schematic of their configuration is shown in
Fig. 10. In their case, the vector
phase-matching condition required an angle of $\sim 61\xb0$ between the FF beam and the grating wave vector of the poling,
as shown in Fig. 10(b). The beam was focused
elliptically to achieve a spot $530\text{\hspace{0.17em}}\mu \mathrm{m}$ wide in the *x* direction in which the
pulse-front tilt was introduced, and $180\text{\hspace{0.17em}}\mu \mathrm{m}$ wide in the *y* direction. Figure 11 shows the main results. $8.3\text{\hspace{0.17em}}\mathrm{nm}$ wide SH pulses at $775\text{\hspace{0.17em}}\mathrm{nm}$ were generated that correspond to a pulse duration of $170\text{\hspace{0.17em}}\mathrm{fs}$ if measured by autocorrelation, considering a Gaussian pulse.
For the sake of comparison, the spectrum of the upconverted wave in collinear SHG
with a crystal of identical length without GVM compensation was also measured. The
measured spectrum was $0.61\text{\hspace{0.17em}}\mathrm{nm}$, which corresponds to $2.9\text{\hspace{0.17em}}\mathrm{ps}$ in time. A noncollinear quasi-phase-matched configuration in
combination with angular dispersion demonstrated a 14-fold increase of the
spectral acceptance.

### 3.2b. Evolution Equations

Let us consider type-I SHG pumped by pulsed FF light. The electric field of the FF and SH waves can be written as ${E}_{i}(x,y,t,z)=1\u22152{A}_{i}(x,y,t,z)\mathrm{exp}(i{k}_{i}z-i{\omega}_{i}t)+\mathrm{h.c.}$ with $i=1,2$ and h.c. standing for Hermitian conjugate. The evolution of the slowly varying amplitudes ${A}_{i}$ can be described by two coupled equations [12]

*z*is the longitudinal coordinate, ${K}_{1}={\omega}_{1}^{2}{\chi}^{\left(2\right)}\u2215\left({k}_{1}{c}^{2}\right)$, and ${K}_{2}={\omega}_{2}^{2}{\chi}^{\left(2\right)}\u2215\left({k}_{2}{c}^{2}\right)$, ${\chi}^{\left(2\right)}$ is the second-order nonlinear coefficient, $\Delta k=2{k}_{1}-{k}_{2}$ is the wave vector mismatch, and ${\Gamma}_{i}$ are the absorption coefficients. The spatial walk-off parameters are given by the angles ${\rho}_{i}$, ${k}_{i}^{\prime}$ are the inverse group velocities, and ${k}_{i}^{\u2033}$ are the GVD parameters.

If the input FF signal has a large beam waist and is tilted, it can be shown that the temporal evolution of the FF and SH waves can be described by [12]

#### 3.3. Solitons in ${\chi}^{\left(2\right)}$ Media

An initially short pulse propagating along the longitudinal direction
*z* in a medium will broaden in time owing to the chromatic
dispersion (GVD) of the material. The pulse will also broaden in the transverse
spatial dimensions $(x,y)$ because of diffraction. The nonlinear interaction of the light beam
with the atoms of the material modifies the main features of light propagation and
can even allow, under appropriate circumstances, the temporal and spatial broadening
caused by dispersion and diffraction to be overcome. The interplay among dispersion,
diffraction, and nonlinearity can produce localized wave packets whose features in
space or time (or both) do not change during propagation. These localized objects are
called spatial or temporal solitons, respectively. When solitons present a
bell-shaped intensity profile, they are called bright solitons. The localized
structures that correspond to a null of the optical signal are called dark solitons.
The existence of bright or dark solitons depends on the specific dispersive
properties of the nonlinear media and on the type of nonlinearity that produce
them.

In certain cases, when wave propagation can be described by very specific equations, such as the Korteweg–de Vries equation or the nonlinear Schrödinger equation, one can give a very precise mathematical definition of what a soliton is [43, 44]. In more real physical scenarios, many effects should be considered that make the evolution equations more complicated. Nevertheless, the equations that describe the evolution of the relevant physical parameters still predict the existence of localized waves with solitonlike behavior. To encompass such solitary waves, one only needs to generalize the definition of what a soliton is [45, 46]. In general, a soliton is a localized structure that propagates undistorted over long distances because of the balance between dispersion and diffraction on the one side and the nonlinearity of the material on the other side.

As has already been said, optical solitons can be considered in the temporal or
spatial domains. Consider a light beam that propagates in a medium. If in some way
the transverse spatial shape of the optical field is confined (for example in a
waveguide), the degrees of freedom of interest that describe the light propagation
are the direction *z* and the temporal variable *t*. In
this case, the solitons are referred to as temporal. On the other hand, if the
temporal variable can be considered constant, for example by using long pulses or
continuous-wave (cw) lasers, the spatial variables will govern the wave propagation
and the solitons are referred to as spatial.

The nonlinear effect responsible for the generation of optical solitons depends on the particular scheme being used. For example, parametric interactions in ${\chi}^{\left(2\right)}$ media allow the generation of multicolor solitons formed by waves with different frequencies. The formation of these so-called quadratic solitons is mediated by the interaction of the FF and the SH waves in a SHG geometry. On the other hand, self-focusing due to cubic ${\chi}^{\left(3\right)}$ Kerr nonlinearity allows the observation of temporal solitons when single-mode optical fibers are used [47], and the photorefractive effect in electro-optic materials can create a saturable nonlinear refractive index, where photorefractive solitons can be observed [48].

In this subsection of the tutorial, we will concentrate on quadratic solitons, i.e.,
the solitons produced in ${\chi}^{\left(2\right)}$ materials in a SHG geometry. The key point to understand the
enabling role of angular dispersion in the generation of solitons can be seen by
defining the effective lengths that determine the relevant quantities that enter into
play and by writing the equations that describe the evolution of the FF and SH waves
as a function of these new variables [49]. If
we consider a typical temporal width ${T}_{0}$, beam waists ${W}_{0x}$ and ${W}_{0y}$, and a value of the peak amplitude *N*, it is
possible to define a normalized field amplitude, ${a}_{i}={A}_{i}/N$, and spatial and temporal dimensionless variables $\tau =t\u2215{T}_{0}$, $s=x\u2215{W}_{0x}$, $\eta =y\u2215{W}_{0x}$, $\xi =z\u2215\left(2{L}_{\mathrm{dis}}\right)$. With these definitions, and using the subindex 1 for the FF wave
and the subindex 2 for the SH, the evolution equations (60) become

- Dispersion length (FF), ${L}_{\mathrm{dis}}={T}_{0}^{2}\u2215\left(2|{k}_{1}^{\u2033}|\right)$
- Dispersion length (SH), ${L}_{\mathrm{dis}}^{\prime}={T}_{0}^{2}\u2215\left(2|{k}_{2}^{\u2033}|\right)$
- Diffraction length (FF), ${L}_{\mathrm{dif}}={k}_{1}{W}_{0y}^{2}\u22152$
- Diffraction length (SH), ${L}_{\mathrm{dif}}^{\prime}={k}_{2}{W}_{0y}^{2}\u22152=2{L}_{\mathrm{dif}}$
- Spatial walk-off length, ${L}_{w}={W}_{0y}\u2215|\mathrm{tan}\phantom{\rule{0.2em}{0ex}}{\rho}_{2}-\mathrm{tan}\phantom{\rule{0.2em}{0ex}}{\rho}_{1}|$
- Temporal walk-off length, ${L}_{\mathrm{gvm}}={T}_{0}\u2215|{k}_{1}^{\prime}-{k}_{2}^{\prime}|$
- Absorption length (FF), ${L}_{\mathrm{abs}}=1\u2215{\Gamma}_{1}$
- Absorption length (SH), ${L}_{\mathrm{abs}}^{\prime}=1\u2215{\Gamma}_{2}$
- Nonlinear length (FF), ${L}_{\mathrm{nl}}=1\u2215{K}_{1}N$
- Nonlinear length (SH), ${L}_{\mathrm{nl}}^{\prime}=1\u2215{K}_{2}N$
- Coherence length, ${L}_{\mathrm{coh}}=\pi \u2215\left|\Delta k\right|$
- $\alpha ={W}_{0y}\u2215{W}_{0x}$

For the sake of simplicity, notice that the diffraction length ${L}_{\mathrm{dif}}$ is defined with the value of the waist ${W}_{0y}$, corresponding to the size in the dimension in which the soliton
will be formed. Angular dispersion is introduced in the transverse dimension
*x*. On the other hand, the spatial walk-off length ${L}_{w}$ is defined with the same value of the waist, because the walk-off
angle lies in the plane of angular dispersion, i.e., in the direction
*x*. For both ${L}_{\mathrm{dif}}$ and ${L}_{w}$, the quantity of interest is ${W}_{0y}$.

The balance between the linear effects included in Eqs. (62) and the nonlinear parametric interaction between the FF and SH waves is mediated by the ratio of the characteristic lengths corresponding to the dispersion and diffraction lengths. The observation of different types of solitons implies that the relationship between all of these characteristic lengths is tailored accordingly.

In order to observe one-dimensional and two-dimensional spatial quadratic solitons, there are three main requisites that have to be satisfied: (a) material dispersion must be small enough to render the temporal effects on the propagation negligible, (b) the crystal length must be larger than, or at least comparable with the diffraction length in order make the effects of nonlinear induced focusing observable, and (c) the spatial walk-off length must be larger than, or at most comparable with the diffraction length $(2{L}_{\mathrm{dif}}\u2215{L}_{w}\sim 1)$ so that solitons can be excited with the currently available peak powers [50, 51].

Spatial quadratic solitons have been observed in one dimension [52]. Materials with appropriate dispersive properties were chosen such that the use of cw, or even picosecond pulses, renders negligible all the temporal effects. For the case of one dimension, a $47\text{\hspace{0.17em}}\mathrm{mm}$ planar waveguide of $\mathrm{Li}\mathrm{Nb}{\mathrm{O}}_{3}$ was used to confine the beam in one transverse dimension. A cw beam with a waist in the nonconfined dimension of $70\text{\hspace{0.17em}}\mu \mathrm{m}$ yields a diffraction length of ${L}_{\mathrm{dif}}=19\text{\hspace{0.17em}}\mathrm{mm}$ and $L\u2215{L}_{\mathrm{dif}}\sim 2.5$. In addition, the spatial walk-off is negligible, and hence the conditions for the generation of spatial solitons are satisfied.

Spatial quadratic solitons have also been observed in two dimensions [53]. They were observed in a $1\text{\hspace{0.17em}}\mathrm{cm}$ type-II KTP bulk sample pumped by a $15\text{\hspace{0.17em}}\mathrm{ps}$ laser pulse at $1064\text{\hspace{0.17em}}\mathrm{nm}$. The beam waist of the input FF beam was ${W}_{0}=20\text{\hspace{0.17em}}\mu \mathrm{m}$, leading to ${L}_{\mathrm{dif}}=2\text{\hspace{0.17em}}\mathrm{mm}$, and the walk-off angles were ${\rho}_{1}=0.19\xb0$ and ${\rho}_{2}=0.28\xb0$, giving ${L}_{w}\sim 13\text{\hspace{0.17em}}\mathrm{mm}$. With these values it is easy to see that it is possible to observe spatial solitons, since $L\u2215{L}_{\mathrm{dif}}\sim 5$ and $2{L}_{\mathrm{dif}}\u2215{L}_{w}\sim 0.3$.

For observing temporal solitons, restrictions analogous to the ones described for spatial solitons have to be satisfied: (a) spatial effects on the beam propagation must be minimized, for example, by using beams with a large waist size, (b) the length of the nonlinear crystal must be larger than, or at least comparable with the dispersion length, (c) the temporal walk-off length must be larger than, or at most comparable with [50, 51] the dispersion length $(2{L}_{\mathrm{dis}}\u2215{L}_{\mathrm{gvm}}\sim 1)$, and (d) the dispersion must have the appropriate sign of anomalous dispersion (i.e., ${k}_{1}^{\u2033},{k}_{2}^{\u2033}<0$) to support the existence of bright solitons. The conditions to obtain spatial and temporal solitons are summarized in Table 1.

From the conditions described above for the observation of temporal solitons, it is clear that the possibility of controlling the group velocities ${k}_{1}^{\prime}$ and ${k}_{2}^{\prime}$ and GVDs ${k}_{1}^{\u2033}$ and ${k}_{2}^{\u2033}$ is crucial for the observation of temporal solitons. As shown in Subsection 2.2, this is precisely what angular dispersion allows us to do, and it thus gives us a tool for the generation of temporal quadratic solitons.

For a quantitative explanation of the use of angular dispersion for the generation of
solitons, let us consider the generation of spatiotemporal solitons in a type-I $\mathrm{Li}\mathrm{I}{\mathrm{O}}_{3}$ crystal pumped by a ${T}_{0}=100\text{\hspace{0.17em}}\mathrm{fs}$ laser pulse centered at $800\text{\hspace{0.17em}}\mathrm{nm}$ [54]. For $\mathrm{Li}\mathrm{I}{\mathrm{O}}_{3}$, the refractive index at the phase-matching angle is ${n}_{1}={n}_{2}=1.87$; the GVM between the fundamental and the SH waves is ${k}_{2}^{\prime}-{k}_{1}^{\prime}=566.9\text{\hspace{0.17em}}\mathrm{fs}\u2215\mathrm{mm}$; the GVD for the FF wave is ${k}_{1}^{\u2033}=197.3\text{\hspace{0.17em}}{\mathrm{fs}}^{2}\u2215\mathrm{mm}$; the GVD for the SH wave is ${k}_{2}^{\u2033}=600.5\text{\hspace{0.17em}}{\mathrm{fs}}^{2}\u2215\mathrm{mm}$; and, finally, the FF and SH walk-off angles that lie in the
*x* direction are ${\rho}_{1}=0\xb0$ and ${\rho}_{2}=4.9\xb0$, respectively. For the generation of spatiotemporal solitons, the
spatial pump profile is elliptical. In the spatial dimension in which the soliton
will be formed, let us say *y*, the waist is chosen such that the
diffraction length is smaller than the length of the crystal. For this type of
experiment, typical crystal lengths are of the order of a few millimeters; so the
waist in the *y* direction is set to be ${W}_{0y}\sim 20\text{\hspace{0.17em}}\mu \mathrm{m}$.

All of these values allow us to calculate the effective length parameters and determine the feasibility of observing quadratic spatiotemporal solitons. The numerical values for the effective lengths relevant to Table 1 are

- ${L}_{\mathrm{dis}}=25.3\text{\hspace{0.17em}}\mathrm{mm}$
- ${L}_{\mathrm{dis}}^{\prime}=8.3\text{\hspace{0.17em}}\mathrm{mm}$
- ${L}_{\mathrm{dif}}=2.9\text{\hspace{0.17em}}\mathrm{mm}$
- ${L}_{\mathrm{dif}}^{\prime}=5.8\text{\hspace{0.17em}}\mathrm{mm}$
- ${L}_{\mathrm{gvm}}=0.18\text{\hspace{0.17em}}\mathrm{mm}$

*x*direction where the beam has a very large beam width, so the condition $2{L}_{\mathrm{dif}}\u2215{L}_{w}\le 1$ is not relevant for soliton formation in the

*y*transverse dimension.

With these numbers, it is possible to see that for a $10\text{\hspace{0.17em}}\mathrm{mm}$ long crystal, $L\u2215{L}_{\mathrm{dif}}=3.4$ and condition (b) for spatial solitons is satisfied. On the other hand, in the temporal domain, $L\u2215{L}_{\mathrm{dis}}=0.4$, $2{L}_{\mathrm{dis}}\u2215{L}_{\mathrm{gvm}}\sim 281$, and the dispersion is normal $({k}_{1}^{\u2033},{k}_{2}^{\u2033}>0)$. It is not then possible to observe quadratic temporal solitons, since the conditions listed above for the observation of temporal solitons are not satisfied.

To observe temporal solitons in the discussed example, it is necessary to reverse the
sign of the GVD parameters, reduce the effects of the GVM, and make the dispersion
length smaller than the length of the material. As was mentioned before, and from the
discussion in Subsection 2.2, one way to
accomplish this is by using angular dispersion. Consider the case in which the $\mathrm{Li}\mathrm{I}{\mathrm{O}}_{3}$ crystal is flanked by two gratings of $1400\text{\hspace{0.17em}lines}\u2215\mathrm{mm}$. When the pump beam impinges on the grating at an angle ${\theta}_{0}=20\xb0$ (the output angle of the first-order diffraction would be ${\u03f5}_{0}=51\xb0$), angular dispersion is introduced in the *x*
dimension, and the front of the pulse becomes tilted by a tilt angle $\Phi =-60.7\xb0$. In order for Eqs. (23, 24) to be applicable,
the pump beam has an elliptical spatial distribution: ${W}_{0x}$ is a few millimeters, and ${W}_{0y}\sim 20\text{\hspace{0.17em}}\mu \mathrm{m}$.

According to the second term of Eq. (24), the tilt angle introduces an additional anomalous dispersion of $-2400\text{\hspace{0.17em}}{\mathrm{fs}}^{2}\u2215\mathrm{mm}$. The new effective GVM is ${k}_{2,\mathrm{eff}}^{\prime}-{k}_{1,\mathrm{eff}}^{\prime}=55.6\text{\hspace{0.17em}}\mathrm{fs}\u2215\mathrm{mm}$. And the new GVDs at the FF and SH waves are ${k}_{1,\mathrm{eff}}^{\u2033}=-2213.8\text{\hspace{0.17em}}{\mathrm{fs}}^{2}\u2215\mathrm{mm}$ and ${k}_{2,\mathrm{eff}}^{\u2033}=-605\text{\hspace{0.17em}}{\mathrm{fs}}^{2}\u2215\mathrm{mm}$, respectively. The new effective characteristic lengths obtained by inserting the gratings are

- ${L}_{\mathrm{dis}}^{\mathrm{eff}}=2.3\text{\hspace{0.17em}}\mathrm{mm}\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}({L}_{\mathrm{dis}}=25.3\text{\hspace{0.17em}}\mathrm{mm})$
- ${L}_{\mathrm{dis}}^{\prime \mathrm{eff}}=8.3\text{\hspace{0.17em}}\mathrm{mm}\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}({L}_{\mathrm{dis}}^{\prime}=8.3\text{\hspace{0.17em}}\mathrm{mm})$
- ${L}_{\mathrm{dif}}=2.9\text{\hspace{0.17em}}\mathrm{mm}\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}({L}_{\mathrm{dif}}=2.9\text{\hspace{0.17em}}\mathrm{mm})$
- ${L}_{\mathrm{dif}}^{\prime}=5.8\text{\hspace{0.17em}}\mathrm{mm}\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}({L}_{\mathrm{dif}}^{\prime}=5.8\text{\hspace{0.17em}}\mathrm{mm})$
- ${L}_{\mathrm{gvm}}^{\mathrm{eff}}=1.8\text{\hspace{0.17em}}\mathrm{mm}\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}({L}_{\mathrm{gvm}}=0.18\text{\hspace{0.17em}}\mathrm{mm})$

The values previously obtained with no pulse-front tilt are shown again for the sake of comparison. It can now be seen that the conditions for the observation of spatiotemporal solitons are satisfied: $L\u2215{L}_{\mathrm{dis}}=4.3$, the magnitude of the GVM between the FF and SH waves has been highly reduced so that $2{L}_{\mathrm{dis}}\u2215{L}_{\mathrm{gvm}}\sim 2.5$, and the dispersion has become anomalous, thus enabling the excitation of bright solitons.

In the experimental implementations of spatiotemporal quadratic solitons, BBO and $\mathrm{Li}\mathrm{I}{\mathrm{O}}_{3}$ are some of the materials most widely used. The first temporal solitons were observed in a $7\text{\hspace{0.17em}}\mathrm{mm}$ type-I BBO crystal, where pulses of $200\text{\hspace{0.17em}}\mathrm{fs}$ duration at $527\text{\hspace{0.17em}}\mathrm{nm}$ were injected [55]. Shortly after [54], spatiotemporal quadratic solitons were observed in a $1\text{\hspace{0.17em}}\mathrm{cm}$ type-I $\mathrm{Li}\mathrm{I}{\mathrm{O}}_{3}$ crystal with highly elliptical $110\text{\hspace{0.17em}}\mathrm{fs}$ pulses at $795\text{\hspace{0.17em}}\mathrm{nm}$. In later work [56], the generation of spatiotemporal solitons in BBO and $\mathrm{Li}\mathrm{I}{\mathrm{O}}_{3}$ was studied in detail.

A typical experimental setup used to generate spatiotemporal solitons with the help of angular dispersion [56] is shown in Fig. 13. In this setup, the gratings that introduce angular dispersion and modify the dispersive properties of the nonlinear crystal are clearly seen. Cylindrical lenses to control the spatial shape of the beam and guarantee the necessary ellipticity that allows the tailoring of group velocities and GVD are also depicted.

The presence of spatiotemporal solitons generated with the setup of Fig. 13 was demonstrated by measuring the temporal
pulse duration and the beam waist of the soliton wave in the *y*
transverse dimension at different propagation distances [57]. Figure 14 shows the
experimental results: the dashed curves represent the temporal and spatial broadening
that the optical wave would suffer if the ${\chi}^{2}$ nonlinearity were not active, i.e., working at a low pump power. On
the other hand, the experimental black points were measured for a high pump power and
reflect the fact that the temporal and spatial widths of the soliton wave do not
change during propagation thanks to the interplay among the nonlinear effect,
dispersion, and diffraction. This is precisely the fact that reveals the presence of
spatiotemporal solitons.

#### 3.4. Generation of Terahertz Waves

The generation of electromagnetic pulses at THz frequencies is of great interest in various fields. One of the main areas of application of THz waves is the probing and detection of materials, since the characteristic energies of many interactions in molecules occur in this region [58]. These waves with submillimeter wavelengths at the crossing of the far-infrared and microwaves are, in general, not that easy to produce because their wavelength is too long for optical devices and too short for electronic circuits.

One of the ways to produce THz waves is to use a special case of frequency difference generation, called “optical rectification.” In this process, a photon at frequency ${\omega}_{1}$ is absorbed by an atom of a nonlinear medium, and two new photons are generated: one at the optical frequency ${\omega}_{2}$, and another one at a much lower frequency ${\omega}_{1}-{\omega}_{2}$ (THz). For the process of optical rectification to be efficient, the phase-matching conditions between all the interacting waves have to be satisfied.

Let us first consider a case in which the optical and the THz waves propagate in the
same direction *z* (collinear configuration). The pump beam is an
intense optical pulse with central frequency ${\omega}_{0}$, and a large beam area (plane wave). The electric field of the pump
writes $E(z,t)=1\u22152A(z,t)\mathrm{exp}(i{k}_{\mathrm{opt}}^{0}z-i{\omega}_{0}t)+\mathrm{h.c.}$, where ${k}_{\mathrm{opt}}^{0}={\omega}_{0}{n}_{\mathrm{opt}}^{\mathrm{ph}}\u2215c$ is the wavenumber of the optical pulse, ${n}_{\mathrm{opt}}^{\mathrm{ph}}$ is the refractive index at the central frequency, and h.c. stands
for Hermitian conjugate. The slowly varying amplitude $A(z,t)$ can be written as

*ω*is the optical frequency deviation from the central frequency ${\omega}_{0}$. Notice that here we slightly modify the notation with respect to the previous sections. The optical frequency deviation is now denoted

*ω*instead of

*Ω*, the refractive index at the central frequency is denoted ${n}_{\mathrm{opt}}^{\mathrm{ph}}$ instead of ${n}_{0}$, and the optical wavenumber at ${\omega}_{0}$ is written as ${k}_{\mathrm{opt}}^{0}$ instead of ${k}_{0}$. The goal is not to confuse the reader, but to adapt our notation to the symbols generally used in the literature when dealing with combinations of optical and THz waves.

When the conditions for optical rectification are satisfied, the pump beam that
interacts with a nonlinear crystal generates a nonlinear polarization at the THz
frequency *Ω*, which is written as [59]

In materials with a large nonlinear coefficient, such as $\mathrm{Li}\mathrm{Nb}{\mathrm{O}}_{3}$, ${n}_{\mathrm{THz}}^{\mathrm{ph}}$ is much larger than ${n}_{\mathrm{opt}}^{\mathrm{gr}}$ in the frequency band of interest [61]. For instance, for a pump at $800\text{\hspace{0.17em}}\mathrm{nm}$, ${n}_{\mathrm{opt}}^{\mathrm{gr}}=2.23$ and ${n}_{\mathrm{THz}}^{\mathrm{ph}}=5.16$ [62]; so collinear phase matching is not possible in this case. However, if we could modify the optical group index it would then be possible to satisfy the phase-matching condition given by Eq. (67). This is precisely what can be achieved by introducing angular dispersion in the pump beam as described in Subsection 2.2: if a laser pulse acquires pulse-front tilt, the group index ${n}_{\mathrm{opt}}^{\mathrm{gr}}$ is changed. To clarify this idea we can rewrite Eq. (23) in terms of the group index. Noticing that ${n}_{\mathrm{opt}}^{\mathrm{gr}}=c{k}_{\mathrm{opt}}^{\prime}$, we obtain

In a more general case, THz generation occurs in a noncollinear geometry; i.e., the
optical and THz waves do not propagate along the same direction. Figure 15 depicts the general configuration considered
where the THz wave is generated at an angle $\overline{\gamma}$ with respect to the pump beam direction of propagation. Different
optical frequencies, *ω* and $\omega +\Omega $, propagate inside the nonlinear crystal in different directions, ${\mathbf{k}}_{\mathrm{opt}}\left(\omega \right)$ and ${\mathbf{k}}_{\mathrm{opt}}(\omega +\Omega )$, determined by the amount of angular dispersion
*ν* introduced in the pump. In the noncollinear case, the
phase-matching condition must be written in the vector format [10]

*x*and

*z*components, we obtain [60]

Equation (71) is the key expression that highlights the role of angular dispersion in the generation of THz waves by means of optical rectification. By changing ${(\partial \u03f5\u2215\partial \omega )}_{{\omega}_{0}}$ with the help of dispersive elements, we can tune the noncollinear angle $\overline{\gamma}$ and satisfy the phase-matching conditions. Furthermore, the introduction of different amounts of angular dispersion allows the tuning of the frequency of the generated THz wave [62].

In a dispersive medium, the tilt angle *Φ* is given by Eq.
(8), and the relationship between
the angles *ν* and *Φ* is given by Eq.
(14), which if rewritten in terms
of the refractive index and the group index of the optical wave yield [9]

*ν*in the $x-z$ plane (dashed lines in Fig. 15).

The efficiency of the generation of THz waves by means of the process of optical rectification of femtosecond pulses depends on the fulfillment of the phase-matching condition between the optical and THz waves. By using optical pulses with pulse-front tilt, not only do we make it possible to comply with the phase-matching condition, but also we can tune the THz frequency at which this happens, which adds tunability to the scheme.

The full potential of using tilted pulses to tune the frequency of the generated THz
waves was demonstrated in [62]. This is shown
in Figs. 16, 17. The THz output was tuned between 1 and $4.4\text{\hspace{0.17em}}\mathrm{THz}$ in $\mathrm{Li}\mathrm{Nb}{\mathrm{O}}_{3}$ at a temperature of $10\text{\hspace{0.17em}}\mathrm{K}$ by changing the tilt angle *ν* between
59° and 64°. Figure 16 shows
the spectra of the THz waves when the tilt angle *ν* is
changed. Figure 17 plots the change of the
central frequency of the THz waves as a function of the tilt angle. The measured
energy is also shown. The capability of generating even higher frequencies would
require the use of pulses with pulse durations below $100\text{\hspace{0.17em}}\mathrm{fs}$ in less absorbing materials.

We remark that the use of pulse-front techniques allows the use of pumps in different frequency bands that may be more optimal. For instance, pulse energies of up to $100\text{\hspace{0.17em}}\mathrm{nJ}$ with a spectral bandwidth of up to $2.5\text{\hspace{0.17em}}\mathrm{THz}$ were obtained by optical rectification of $1030\text{\hspace{0.17em}}\mathrm{nm}$ laser pulses with $400\text{\hspace{0.17em}}\mu \mathrm{J}$ energy and $300\text{\hspace{0.17em}}\mathrm{fs}$ pulse duration [63], achieving a conversion efficiency of $2.5\times {10}^{-4}$, an order of magnitude higher than the one measured when using other materials in an optimized geometry.

## 4. Angular Dispersion as an Enabling Tool in Quantum Optics

Until now, we have seen the role of angular dispersion in applications in the field of nonlinear optics. We may regard these applications as belonging to classical optics in the sense that we are dealing with large amounts of photons whose properties can accurately be described by the classical Maxwell equations. In this section, on the other hand, we will see that angular dispersion is also useful in quantum optics when, for example, pairs of photons are created.

We will focus on the generation of pairs of photons by means of the process of spontaneous parametric downconversion (SPDC). SPDC is a nonlinear optical process in which an intense pump impinges on a nonlinear material and occasionally creates a pair of photons of lower frequencies. The correlations of the photon pairs produced by SPDC are of particular interest from both the fundamental and the practical points of view. Fundamentally, SPDC photon pairs are present at the core of many experiments to test the validity of the foundations of quantum mechanics. And, practically, new technologies based on their correlations promise improvements over their classical counterparts, among them quantum communications and information processing or clock synchronization.

In this tutorial, we will concentrate on the frequency properties of the SPDC photons, namely, on the bandwidth and the type of frequency correlations between the two photons. The appropriate frequency content of paired photons depends on the particular application under consideration [64, 65, 66, 67]. Some applications require frequency-correlated photons, some require frequency-anticorrelated photons, and some require frequency-uncorrelated photons. For this reason in recent years various methods to tailor at will the spectral properties of paired photons have been developed [68, 69, 70, 71].

In this section, we will see how the use of light beams with pulse-front tilt in SPDC allows us to control the bandwidth and the type of frequency correlation. These two frequency properties can be tuned by a proper tailoring of the group velocities and GVD parameters of all the waves that interact in the nonlinear process [72, 73]. As we have seen throughout the tutorial, this tailoring is precisely what angular dispersion enables. Indeed, it will be the pulse-front tilt angle that will play the role of a control parameter to tune, at will, the bandwidth and the type of frequency correlations of the SPDC photons. Remarkably, when using pulse-front tilt, there is no need for any particular engineering of the SPDC source. Moreover, the method is independent of the material and can be used in any frequency range where other methods do not work.

As we will see in what follows, angular dispersion allows us to increase the bandwidth of the joint spectrum of paired photons, an important point for the generation of very narrow temporal biphotons. In addition, the use of pump beams with pulse-front tilt makes possible the generation of frequency-correlated, frequency-anticorrelated, and even uncorrelated photon pairs. The latter case offers a very attractive applicability of angular dispersion for the generation of heralded indistinguishable and pure single photons with a tunable frequency bandwidth [74].

#### 4.1. Angular Dispersion in Spontaneous Parametric Downconversion

Let us consider the generation of SPDC photons in a collinear configuration depicted in Fig. 18 when the two downconverted photons propagate in the same direction. Unlike the typical SPDC setups, the nonlinear medium is placed between two diffraction gratings that introduce angular dispersion. It can be shown that after the two gratings, the quantum state of the downconverted photons can be written as [75]

*L*, by the spectral characteristics of the pump beam ${E}_{p}({\omega}_{p}^{0}+{\Omega}_{p})$, and by the dispersive properties of the nonlinear material expressed by the phase-mismatch term along the longitudinal direction, $\Delta k={k}_{p}-{k}_{s}-{k}_{i}$, with ${k}_{j}={[{({n}_{j}\left({\omega}_{j}\right){\omega}_{j}\u2215c)}^{2}-{\left|{\mathbf{q}}_{j}\right|}^{2}]}^{1\u22152}$, where ${n}_{j}$ is the refractive index and ${s}_{k}={k}_{p}+{k}_{s}+{k}_{i}$.

The role of angular dispersion is to modify $\Psi ({\Omega}_{s},{\Omega}_{i},{\mathbf{q}}_{s},{\mathbf{q}}_{i})$ due to the dependence of $\Delta k$ on **q**. Inspection of Eq. (20) shows us that angular dispersion can thus modify the
frequency shape of the phase-matching function $\Delta k$. After the second grating, downconverted photons are detected by
using strong spatial filters, e.g., single-mode fibers, so that ${\mathbf{q}}_{s},{\mathbf{q}}_{i}\simeq 0$. To get a further physical insight, let us expand $\Delta k$ up to second order. Considering walk-off for the pump, the signal,
and the idler,

*Φ*in SPDC configurations, it is possible to control the frequency properties of paired photons.

#### 4.2. Tunable Control of Frequency Correlations of Paired Photons

To see how angular dispersion can be used to control the type of frequency correlation of SPDC photons, let us consider the first-order terms of Eq. (75). One obtains frequency-anticorrelated photons $({\Omega}_{s}=-{\Omega}_{i})$ when ${k}_{s,\mathrm{eff}}^{\prime}={k}_{i,\mathrm{eff}}^{\prime}$ and frequency-correlated photons $({\Omega}_{s}={\Omega}_{i})$ when ${k}_{p,\mathrm{eff}}^{\prime}=({k}_{s,\mathrm{eff}}^{\prime}+{k}_{i,\mathrm{eff}}^{\prime})\u22152$. Frequency-uncorrelated photons are obtained if the effective group velocity of the pump is equal to that of the signal or the idler, ${k}_{p,\mathrm{eff}}^{\prime}={k}_{s,\mathrm{eff}}^{\prime}$ and/or ${k}_{p,\mathrm{eff}}^{\prime}={k}_{i,\mathrm{eff}}^{\prime}$.

Figure 18 shows the experimental arrangement used to measure the joint spectrum $S({\omega}_{s},{\omega}_{i})={\left|\Psi ({\Omega}_{s},{\Omega}_{i})\right|}^{2}$ of the entangled paired photons. Before being detected, each of the photons forming a pair passes through its respective monochromator, which is scanned to measure the joint spectrum $S({\omega}_{s},{\omega}_{i})$. Figure 19 shows the experimental results that demonstrate the feasibility to fully control the frequency correlations in SPDC via angular dispersion. In particular, the measurements were performed by using a $3.5\text{\hspace{0.17em}}\mathrm{mm}$ type-II BBO crystal cut for collinear degenerate phase matching.

In Fig. 19, different types of frequency
correlation are observed by varying the tilt angle *Φ*. The
first row of Fig. 19 shows the case with no
tilt. As expected for a pulsed pump and type-II phase-matching, the spectra of the
signal and idler photons are different, one being narrower than the other, which is a
consequence of their different group velocities. The following rows correspond to
different values of the tilt angle. The second and fourth rows of Fig. 19 depict the cases of highly
frequency-anticorrelated and highly frequency-correlated photons. The third row shows
the interesting case of frequency-uncorrelated pairs.

The frequency uncorrelation observed in the third row of Fig. 19 is indeed a signature of the presence of a separable quantum state. To demonstrate full separability of the two-photon state, it is also required that there be no phase entanglement [76]. Theoretically, we can calculate the Schmidt decomposition of the state given by Eq. (73) when the separability condition of the group velocities is fulfilled. In that case, the entropy of entanglement is nearly 1, and the Schmidt decomposition contains only one mode, revealing the separability of the quantum state [68]. In this way, the scheme offers a possibility to generate paired photons in a separable state, a quantum state so desired in quantum information processing applications. Frequency-uncorrelated photons can well serve as a source of heralded single photons: the detection of one of the photons heralds the presence of its twin photon in the setup without in any way changing its state. Experimentally, a tomographic analysis or a four-photon experiment such as the one described in [71] would be needed to fully demonstrate the separability and therefore the purity of the generated single photons.

The frequency correlations of entangled paired photons generated in the process of spontaneous parametric downconversion can be tuned independently of the frequency band and of the nonlinear crystal used. The photons can exhibit frequency anticorrelation, frequency correlation, and even frequency uncorrelation only through control of the amount of angular dispersion.

#### 4.3. Controlling Bandwidth of Paired Photons

As we have mentioned before, Eq. (74) tells us that the bandwidth of the downconverted light is determined by the length and the dispersive properties of the nonlinear material, by the geometry of the SPDC configuration (collinear or noncollinear, type I or type II) and by the spectral characteristics of the pump beam. In the following, we will see that the use of SPDC with a pump beam with angular dispersion allows us to control the SPDC bandwidth as well [77].

To further clarify the ideas, we consider the case of a narrowband pump, for example, a picosecond pulsed laser. In this case, energy conservation dictates ${\Omega}_{s}\simeq -{\Omega}_{i}$, and Eq. (75) reduces to

*Φ*such that ${k}_{i,\mathrm{eff}}^{\prime}-{k}_{s,\mathrm{eff}}^{\prime}=0$, the bandwidth will increase because the first nonzero terms that contribute to $\Delta k$ are terms of second order or higher. For example, in a type-II process the bandwidth of generated photons is inversely proportional to the length of the nonlinear material

*L*[72]. If the first-order terms of $\Delta k$ are removed by using angular dispersion, the dependence of the bandwidth on the length will go as $1\u2215\sqrt{L}$. In addition, the higher dispersion terms are much weaker, which further broadens the spectrum. In a type-I SPDC process, if the tilt

*Φ*is such that ${k}_{s,\mathrm{eff}}^{\u2033}=-{k}_{i,\mathrm{eff}}^{\u2033}$, the first nonzero terms in $\Delta k$ are of fourth order $\propto {\Omega}_{s}^{4}$, and the dependence of the bandwidth on the length of the crystal goes as $1\u2215{L}^{1\u22154}$.

The values of the tilt angle that maximize the bandwidth for type-II and type-I processes are [77]

The effect of introducing angular dispersion to increase the bandwidth of the SPDC photons was demonstrated experimentally by using the setup of Fig. 18 with a $2\text{\hspace{0.17em}}\mathrm{mm}$ BBO crystal cut for degenerate type-II collinear phase matching. Figure 20(a) depicts the joint spectrum for the case without angular dispersion, and Fig. 20(b) shows the joint spectrum when a tilt $\Phi ={\Phi}_{\mathrm{II}}^{\mathrm{max}}=38\xb0$ was introduced.

To get a quantitative value of the increase in the bandwidth, let us examine Fig. 21. The solid curves represent the theoretical predictions, and the dots are the experimental results. Figures 21(a), 21(b) correspond to the spectra of single counts of the signal and idler photons. The curve on the left corresponds to no tilt, $\Phi =0\xb0$, and the one on the right to the case of $\Phi =38\xb0$. For the case with no gratings, a FWHM bandwidth of $\Delta {\lambda}_{s}\sim 5.2\text{\hspace{0.17em}}\mathrm{nm}$ is obtained, while for the case with gratings, the FWHM bandwidth is $\Delta {\lambda}_{s}\sim 37\text{\hspace{0.17em}}\mathrm{nm}$, leading to a sevenfold increase of the bandwidth. Figures 21(c), 21(d) depict the coincidence counts. The widths along the antidiagonal line (at $-45\xb0$) of the joint spectra, Figs. 20(a), 20(b), are plotted. The coincidence width of $7.5\text{\hspace{0.17em}}\mathrm{nm}$ broadens to $52\text{\hspace{0.17em}}\mathrm{nm}$.

The effect of spectrum broadening can be employed to generate a very short temporal biphoton that is given by the Fourier transform of the joint spectral amplitude [72]. It was shown in [77] that when the pulse-front tilt is used, the spectral phase profile remains smooth and very flat, which translates into ultrashort nearly transform-limited biphotons with a temporal correlation of a few femtoseconds. The detection of one of the photons of the pair determines the detection of the other photon located in a distant place within a time window given by the biphoton’s temporal width. This phenomenon could expediently be used, e.g., for clock synchronization. The pulse-front-tilt technique contrasts with other methods in which the increase in bandwidth is not directly accompanied by a decrease of the correlation time. That can be caused by a particular shape of the spectral phase or when the phase relationship between individual frequencies, and thus coherence, is lost. This is, for example, the case of white-light continuum generated by Kerr self-phase modulation.

Up to this point we have considered the effects of angular dispersion when a collinear geometry is used. However, it is worth mentioning that the combination of the pulse-tilt techniques described above with the use of noncollinear geometries further expands the possibilities to control the joint spectrum of paired photons [74]. In noncollinear geometries it is possible to map the spatial characteristics of the pump beam into the spectra (spatial-to-spectral mapping) [78, 79], providing another way to manipulate the joint spectral amplitude of the biphoton.

The bandwidth of paired photons generated in the process of spontaneous parametric downconversion can be enhanced or reduced, independently of the frequency band and of the nonlinear crystal used. It is possible to generate spectral bandwidths of up to hundreds of nanometers in the visible optical range that translate into temporal correlations between the two photons of just a few femtoseconds.

## 5. Conclusions

The angular dispersion of light is an old physical phenomenon that was already discussed
at the beginning of the 18th century by Isaac Newton in his book
*Opticks* [1], where he
describes how white light decomposes into colors and diverges after passing through a
prism.

In this tutorial we have shown how, with the appearance of the laser, angular dispersion has become an important enabling tool in different areas of nonlinear and quantum optics. The key tool is the possibility to modify the dispersive properties of materials by using light pulses with suitable amounts of angular dispersion. The use of these pulses in many applications has been described with such an unifying view.

Most times, this common perspective is absent in scientific papers and technical reports, or at least is not clearly seen, because of the use of different notation in each field or because emphasis is put on diverse aspects each time.

Section 2 of this tutorial hopefully offers such a view, and each application considered in Sections 3, 4 is analyzed under the general unifying framework developed in Section 2. The first two applications, pulse compression and CPA, described in Subsection 3.1, are nowadays routinely used in commercial systems for compressing and stretching optical pulses. The next application is achromatic phase matching, considered in Subsection 3.2, which enables us to enhance the capability of frequency doublers for efficiently doubling ultrashort pulses.

The excitation of quadratic temporal solitons is described in Subsection 3.3, where it is shown that the introduction of angular dispersion permits the observation of temporal solitons. In Subsection 3.4, we have described how the use of pulses with pulse-front tilt allows us to satisfy the condition of phase matching, a requisite not easily achievable, but notwithstanding necessary, for the efficient generation of THz waves in the process of optical rectification of femtosecond laser pulses. Even more, the method allows us to tune the frequency of the generated THz wave, allowing the implementation of tunable generators of THz waves. Finally, the generation of entangled paired photons with tunable bandwidth and frequency correlations is analyzed in Section 4.

Subsection 3.3 and Section 4 are two outstanding cases that exemplify the role of light beams with angular dispersion. By altering the unfavorable conditions offered by most natural materials, the use of angular dispersion makes it possible to observe physical effects that would not be possible otherwise.

The necessary conditions for the observation of temporal solitons in quadratic nonlinear media are not met in commonly used nonlinear crystals. This is also the case for the observation of frequency-entangled photons that show frequency correlation or frequency uncorrelation. In both cases, pulses with angular dispersion allow us to modify the dispersive properties of media, effectively engineering new materials that meet the necessary requirements in terms of new effective group velocity and GVD parameters.

Summarizing, light beams with angular dispersion, or pulse-front tilt, allow us to perform tasks in nonlinear and quantum optics not possible otherwise, highlighting the role of angular dispersion as enabling tool.

## Acknowledgments

We thank F. Wise, A. Schober, and J. Hebling for providing some of the figures that appear in this tutorial, helping to greatly improve its clarity. We give special thanks to L. Torner and S. Carrasco for long and helpful discussions during many years about the role of angular dispersion in nonlinear optics, especially concerning the generation of quadratic spatiotemporal solitons and the control of the frequency correlations of entangled photons. We thank X. Liu, F. Wise, and P. Di Trapani for occasional, but illuminating, discussions about solitons. We thank P. Loza and D. Artigas for helpful discussions concerning the process of SHG. Finally, we also thank R. Trebino and S. Akturk for discussions about the relationship between angular dispersion and pulse-front tilt. This work was supported by the European Commission (Qubit Applications, contract 015848), by the Government of Spain (Consolider Ingenio CSD2006-00019, FIS2007-60179), and was supported in part by FONCICYT project 94142.

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**Juan P. Torres** leads a research group in Optics at the ICFO-Institute of
Photonic Sciences in Barcelona, Spain, since 2003. He is also associate professor at the
Universitat Politècnica de Catalunya since 1996, where he teaches courses on
electromagnetic theory and several aspects of Photonics. He did his undergraduate
studies in physics at the Universitat de Barcelona from 1982 to 1987, and received a
Ph.D. in Sciences from the Universitat Politècnica de Catalunya in 1994. Juan P.
Torres' main area of interest is Nonlinear and Quantum optics. In particular, he is
interested in exploring theoretically and experimentally the unique features of new
types of optical waves, such as solitons and vortex beams, and basic concepts of quantum
theory, such as entanglement and decoherence. Juan P. Torres has co-authored numerous
scientific papers published in international peer-reviewed journals and has established
collaborations with research groups in many different countries. Nowadays, he is very
interested in applying concepts and techniques born and developed in nonlinear and
quantum optics to the life sciences.

**Martin Hendrych** received his Ph.D. degree in quantum optics from Palacky
University, Olomouc, Czech Republic, in 2003. During his study, he experimentally
implemented quantum key distribution, quantum identification, and quantum secret-sharing
schemes. In 2000, he was awarded a NATO Advanced Science Fellowship to fund his stay in
the Quantum Imaging Laboratory at Boston University, Boston. Upon completion of his
degree, he worked as a Research Scientist at the Joint Laboratory of Optics of Palacky
University and the Institute of Physics of the Czech Academy of Sciences, Olomouc, Czech
Republic. Since 2005, he is a Research Fellow at ICFO—Institute of Photonic
Sciences, Barcelona, Spain. Among his main research areas are quantum and nonlinear
optics, design and implementation of sources of entangled photons in the fields of
quantum information and quantum communications, frequency entanglement, dispersion
control, and Bragg-reflector waveguides.

**Alejandra Valencia** works at ICFO—The Institute of Photonic Sciences
in Barcelona, Spain, since October 2005. She did her undergraduate studies in physics at
Universidad de los Andes, Bogotá, Colombia (1994–1999). In 2002, A.
Valencia received her Master in Sciences degree and in 2005 her Ph.D., both from the
University of Maryland Baltimore County (UMBC), USA. The topic of her dissertation was
the study of protocols for clock synchronization based on the characteristics of
entangled photon pairs. As a postdoctoral researcher, her interest has been mainly
oriented towards the engineering of the frequency correlations of entangled photon pairs
and the generation of pure single photons. A. Valencia has coauthored various scientific
papers published in international peer-reviewed journals and has established
collaborations with research groups in different countries. Nowadays, she works in the
knowledge and technology transfer unit (KTT) of ICFO developing all the outreach and
scientific divulgation activities of the institute.